compilation of thermal properties of hydrogen in its various

U.S. Department ot CommerceNational Bureau of Standards

Research Paper RP1932Volume 41, November 1948

Part of the Journal of Research of the National Bureau of Standards

Compilation of Thermal Properties of Hydrogen in ItsVarious Isotopic and Ortho-Para Modifications

By Harold W. Woolley, Russell B. Scott, and F. G. Brickwedde

New developments in science and industry are aided by accurate knowledge of thebehavior of important substances. The great abundance of chemical processes and com- Xpounds in which hydrogen is involved make it of particular interest. The experimentaland derived data presented here for hydrogen extend over a large range of temperature.Low temperatures are required for the liquid and solid, and moderate and high temperaturesoccur in chemical reactions.

The available thermal data for H2, HD, and D2 in solid, liquid, and gaseous stateshave been brought together, including the distinctive properties of ortho and para formsof H2 and D2. Some data not previously published have been added. The thermal datainclude thermodynamic functions for the ideal gas state, equilibrium constants, data ofstate, viscosity, and thermal conductivity with dependence on the pressure, vapor pressure*solid-liquid equilibria, specific heats, and latent heats. Values of state derivatives usefulin thermodynamic calculations have been given for normal hydrogen, and. the related dif-ferences between thermodynamic functions for real and ideal gas states have been evaluated-A temperature entropy diagram for normal H2 in the range of experimental data is alsogiven. The compiled thermal properties of hydrogen are presented in 38 tables, 33 graphs,and numerous equations. The sources of the data have been given in an extensivebibliography.

I. Introduction

It was recommended by the National ResearchCouncil Committee on Thermal Data for Chemi-cal Industries *2 and by others that the thermaldata on substances of industrial importanceshould be reexamined with the intention of pre-paring consistent tables of thermal data of especialinterest to chemical engineers and investigators.

In this paper thermal data on hydrogen in itsvarious isotopic and ortho-para modifications arecompiled and correlated. Data on properties ofthe gaseous, liquid, and solid states are presentedin tables and graphs, and by use of formulas.Thermodynamic properties are given for the idealgas state. In addition, tables based on the PVTdata for the real gas furnish the additional infor-mation required for the calculation of the thermo-dynamic properties of the real gas. For the con-

1 Division of Chemistry and Chemical Technology, National ResearchCouncil.

2 F . Russell Bichowsky, Chairman, 1938 to 1947.

densed phases, directly observable properties aregiven. Because of the industrial importance offlow and heat-transfer problems, correlations ofviscosity and of thermal conductivity are in-cluded and their dependence upon pressure dis-cussed briefly. A number of topics are discussedin detail to explain the fundamental principlesinvolved. Most of the data included were takenfrom published papers. However, a small pro-portion are based on unpublished measurementsmade at the Bureau.

The following are the symbols and values ofphysical constants and conversion factors used inthis paper.

1. SymbolsMany symbols that are not used extensively in

this paper have been omitted from this list.A, constant in an equation for a PVT isotherm.B, second virial coefficient in equation oKstate of

Bv, rotational spectroscopic constant.

Properties oi Hydrogen

b, b, constant in an equation for a PVT isotherm;also, a constant in an equation of state.

C, C, constant in an equation for a PVT isotherm;also, the Sutherland constant in a viscosityformula.

C, constant in an equation for a PVT isotherm.C°, molar heat capacity (molar specific heat) at

constant pressure for ideal gas.C8, molar heat capacity (molar specific heat) along

a saturation curve.C°, molar heat capacity (molar specific heat) at

constant volume for ideal gas.c, c, velocity of light; also a constant in an equa-

tion for a PVT isotherm.c2, radiation constant hc/k.Dv, rotational spectroscopic constant.E, a thermodynamic function, internal energy per

mole.E°, E for a substance in the ideal gaseous state.EQ, E° at the absolute zero of temperature when

for each molecule the energy associatedwith internal degrees of freedom is at itslowest quantized value.

F, a thermodynamic function, molar free energyF=E+PV-TS.

F°, F for a substance in the ideal gaseous state ata pressure of 1 atmosphere.

Fv, rotational spectroscopic constant.FV>J, or F, term value./ , a thermodynamic function, fugacity:Gv, vibrational term value.g, statistical weight of a quantum level.H, a thermodynamic function, molar heat content

or enthalpy, H=E+PV.H°, H for a substance in the ideal gaseous state.Hv, rotational spectroscopic constant.h, Planck's constant.i, nuclear spin.J, rotational quantum number.K, equilibrium constant.k, k, Boltzmann constant; also, thermal^ con-

ductivity.Lv, latent heat of vaporization.M, molecular weight.m, reduced mass for molecule.N, total number of molecules considered.Njy number of molecules in a given quantum level.No, Avogadro's number.P, pressure.Pc, pressure at the critical point.

Po, pressure of 1 standard atmosphere, 1.01325 X106 dynes cm"2 by definition.

p, momentum corresponding to generalized co-ordinate q.

q, a generalized coordinate.R, molar gas constant.r, atomic separation.re, atomic separation r for minimum potential

energy.5, a thermodynamic function, molar entropy.S°, S for a substance in the ideal gaseous state at

a pressure of 1 atmosphere.T, absolute temperature on the Kelvin scale.Tc, temperature T at critical point.JT0, Kelvin temperature T of the ice point, that is,

of 0° C.U, intramolecular potential energy.Uu, ratio of mean free path lengths for diffusion

and viscosity.V, molar volume.Vc, molar volume at the critical point.VOj molar volume of gas at 1-atmosphere pressure

and the ice point.v0, molar volume of liquid at zero pressure.v, vibrational quantum number.Z, abbreviation for PV/RT.7, ratio of specific heats, Cp/Cv.e, energy for a quantum state.r), viscosity.6, a characteristic Kelvin temperature for a

crystal lattice in Debye's theory of specificheats.

A, length of mean free path.JU, Joule-Thomson coefficient.£, fractional increase in atomic separation beyond

that for minimum potential energy,p, density in Amagat units,or, a correlation function for PVT data.X, a function in one equation of state.<Pj a correlation function for PVT data.

2. Values Used for Some Physical Constants andConversion Factors

(Numbers in parentheses refer to the references givenbelow)

c (velocity of light=2.99776X1010 cm sec"1 (1).

c2 (radiation constant) =j7= p '=-1.4384 cmdeg(2).

h (Planck's constant=6.624X1027 cm sec (1).No (Avogadronumber) = 6.0228XI(Pinole"1 (1).

380 Journal of Research

Po (pressure of standard atmosphere) = 1.01325X106 dynes cm"2 (3).

R (molar gas constant)=iVofc=8.3144X107 ergmole"1 deg"1 (1).

= 1.98714 thermochemical cal mole"1 deg"1 (4).To (Kelvin temperature of ice point) = 273.16° K

.(5). -Atomic weight of hydrogen (H1) on chemical

scale = 1.000786 (1).Atomic weight of deuterium (D or H2) on chemical

scale=2.01418 (1).1 thermochemical calorie=4.1833 international

joules (5).1 international joule (NBS) = 1.000165 absolute

joules (6).(1) Raymond T. Birge, Rev. Modern Phys. 13, 233

(1941).(2) Birge's value (Rev. Modern Phys. 13, 233 (1941))

adjusted for later NBS value of the ratio internationalcoulomb/absolute coulomb = 0.99985; see also reference (7).

(3) Definition.(4) Birge's value (Rev. Modern Phys. 13, 233 (1941))

adjusted to thermochemical calorie and NBS value forratio international joule/absolute joule.

(5) Definition.(6) NBS Technical News Bulletin 31, 49 (1947).(7) R. W. Curtis, R. L. Driscoll, and C. L. Critchfield,

J. Research NBS 28, 133 (1942).

II. Thermodynamic Properties for theHydrogens in the Ideal Gas State

1. General Principles of Calculation

For a gas in a state of extreme rarefaction theenergy of interaction between molecules forms aminute part of the total energy of the gas. Atsuch low pressures the thermodynamic propertiesof the gas may be calculated from the spectro-scopically determined energies of the single mole-cules and the general physical constants withoutconsidering the energy of interaction of one mole-cule with another. Some thermodynamic prop-erties, as for example molar entropy and freeenergy, do not approach a definite value as thepressure of the gas goes to zero. For this reason,values of thermodynamic functions of a gas at lowpressure are often indicated by giving values for apressure of 1 atm for a fictitious ideal gas havingin the limit of low pressure the same thermody-namic functions as the actual gas. The result isthen said to be for the gas at a pressure of 1 at-mosphere in the hypothetical ideal gas state.Data of state may be used to calculate the differ-

ences between properties in the real and idealgas states.

The procedure for calculating the thermody-namic properties of a substance in the ideal gasstate has been discussed by many writers [3, 30,31, 32].3

In outline, it involves the following ideas: Theaverage number ni of molecules in a quantumstate of energy ex is related to the average number,n2 of molecules in another state of energy e2 by theBoltzmann distribution law

n1/n2=e-<i/kT/e-v/kT=e-^-**)/7cT, (2.1)

where k is the Boltzmann constant, and Tis theabsolute temperature.

As there are often several states having the sameenergy, the number of molecules in a given energylevel4 is also proportional to the number of states,g. If Ni, N2, N3, . . . are the numbers of mole-cules in the levels ely e2, e3, . . ., respectively, thenumber of molecules in any one level is

where N, the total number of molecules beingconsidered, is equal to 2Njm If properties are tobe expressed on the basis of 1 mole, N is takenequal to Avogadro's number, iV0.

The quantum states are specified by means ofquantum numbers, the integer values whichcertain natural variables have when a moleculehas a stationary value of energy. The magnitudeof the energy is generally expressed in terms ofthese numbers. In diatomic molecules, the quan-tum numbers of interest are J, the rotationalquantum number, K, the rotational quantumnumber apart from spin, and v, the vibrationalquantum number. The electronic state is alsosimilarly quantized, and quantum numbers ap-propriate to it may likewise be assigned. Thenuclear spins of the two constituent atoms aredesignated by ix and i2. In terms of these num-bers, the statistical weight, g, of a level of a dia-tomic molecule composed of unlike atoms, as forexample HD, is ge(2il+l)(2i2+l)(2J+l)) wherege is the weight of the electronic level of the mole-

3 Figures in brackets indicate the literature references at the end of thispaper.

4 The term stale is used in the sense that two states differ if any of all thequantum numbers associated with the states are different. The term levelis used to express the idea that the energy has a definite value. The statisticalweight, g, of a level is the number of states having the energy which definethe level. A level with more than one state is said to be degenerate.

Properties of Hydrogen 381

cule. The ground electronic level of HD, and ofH2 and D2, also, is a singlet state, and accordingly

The proton and deuetron spins are % and 1,respectively. For diatomic molecules composed oflike atoms, as for example, H2 and D2, there is adivision of the rotational levels of the molecule intotwo groups referred to as the ortho and para series,one of which is composed of the even numberedand the other of the odd numbered rotationallevels. Ordinarily, transitions between ortho andpara levels are relatively rare, so that the gas canbe considered as a mixture of two distinct com-ponents. The high temperature equilibrium mix-ture of the two forms is called the normal mixture,and the more abundant component of the normalmixture is called the ortho component. The sta-tistical weights of the two series depends upon thequantum statistics applicable to the nuclei. ForH2 it is the Fermi-Dirac statistics, for D2 theBose-Einstein statistics.

Fermi-Dirac statistics:

g (para series, even J's) —ge (

g (ortho series, odd J's) =(2.3)

Bose-Einstein statistics:g (ortho series, even J's) =

g (para series, odd J's) =ge (

(2.4)

The energy per mole due to molecular rotationand intramolecular vibration is

where the e's are the energies of the rotational-vibrational levels relative to the lowest energylevel of the molecule. The translational energy,3/2 NokT or 3/2 RT, is added to this to getE°—E°o, the total internal energy per mole forthe ideal gas above the chosen zero in which therewould be no translational energy and each mole-

cule would be in the lowest energy state availableto any form of the molecule.5

E°-E°o=3/2 RT+N0 (2.6)

The superscript zero is used to indicate the idealgas state.

The enthalpy H°, the specific heats O°v and C£,the entropy S°, and the free energy F° for theideal gas state are derivable in accordance withfamiliar methods of thermodynamics from (1) theinternal energy E°—E°o, (2) the equation of statePV=RT, and (3) the translational entropy S°t ofan ideal gas of molecular weight M. The equa-tions for these properties as functions ofare

E°-E\

R~~~

S°

=|ln T+3/2 lnM-ln(P/P0)+ln

§ = | In T+3/2 inM+ln V+ln

"%•• (2.H)

F0-El_H°-El-TS°_ ,HI Kl

(2.12)

4(2.13)

+2 R(2.14)

5 Accordingly for orthohydrogen and paradeuterium EQ is not the internalenergy at 0° K. For these substances at 0° K the internal energy above thechosen zero (J=0, v=0) is the rotational energy per mole of molecules in therotational level J=\. At 0° K internal energies of normal hydrogen andnormal deuterium are respectively three-fourths the internal energy oforthohydrogen and one-third the internal energy of paradeuterium.


In eq 2.12, P and Po are the pressure of the gasand standard atmospheric pressure, respectively,with both expressed in dynes cm"2. The ratioP/Po is the pressure expressed in atmospheres.

For a monatomic gas in which the ground stateis so far below the others in energy that it alonemakes appreciable contribution to the state-sum,

*9je e' j eq 2.7 to 2.14 are simplified consider-ably. With ei, the energy of the ground state,taken as zero, the state-sum reduces to the con-stant #iAs a result, (E°-Eo)/RT=S/2; (H°-E°0)/RT=5/2; C°v/R=3/2; C£/fi=5/2; S°/R=ln g^Sl/R,and (F° - Eo

0)/R T= - In gx + 5/2 - St°/R. When thenuclear spin is included, gx contains (2^+1) as afactor.

Normal hydrogen is a mixture 75 percent oforthohydrogen and 25 percent of parahydrogen,and normal deuterium 66% percent of ortho-deuterium, and 33 % percent of paradeuterium.The molar entropy and free energy of a mixture ofideal gases present in the mole fractions xiy x2,. . . are

(2.15)

(2.16)j .1

where S°j and F°j, the molar entropy and freeenergy of the ideal gas j in a pure state at thepressure of the mixture, are given by eq 2.11 and2.14, using eq 2.12 for the evaluation of S°t.The summation —RXxj In Xjis called the entropyof mixing. Using eq 2.13 for the evaluation ofSt, and setting V equal to the molar volume of theconstituent, that is, the volume of the mixturedivided by the moles of constituent present, isequivalent to using partial pressures in eq 2.12,in which case the entropy and free energy of themixture are equal simply to ^XjS°j and ^x3F

Oj.

j—R'ExJ In

2. Energy Values From Spectroscopic Data

The values of ej to be used in evaluating theequations of the preceding section are derivedfrom analysis of molecular spectra. In general,banded electronic absorption and emission spectra,infrared, rotation-vibration absorption spectra,and Raman spectra are considered. But as theH2 and D2 molecules have no electric dipolemoments in their normal states, they have norotation-vibration absorption spectra. Similarly,no such spectra have been observed for HD,although lack of symmetry permits it to have avery weak dipole moment.

The spectroscopic energy level data for hydrogenare represented by a series in which the energiesof the levels relative to the ground level, v=0,J=0, divided by he are expressed as a function ofthe rotational and vibrational quantum numbersJ and v, see eq 2.17. The quantity ej/hc is calledthe term value of the level and is designated bythe symbol F. Term values are determinedexperimentally from differences between the wavenumbers of spectrum lines and are expressed interms of reciprocal centimeters as a unit. HereFV)J is the term value for the level v, J; Fo,o for theground state being zero.

Up to 25,000 cm"1, the term values on whichtables 4,7, and 8 are based, can be represented by

(2.17)

where the subscripts used indicate the quantumnumbers on which the different symbols dependfor their values.

The functions Gv, Bv, Dv, Fv, and Hv for H2; HD and D2 are as follows:

ForH2 :

Dv= — 0. .0014904(z;+}/2) — 0.000063648 (

"5+0.02800X10-5(z;+K)

2.18


ForHD:

Gv=c

2?,=45.6549—1.

Dv= -0.026136 + 0.000726610+ }0-0.0000268773 0+K)2

^=-2 .1295X10~ 8

For D2:

2.19

5P=30.4286-1.049170+K)+0.00579340+K)2-0.000274860+K)3

Z>,= ~0.011586 + 0.0001510+H)+ 0.0000580+K)2

^=6.22X10- 6 +

Hv= — 0.442 XI0"8

2.20

The numerical values of the coefficients in eq2.18 to 2.20 are based on the latest available spec-troscopic measurements due principally to Rasetti[2], Hyman [5, 6], Jeppesen [6, 7, 12, 15, 24],Beutler [20, 21], and Teal and Mac Wood [22].The data of Fujioka and Wada [23] were not usedand the data of Mie [16] on HD only through itsinfluence on the formula for Gv. The. equationsGv for H2 and HD are those given by Teal andMac Wood [22], and that for D2 by Jeppesen [24].The equations for Bv are essentially Jeppesen's[12, 24] equations expressed for use with J(J-\-l).The constants in the equations for Dv, Fv, andHv were obtained from theory using the equa-tions for Gv and Bv and the formulas of Dunham[10] without his correction terms.

In the case of hydrogen as for many other sub-stances, extrapolations of spectroscopic formulashave to be made into regions of large rotationalquantum numbers for which no wavelengthmeasurements are available in order to obtainvalues for the energies ej of the higher quantumstates. The energy values for large rotational andvibrational quantum numbers are influenced bythe law of internuclear force of the molecule forlarge separations of the nuclei. Special con-sideration has been given to this point in the pres-ent work and two methods were developed where-by more reliable values of the energies of theunobserved higher rotational levels were obtained.

The first improvement was the addition of the

final term in eq 2.17, [HVJ\J+iy]2/[(FtJ*(J+1)3-fl;j4(J+l)4)]. Without the final term, eq 2.17 isof the form in which spectroscopic data have here-tofore been represented, but in that form it is nota good approximation for large values of J. Thethird, fourth, fifth, and sixth terms of eq 2.17 areof alternate sign and for H2 the third, fourth, andfifth terms are approximately equal for J=2&.This suggested that the series be extended withsuccessive terms in constant ratio. The final termof eq 2.17 is the sum of the geometric series ofadded terms in which the term to term ratio isthat between the fifth and sixth terms of eq 2.17.

This change in the formula for the energies ofthe rotational-vibrational levels of the normal(ls1^) electronic state of hydrogen has only asmall effect on the energy values of the observedspectrum lines. Thus the mean difference be-tween Jeppesen's [12] observed and calculatedterm values for the 2p12—ls12 band for H2

was 1.032 cm"1, whereas using eq 2.17 in place ofJeppesen's equation for the Is1?} state the meandifference between observed and calculated valuesis 1.030 cm"1.

As a second improvement, for the calculation ofthermodynamic properties above 2,000° K, analternative determination of the highest rota-tional levels was made. Instead of using thepower series eq 2.17, the energies correspondingto any degree of rotation and vibration were deter-mined from the potential energy. This was


carried out in effect by (1) determining the poten-tial energy U of the nonrotating H2 molecule as afunction of the internuclear separation, (2) addingthe rotational energy h2 J (J + 1)/ST2 I e(r/re)

2 to Uto obtain an effective potential energy, Ur, for amolecule with rotational quantum number J, and

(3) using the quantum condition & pdq= I (2m)1/2

(eVtj—U'y/2dr=(v+ l/2)h to determine the energyeViJ of the quantum state v, J.

The coefficients of a power series used to repre-sent the molecular potential energy were evaluatedfor the H2 molecule using Dunham's [10] theoreti-cal relations and the rotational and vibrationaldata for H2:

1-1.6082?+1.8598£2-1.8882£3 +

, (2.21)where £ is (r—rc)/re,re being the equilibrium valueof the internuclear separation, and U is expressedin reciprocal centimeters. Although this seriesis a poor representation of U for internuclearseparations twice the equilibrium value (i. e., at£= 1), it is very good for small values of £. There-fore, this series was not used for the potential

energy function finally accepted for internuclearseparations much greater than the equilibriumvalue, but it was used for internuclear separationsless than the equilibrium value. At dissociationthe minimum value of r for classical motion ismore than half of re (i. e., |£|<^0.5), and the seriesdetermines the inner portion of the potentialenergy curve with sufficient reliability for thepresent purposes.

The ranges of internuclear oscillation, £max— £min,for different values of the energy needed to fix theouter portion of the potential energy curve, weredetermined from (1) the vibrational levels of thenonrotating molecule, symbolized by Gv in eq 2.17to eq 2.20, which have been accurately measuredto within 140 cm"1 of dissociation [5, 12, 20, 21]and (2) the quantum condition.

{Gv-U)1/2d£=(v+l/2)h.

(2.22)

The method used to obtain (%m&x— £mln) by satis-fying eq 2.22 was essentially that of Eydberg [8]and Klein [9]. Calculated values of the potentialenergy U in wave numbers are given in table 1.

J(J+I)=O 300 600 900 1200 1500

FIGURE 1. Potential-energy curves for H2.


TABLE 1. Molecular potential energy Ufor H2 as a function

of £=(r—re)lre, the change in internuclear separation

- 0 . 5- . 4- . 3- . 2- . 1

0. 1. 2. 3.4. 5.6.7. 8

U

cm-i53, 64827,15012,3384,511

9420

6832,3604,6287,2239,968

12, 74415,46618,079

0.9L.OL.IL.2L3L.41.51.61.71.81.92.02.12.2

U

cm—1

20, 54022, 82224, 91526,81028, 50530,00931,32932,47233,45434, 29235,00135, 59936,09236, 496

2.32.42.52.62.72.82.93.03.13.23.33.43.5

00

u

cm—1

36, 82837,10037,32237,50337, 65037, 77037, 86737, 94638,00938,06138,10238,13638,16338, 296

re=0.7414X10-8cm

The effective potential energy curves for rotat-ing molecules obtained by adding to U for thenonrotating molecule the energy of rotation,J(J+l)Be/(l+£)2, in cm"1, are illustrated infigure 1. By applying the quantum integral,

(F-U'y'2d£=(v+l/2)h,

(2.23)

to the effective potential energy curves, U', a set ofcorresponding values of energy (F) and vibra-tional quantum number was determined for each

of a few large values of the rotational quantumnumber. In table 2 these corresponding valuesare given together wdth the maximum and mini-mum values of the energy (F) for different valuesof e/(e/+l). The data were used to determinethe constant energy lines in the v versus J dia-gram in figure 2.

TABLE 2. Corresponding values of v, J(J+1), and F

obtained by evaluating (j) pdq= (v-\-1 j2)h

[The values of v and J are not integral and so do not represent stationarystates, yet the table values indicate how F depends on v and Jover a rangeincluding many stationary states.]

F(above Uat £=0)

cm-1

38,26934, 26930,26926, 26922,269

38, 26934, 26930, 269

38,269

42,269

Maximum

300300300300300

600600600

900

1,200

8. 84836. 28744.50152. 98811. 6461

4. 83782. 72921.0757

1.4032

0.4845

i values of F and v forgiven values of J(J+1)

38,28839,09840,32341,85843, 71245, 989

Minimum

0300600900

1, 2001, 500

15.0539.9196.6153.9291.703

-0 .072

values of F and v forgiven values of J(J+1)

015,02725, 84734,11140, 60645, 601

0300600900

1,2001,500

- 0 . 5- . 5- . 5- . 5- . 5- . 5

10 15 20 25 30 35 40

J

FIGURE 2. Energy contour diagram for H2.

Table 3 shows that over a wide range of Jvalues the results of the numerical integrationjust described are in good agreement with therotational energy formula (eq 2.17) when the lastterm, corresponding to a geometric series con-tinuation, is included. For the larger values ofJ there are appreciable differences; yet, when it


is observed how large the final term of eq 2.17is in these cases, it seems surprising that thediscrepancies between F (table 2) and F (eq 2.17are as small as they are. In another publication[27] a more rapidly converging series representingJ (*/+l) as a function of the rotational energyhas been suggested.

TABLE 3. Comparison of rotational-vibrational energiesF from table 2 and from equation 2.17

300300600900

1,200

V

4. 5015

6. 2874

1.0757

1. 4032

0. 4845

F (table2)

cm -1

30, 269

34,269

30, 269

38, 269

42, 269

F (table2)-F

(eq 2.17)

cm -1

-54-34-78-300761

F i n a l t e r mo f e q 2 . 1 7

c m - 1

155154

3,904

24,192

86,345

3. Details of the Calculations and Results

In the evaluation of the series of section II, 1for the calculation of the thermal properties,direct summation was employed for temperaturesbelow 2,000° K. The resulting values to 2,000°K for the various thermodynamic functions £°,H°-E°o, -(F°-E°o)/T, and Cv° for the idealgas state at one atmosphere pressure are tabu-lated in tables 4, 5, and 6, for H2, HD, and D2.For n-H2 for temperatures above 2,000° K, thecontributions due to levels below 25,000 cm"1

were calculated by direct summation, whereas forlevels above 25,000 cm"1 a less laborious methodwas used involving the determination of thenumber of levels within successive equal steps of2,000 cm"1 in the rotational vibrational energy,using the results of the calculations of the lastsection which led to figures 1 and 2. For these

TABLE 4. Thermodynamic functions for H2 in the ideal gaseous state

Values for S° and -(F°-Eo)/T include nuclear spin

Temperature

°K10

2020 39

3033.1

4050

60708090100

120150200250298 16

300

350400500600700

1 000

1 500

2 0003 000

4 000

5 000

S°, cal mole -1 (

P-H2

11. 215

14. 658

14. 754

16. 672

1 7 . 1 6 118.102

19. 214

20.135

20.938

21.669

22.356

23.014

24. 259

25.945

28 202

29.889

31.168

31. 212

32.306

33.244

34.806

36.083

37.165

39. 701

42. 720

45.007

0-H2

15. 58119.024 .

19.120

21.039

21. 527

22. 468

23. 576

24.492

25.248

25.913

26. 500

27.029

27.959

29.143

30 808

32. 225

33.404

33.446

34. 505

35.432

36.990

38. 266

39.348

41. 884

44. 903

47.190

Ieg-i

71-B.2

15.607

19.050

19.146

21.064

21. 553

22. 494

23. 603

24. 513

25. 288

25. 969

26. 581

27.142

28.151

29.461

31. 275

32. 758

33.963

34.005

35.073

36.003

37. 561

38. 838

39.920

42.455

45.475

47. 762

51. 221

53. 839

55. 969

H°-Eo, cal mole -i

P-H2

49. 6785

99.357

101.295149.036

164. 437

198. 729

248. 581

299.106

351. 222

406.015464.385

526. 837

663.752

890. 605

1,282. 70

1, 660. 49

2,009. 99

2,023.16

2,377.84

2,729.19

3, 429. 24

4,129.48

4, 831.65

0-H2

388.327

438.006

439.943

487. 684

503.085

537.363

587.041

636. 722

686. 422

736.179

786.085

836. 277

938. 227

1,097. 78

1,387.90

1, 705.80

2,028.34

2,040.87

2,384.39

2, 731. 54

3,429. 53

4,129. 52

4,831. 66

TC-H2

303. 665

353.344

355.281

403.022

418.423452. 705

502. 426

552.318

602. 622

653. 638

705. 660

758. 916

869. 609

1,045.99

1,361.61

1, 694.47

2,023. 75

2,036.44

2,382. 75

2, 730.95

3,429. 46

4,129. 51

4, 831. 66

6,966. 23

10, 697. 2014, 679. 2

23, 230.9

32,345.

41,895.

F--E8

P"H2

6.247

9.690

9.786

11.705

12.193

13.134

14. 243

15.150

15. 921

16. 594

17.197

17. 745

18. 729

20.007

21. 788

23. 246

24.426

24.468

25. 512

26.421

27.948

29. 200

30.263

32. 735

35. 589

37. 668

0-H2

-23. 252

-2.876

-2 . 457

4.783

6.328

9.034

11.836

13. 870

15. 442

16. 710

17. 766

18. 667

20.140

21.825

23.869

25.402

26.602

26.643

27.693

28.603

30.131

31.383

32.446

34.918

37. 770

39. 851

-1 deg -1

n-H.2

-14.7601.382

1.721

7.630

8.911

11.17613. 554

15. 307

16. 679

17. 799

18. 741

19. 554

20.904

22.488

24.466

25.981

27.175

27. 217

28.26529.175

30. 702

31. 955

33 018

35.49038.343

40.422

43. 478

45. 753

47. 590

Cp cal mole -1 <

P-H2

4.968

4.968

4.968

4.968

4.968

4.973

5.007

5.115

5.330

5.646

6.036

6.455

7.204

7.807

7.742

7.380

7.158

7.152

7.049

7 010

6.998

7.010

7 037

0-H2

4.968

4.968

4.968

4.968

4.968

4.968

4.968

4.969

4.972

5.982

5.003

5.039

5.170

5.487

6 110

6.565

6.803

6.809

6.9176 963

6.992

7.009

7 036

ieg-i

n-H2

4.968

4.968

4 968

4.968

4.968

4 969

4.978

5 005

5.061

5.148

5.2615.393

5.678

6.067

6 518

6.770

6.891

6.894

6.951

6 975

6.993

7.009

7 036

7. 219

7 720

8.195

8.859

9.342

9.748


TABLE 5. Thermodynamic functions for HD in the idealgas state

Values for S° and — (F°-Eo0)/7

1 include nuclear spin

Temperature

°K10—2022.133040 _ .

5060708090

100-120-150200 -250-

298.16300400500- . '600

700 .1,0001,5002,000

So

cal mole-1

deg-i15.98219. 49720.05021. 86123. 792

25. 37526. 68027. 77228. 71429. 542

30. 27931. 55433.11235.11936. 676

37. 90537. 94839.95741. 51742. 795

43.88146.44349. 52751. 871

H°-E§

cal mole - 1

49.681100. 600112 234159. 230226. 510

297.472368.910439 914510. 464580. 708

650. 733790. 592

1,002 021,348.821, 697. 62

2,033. 662,046. 502, 744. 723,443. 854,144. 90

4,849. 607,007. 5010,821. 214,898.4

F°-E§T

cal mole - ideg-i11.01414. 46814. 97916. 55318.129

19. 42520. 53121.48822.33323.089

23. 77224.96626. 44528.37529.885

31.08431.12633.09534. 62935. 886

36. 95339. 43642.31344. 421

cal mol -ideg-i4.9715.3655 5646 3676.991

7.1497.1267 0767.0377.013

6.9996.9856 9786.9756.977

6.9796.9796.9866.9997.025

7.0727.3397.9098. 376

higher levels having characteristic temperaturesabove 36,000° K, the exact placement of eachindividual level is not important for calculationsup to 5,000° K.

Figure 1 shows that the effective potentialenergy curves for rotational quantum numbersother than 0 have broad potential energy barriersabove the minimum dissociation energy, 38,296cm"1, for J=0. As a result there are above 38,296cm"1, the minimum dissociation energy, quantizedrotational-vibrational levels belonging to thesequences of levels below 38,296 cm"1. Thesestates are represented by the points in figure 2between the dashed curve and the full line dis-sociation energy curve passing through (t/=0,0=15.1) and (J = 32.5, v= — %).

It seemed proper to include in the calculationsof the thermal properties of hydrogen above 2,000°K these quantized or partially quantized rotation-al-vibrational states. The values of the thermo-dynamic functions for n-H2 from 2,000° to 5,000° Kin table 4 are based on this convention.

The effect of the quantized rotational-vibration-al levels above the minimum dissociation energyof H2 on the most sensitive of the functions cal-culated, namely the molecular heat capacity, isrepresented in figure 3. Curve A represents the

TABLE 6. Thermodynamic functions for D2 in the ideal gaseous stateValues for S° and ~(F°-E% )/T include nuclear spin

Temperature

°K102023.57304050

60—70809 0 -100

120150200298.16.—300400

5006007001,0001,500

2,000

S°, cal mole-* deg-i

P-D2

17. 64521.08821. 90423.10224. 53325. 649

26. 57627.38428.11428. 78629.414

30. 55932.04134.02336. 80536. 84838. 857

40. 419

41. 704

42. 802

45.422

48.611

51.027

o-D2

16.839

20. 283

21.101

22.315

23. 843

25.180

26.41827. 56328. 60129. 52730. 353

31. 73933.36635.39538.18238.22540. 234

41. 79643.08144.17946. 80049. 989

52. 405

W-D2

18.372

21.816

22. 633

23.842

25.338

28. 600

27. 73628. 76829. 70430. 54531. 304

32. 61134.18936. 20238.98839.03141.040

42. 60243. 88744. 98547.60550. 794

53. 210

H°—EQ, calmole-i F°-F0 ', cal mole -1 deg - 1

P-D2

220. 505270.183287. 918319. 863369. 584419. 599

470. 480522. 948577. 589634. 706694.306

819. 9961,019. 521,364.062,048.102,080. 93

0-D2

49. 67999.364

117.139149. 514202. 775262. 811

330.843405.192482. 997561. 671639. 875

791. 9081,010.371,362. 902,048.082,060. 92

n-D2

108. 621

156.303

174.085

206. 297

258. 378

315. 048

377.389

444. 444

514. 528

586.016

658.018

801. 270

1,013.42

1, 363.29

2,048.09

2,060. 92

2, 759.18

3, 459.38

4,164.03

4, 876.39

7, 084.30

11,027.3

15, 229

P-D2

- 4 . 406

7.579

9.689

12. 440

15. 293

17.257

18. 734

19. 913

20.893

21. 734

22. 471

23. 725

25. 244

27.202

29. 936

29. 978

31. 959

33. 500

34. 763

35. 835 .

38. 338

41. 259

43.411

0-D2

11. 871

15.315

16.131

17. 331

18. 774

19.923

20.904

21. 775

22. 564

23. 287

23.954

25.139

26. 629

28. 580

31.313

31.355

33.336

34. 877

36.141

37. 212

39. 716

42. 637

44. 789

n-B2

7.710

14. 001

15. 248

16. 965

18.879

20. 299

21. 446

22. 419

23. 272

24.035

24. 724

25. 933

27. 432

29.386

32.119

32.161

34.142

35. 683

36. 946

38.018

40. 521

43.442

45. 594

p, cal mole-* deg-i

P-D2

4.9684.9884.9684.9684.9805.033

5.1565.3485.5865.8386.079

6.4666.7906.9476.9776.977

0-D2

4.9684.9724.9895.1055.6176.412

7.1637.6567.8627.8607.751

7.4547.1496.9966.9786.978

4.968

4.971

4.982

5.059

5.404

5.952

6.495

6.887

7.103

7.187

7.193

7.125

7.029

6.980

6.978

6.978

6.989

7.019

7.079

7.173

7.562

8.178


10

o£ 7"oO

Q.O

/

A

• Brinkworth© Cornish & Eastmane Crofts® Eucken<» Eucken a Muckeo Fenning & Whiffen<» Giacomini* Partington & Howe• Pier a Bjerrumc Scheel a Heuse° Wohl a Mag at

1000 2000 3000 4000 5000T oK

FIGURE 3. Specific heat of normal hydrogen at constant pressure.

TABLE 7. Thermodynamic functions for H2 in ideal gaseousstate

[Based only on levels below minimum dissociation energy]

T

° K3,0004,0005,000

Entropy

cal mole-1

deg-i51. 22153.83855. 960

Enthalpy

cal mole~l

23,230.832,34141, 854

F°-E°QT

cal mole-1

deg-i43.47845. 75347. 589

Specific, heat

cal mole-1

deg-1

8.8599.3419.675

molecular heat capacity if the quantized rota-tional-vibr a tional levels above the minimum dis-sociation energy are included as molecular levels,and curve B represents the molecular heat capac-

Properties of Hydrogen807127—48 3

ity if the molecular levels are regarded as extend-ing only up to the minimum dissociation energy.In table 7 are tabulated the values of the thermo-dynamic functions for n-H2 based on calculationsinvolving only energy levels below the minimumdissociation energy.

For convenience in the calculation of the thermo-dynamic functions of the real gas n-H2, values forn-H2 in the ideal gas state at all temperatures forwhich there are entries in the tables of PVT datawere obtained from table 4 by interpolation andare tabulated in table 8. The interpolated valuesof S°, -(F°-E°o)/T, and C°v agree to within±0.001 with values that would have been ob-

389

tained by direct summation. In the case ofH°—E°o, the agreement is within three in thelast digit carried.

TABLE 8. Thermodynamic functions for normal H2 inthe ideal gaseous state

Values for S° and — (F0—EQ)IT include nuclear spin

TABLE 8. Thermodynamic functions for normal H2 inthe ideal gaseous state—Continued

16-18..20..22..24..

28..30-32..34..

36_.38..40..42..44..

46 -48..50-52..54..

56..58..60 -65..70-

75..80 -8 5 -90-95..

100-105..110..115..120..

125..130..135..140..145..

150-155..160-165..170-

180..190-200-210-220-

> K

s°

cal mole -1

deg -1

17.94218.52719.05019.52419.956

20.35320.72221.06421.38521.686

21.97022.23922.49422. 73722.968

23.18923.40023.60323.79823.986

24.16824.34324.51324.91525. 288

25.63925.96926. 28326. 58126.868

27.14227.40827.66427.91128.151

28.38428.61028.83129.04729. 257

29.46129.661 .29. 85630.04730. 234

30. 59530. 94231. 27531. 59431. 901

H°-E°

cal mole -1

333.473343.408353.344363.280373.215

383.151393.087403.022412.959422.896

432.832442. 767452.705462.643472. 583

482. 527492.474502.426512.384522.351

532.327542.315552.318577.399602. 622

628.022653.638679. 507705.660732.122

758.916786.056813. 549841.400869.609

898.175927.086956.335985.91

1,015.80

1,045.991,076.471,107. 221,138.231,169.49

1, 232. 711. 296. 781,361. 611,427.101.49S. 20

F°-E°T °

cal mole -1deg-i

-2.900-0.551

1.3823.0114.405

5.6166.6837.6308.4809.248

9.94710.58711.17611. 72212.227

12.69913.14013. 55413.94414.313

14.66214.99315.30716.03216.679

17.26517.79918. 28918. 74119.161

19. 55419.92220. 26820. 59520.904

21.19821.47921. 74722.00522. 251

22.48822. 71622. 93623.14923.355

23. 74724.11624.46624. 79825.114

c;

cal mole -1deg-i4.9684.9684.9684.9684. 968

4.9684.9684.9684.9684.968

4.9684.9684.9694.9704.971

4.9734.9754.9784.9824.986

4.9914.9985.0055.0295.061

5.1015.1485.2025.2615.325

5.3935.4635.5345.6065.678

5.7485.8165.8835.9476.008

6.0676.1236.1776.2286.276

6.3666.4466.5186.5816.638

T

0 K230

240 . .

250

260

270 .

280

300

320

340

360

380

400

420

440 - _

460

480

500520

540

560

580

600

650

S°

cal mole -1

deg-i32. i9732.48332. 75833.02433.282

33.53134.00534.45234.87235. 269

35.64636.00336.34436.66836.979

37. 27637. 56137.83738.10038.355

38.60038.83839.399

H°-E°o

cal mole -1

1,559.851,626.961,694.471, 762.331,830.49

1,898. 922,036.442,174. 632,313. 282,452. 29

2, 591. 532, 730. 952,870. 513,010.143,149.85

3,289.623,429.463, 569.343, 709.283,849.30

3,989.364,129. 514,480.19

F°-E°T °

cal mole -1

deg-i25.41525.70425.98126.24626. 502

26. 74927.21727.65628.06828.457

28. 82629.17529.50929.82630.131

30.42230. 70230. 97331.23131.481

31. 72231.95532. 506

cal mole - 1

deg-i6.6886.7316.7706.8036.831

6.8566.8946.9226.9436.957

6.9686.9756.9806.9846.987

6.9906.9936.9966.9997.002

7.005. 7.009

7.021

The contributions to the entropy and to the re-lated free energy functions arising from (1) thenuclear spins, (2) the triple degeneracy of the low-est rotational state of o-H2 and p-D2, and (3) themixing of the ortho and para varieties in n-H2 andn-T>2 have been included through eq 2.3, 2.4, 2.15,and 2.16 in all the tables. A comparison of theentropies and free energies of hydrogen and deuter-ium calculated from calorimetric data with valuesin the tables must take into account the degener-acies existing in the solid state at the lowest temp-erature of the calorimetric measurements. Theremust accordingly be added to the calorimetric val-ues of entropy calculated from data extending from10° K to higher temperatures, the entrooies oftable 9. In calculations concerning chemical re-actions above room temperature nuclear spin en-tropies are customarily omitted for all componentsof the reactions.

To obtain entropies of w-H2, HD, and n-D2

suitable for such use above room temperature,there should be subtracted from table values ofthe entropies R In (2ii + l) (2i'2+2) where ix

and i2 are the two nuclear spins within the mol-ecule [14]. For n-H2 this is equal to R In 4 = 2.755


TABLE 9. Low-temperaiure (10° K) entropy contributions arising from rotational and nuclear-spin degeneracies

VarietyValues of JWeight of lowest rotational level

(2J+1).Nuclear spin weight, see eq 2.3 and

. 2.4.Total added entropy . .

H2

ParaEven1

1

0

—R(xo\nxo+xP In xP)XoSo+XpSpTotal added entropy (XJ8,—RXJ In a:,)

OrthoOdd3

3

R In 9=4.366 calmole -ideg-1.

HD

OnlylBoth odd and even1

6


n-H2

R(ln 4-3A In 3) = 1.117 cal mole-* deg-i% R In 9=3.275 cal mole"1 deg-ii^ln 4+% In 3)=4.392 cal mole-ideg -i

D2

OrthoEven1

6


ParaOdd3

3

R In 9=4.366 caJmole -ideg-1.

n-T>2

R(ln Z-% In 2) = 1.265 cal mole-* deg-iR($£ In Z+% In 2) =3.829 cal mole -* deg-i%R In 3=5.094 cal mole-ideg-i

cal mole"1 deg"1; for HD, E In 6=3.560 cal mole"1

deg"1, and for n-D2, B In 9=4.366 cal mole"1

deg-1.The reliability to be expected in thermody-

namic functions for the ideal gas state calculatedfrom spectroscopic data has been considered byearlier writers on the basis of the reliability ofspectroscopic constants and the gas constant R.The former estimate of one or two hundredthsof a calorie mole"1 deg"1 for the probable errorin the free energy function, specific heat andentropy, appears reasonable. Over much of thetemperature range it is probably a more liberalestimate than necessary, as more recent and pre-sumably better spectroscopic data and valuesfor the physical constants have been used. Alarger allowance may be necessary for the highertemperatures, however, possibly twice as muchat 5,000° K.

The results of the present calculations below2,000° K arQ in fairly close agreement with thoseof Giauque [4], Johnston and Long [18], Davisand Johnston {17], and Wagman, et al. [28].Above 2,000° K the effect of the new calculationsof the high rotational levels of H2 is apparent.

This can be seen in figure 3 in which the resultsof Davis and Johnston (curve G) for the specificheat of hydrogen, the most sensitive propertycalculated, are compared with table values of thispaper (curves A and B). Curve A, correspond-ing to table 4, is based on the inclusion of thequantized rotational-vibrational levels above theminimum dissociation energy as molecular levels,and curve B, corresponding to table 7, is basedonly on levels below the minimum dissociationenergy.

In figure 3 are plotted also a large number ofscattered points representing the experimentalobservations of many investigators. [33 to 37,40 to 46, 50, 51, 56]. In cases where mean specificheats were reported, they have been plotted forthe mean temperatures of the experimental in-tervals. At room temperatures and below, thetheoretical and experimental specific heats are ingood agreement, as has been the case since thecorrect treatment of the ortho and para forms byDennison [1] in 1927. Above 1,200° K the ob-servations obtained by the explosion method lieabove the theoretical curve. The difficulties ofthe explosion method are great and the accuracynot high [53], consequently the authors feel thatthe calculated curve and table are more reliable.

At atmospheric pressure and a temperature of2,000° K, there is a small but perceptible dissocia-tion of H2, HD, and D2. As the heat of dissocia-tion of hydrogen is large there are significantdifferences between the calculated properties ofmolecular H2, HD, and D2, tables 4 to 6, and theproperties of the dissociating gases. At 2,000°K the table value of Cv for molecular H2 is 8.195cal mole"1 deg"1, whereas for an ideal gas mixture ofmolecular and atomic hydrogen in equilibriumat atmospheric pressure the value is 8.797, adifference of 0.60 cal mole"1 deg -1. For HD andD2 the differences between the two specific heatsare 0.41 and 0.57 cal mole"1 deg"1, respectively.The effect of pressure upon the specific heat ofdissociating hydrogen is illustrated in figure 4 anddiscussed in section III. At temperatures wherethere is appreciable dissociation of HD, equilib-rium mixtures of H2, HD, and D2, are established.


III. Equilibrium Constants for Dissociation,Isotopic Exchange, and Ortho-ParaConversion

The equilibrium constanttion

of a gaseous reac-

(3.1)

in which each of the participating gases AlyA2,. . ., Bi,B2, . . . has the equation ofstate PV=RT, is related to the partial pressuresof the gases and to their free energies, F*, at unitpressure by the equation

RT InPJEPJgPJg.

^ - A F * (3.2)

Equilibrium constants for dissociation, isotopicexchange,6 and ortho-para conversion of hydrogenmay be calculated by using the —(F°—EQ)/Tvalues of tables 4, 5, and 6. El is the internalenergy per mole of molecules without translationalmotion in the lowest energy level «/=0, v=0 andin the ideal gas state, and F° is for the ideal

gas state and a pressure of 1 atm: Using% instead of F*,

RhiK=AAE°0 (3.3)

The values of A^o0 for the reactions consideredin this section are given by the spectroscopicdata used in the previous section. Using freeenergy values as given in the tables of this paper,the atmosphere is the unit of pressure for K andP in the mass action law,

(3.4)

• Equilibrium H2 and D2.

Deviations from the laws of ideal gases canbe taken into account by use of fugacities oractivities in place of partial pressures and theforms of eq 3.2, 3.3, and 3.4 for K are retained.When fugacities or activities are substituted forpartial pressures, F* becomes the free energy atunit fugacity or activity. For a fuller discussionof the use of fugacities and activities the readeris referred to references [29 to 32],

The entropies of monatomic H and D (see p.383) must include the nuclear and electron spinentropies besides the entropy of translation, eq

80

6 0

40

20

PARA

^ R T H O

f\/D/

/ y

1

A

392

0 1000 2000 3000 4000 5000

T °KFIGURE 4. Curves showing effect of dissociation on specific heat of H2.

Journal of Research

2.12, when used with table values of the entropyand free energy of molecular H2, HD, and D2,in the calculation of equilibrium constants fordissociation. Accordingly for H,

F° _ 5 , T

RT ~2mi•2.2663 and ^ =

and for D,

F°-i

£ln T+ 0.2337,

4 In T-0.8223 and

| In T+1.6777

(3.5)

(3.6)

in the ideal gas state at a pressure of 1 atm forthe range of temperatures covered by the tables.

1. Dissociation of H2, D2/ and HD

The chemical equations for dissociation and thecorresponding mass action equations are

(a)

(b) D2±=>2D; -P-=J

(c) HD*=>H+D;

(3.7)

(3.8)

(3.9)

For these reactions, AE°0 of eq 3.3 is the dif-ference between the internal energy of 2 moles ofdissociated atoms and 1 mole of molecules in therotational-vibrational state J = 0 , v=-0. Beutler'svalue [21], 36,116 ±6 cm"1, was accepted for thedissociation of H2 from its ground state. Assum-ing that the total depth of the potential energycurve is the same for H2, HD, and D2, the disso-ciation energies of HD and D2 were obtained fromthe zero-point vibrational energies. These zeropoint energies were calculated by adding to Go

(see eq 2.17), the term which Dunham [10]included in the energy of the ground state relativeto the bottom of the potential energy curve anddesignated Foo in his system. The values thusobtained for the zero point energies of H2, HD,and D2 were respectively 2,179.6, 1,891.0, 1,546.6cm"1, and the corresponding energies of dissocia-tion for HD and D2 from the ground state 36,404.6and 36,749.o cm"1, respectively.

The heats of dissociation of H2, HD, and D2 inthe ideal gas state at temperature T are equal toAE°0+5RT-(H°-El), where (H°-E°o) is thetable value of the enthalpy at temperature T.The heats of dissociation at 0° and 298.16° K aregiven in table 11. The theoretical value for theheat of dissociation of n-H2 at 298° K agrees wellwith the calorimetric value 105,000 ±3,500 calmole"1 obtained by Bichowsky and Copeland [47].

On the assumption that the atomic and molecu-lar forms of hydrogen and deuterium are individu-ally ideal gases, the fraction of the originally totallynondissociated hydrogen which has dissociated is•yjK/(K-\-4:P), wherein is the dissociation constantand P is the total pressure in atmospheres.

The dissociation constants K and fractions oforiginally undissociated diatomic molecules, dis-sociated at 1-atmosphere pressure, are given intable 10 for H2, HD, and D2.

The experimental values of the equilibrium dis-sociation constants of H2 as determined by Lang-muir and Mackay [32], and by Langmuir [39], arein agreement with the theoretical values of table10. Langmuir's ^-values are 0.17 percent at

TABLE 10. Dissociation constants, K, and fractiondissociated, x, at 1-atm pressure

For H2^2H

TfK

3005001,0001,5002,0003,0004,0005,000

K

atm18.39X10-724.939X10 ~41

5.174X10 -is3.100X10 -I"2.641X10-62.480X10-22. 523641.038

X

21.44X10-373. 514X10 -2i1.137X10 -»8.675X10-68.125X10 ~4

0.07850.6220.9546

For HD#H+D

3005001,0001,5002,000

2. 732X10 -721.265X10 -411.987X10-181.350X10 -io1. 215X10 -6

8.264X10 -371. 779X10 -2i7.048X10-io5.810X10-65. 512X10 -*

For D2?=*2D

3005001,0001,5002 ,000 . . .

1. 319X10 -721.171X10 -4i2.972X10 -is2.330X10-102. 227X10 -e

5.742X10-371.711X10-218.620X10 -ii7.632X10-67.462X10 ~4


2,000° K, 1.6 percent at 2,500° K, 7.2 percent at3,000° K, and 21 percent at 3,500° K.

TABLE 11. Heats of dissociation of H2, HD, and D2 incal mole~l

°K

8.16....

103,239

104,191

o-H2

102,900

104,173

W-H2

102,985

104,177

HD

104,064

104,992

0-D2

105,048

105,962

104,877

105,962104,991105,962

An equation of state for 1 mole of molecularH2, HD, or D2 capable of forming 2 moles ofatoms when completely dissociated, assumingas before that atoms and molecules individuallybehave as ideal gases, is

PVRF

K (3.10)

orK+4P

PV KV / / ~WT\RT=l-8RTV-Al1 + mKv)' ^-U)

where K is a function of T determined by eq 3.3and V is the volume per 2N0 atoms uncombinedor combined as molecules.

The thermodynamic properties of an equilibriummixture of atomic and molecular hydrogen in theideal gas state can in principle be calculated fromthe properties of atomic hydrogen at low pressuresand the equation of state (eq 3.10) or (eq. 3.11).It is simpler, however, to determine the propertiesof the mixture from the properties of the atomicand molecular varieties and the fraction disso-ciated.

The equation given by Epstein [30] for the heatcapacity of a reacting gas mixture, when appliedto the heat capacity of an equilibrium mixtureof atomic and molecular hydrogen, is

V^p) mixture cy (^p/atomic

R ~Ax R(i-^)x r 9 (H°)atomlc

2 ' l

' R/molecular

~RT

where x is the fraction of the originally totallynondissociated hydrogen that has dissociated,(Cl)atomic and (C°)molecular are heat capacities permole of atoms and molecules respectively in theideal gas state, and (C°)mixture is for a mixture

containing 2N0 of atoms combined or uncombined,the components being in the ideal gas state.(Cp)mixture is a function of P as well as T since x isa function of P. In figure 4, curves D, C, and Bshow the variation of (C°JR)mixture for H2 withtemperature for pressures of 0.01, 1, and 100atmospheres, respectively. Curve A drawn forcomparison is the heat capacity of 1 mole ofundissociated H2, that is, (C°JR)molecui&r. Itappears from these curves that when dissociationhas its greatest importance, thermal effectsoriginating in other ways are likely to be dwarfedby comparison. Wildt [19] has calculated theratio of specific heats of hydrogen at high tempera-tures using principles similar to those employedhere. The results obtained have application tostellar atmospheres.

2. Ortho-Para Equilibrium

_/y-H 2 \_>

-K. (3.13)

p - D 2 ± ^ o - Z ) 2 , p ^ = ( ^ _ 2 ) = K (3.14)

The equilibrium constants of the ortho-paraconversion of H2 and D2 in the ideal gas state areindependent of P. Accordingly, pressure does notappreciably change the ortho-para ratio underequilibrium conditions. Although the lowest rota-tional levels of the ortho and para varieties differ,AEQ for the two reactions (eq 3.13 and eq 3.14)is zero, because in the calculations for both theortho and para varieties the ground state of themolecule, J=0 and v=0, was arbitrarily selectedas the origin of energies.

In table 12 are given values of the percentagepara composition in the ideal gas state of equi-librium mixtures of ortho-para varieties calculatedfrom the state-sums, 2<7 ~" ej/kl\ see eq 2.2 and eq2.14. These values are in close agreement withearlier values obtained by Harkness and Deming[11] and are in agreement with the variations inthe relative intensities of the ortho-para spectrallines and with estimates of the ortho-para compo-sitions based on measurements of thermal conduc-tion from heated wires. The success in explainingthe heat capacity of gaseous hydrogen at moderateand low temperatures is also corroborating evi-dence for table 12 [48].

394 Journal o! Research

•

/

/

_ —

•

o

o

o Rittenberg , Bleakney & Urey• Gould, Bledkney a Taylor

Theoretical Curve

oo

100 200 300 400T °K

FIGURE 5. The equilibrium constant for '.

500 600 700 800

TABLE 12. Ortho-para composition at equilibrium

T

°K102020 3923 573033 1040 .5060 . -708090

100120150 .200250298 16300 - - .350400500

Percentagein para form

for H2

99.999999.82199. 789

97.02195.03488. 72777.05465. 56955. 99148. 53742. 88238. 62032. 95928. 60325. 97425.26425.07525.07225.01925.00525.000

Percentagein para form

for D2

0.02771.998

3.7617.864

14. 78420. 71825.13128.16230.14131.39532.16432. 91633. 24633.327

33.33333.333

The equilibrium constant Kex of the isotopicexchange reaction (eq 3.15) is related to the dis-sociation constants K of eq 3.7, 3.8, and 3.9 bythe equation

Ke^1^^' (3.16

3. Isotopic ExchangeThe chemical and mass action equations

isotopic exchange areHD

for

H (3.15)

The equilibrium constant Kex for isotopic exchangein the ideal gas state is independent of P, andaccordingly the relative equilibrium concentra-tions of H2, HD, and D2 are also independent ofpressure in the ideal gas state. For this reactionthe AE'o of eq 3.3, the difference between twicethe energy of the ground state of HD minus thesum of the energies of the ground states of H2

and D2, is equal to twice the zero-point vibra-tional energy of HD minus the sum of the zero-point vibrational energies of H2 and D2. Usingthe values given in section III, 1 for the zero pointenergies, Ai?o is 159.5 cal for the formation of2 moles of HD.

In figure 5 are plotted experimental values ofKex, whereas the curve was derived from spectro-scopic data as has been indicated. The data ofKittenberg, Bleakney, and Urey [54] were ob-tained from measurements on hydrogen-deuteriummixtures prepared by the decomposition of mix-


tures of HI and DI, and those of Gould, Bleakney,and Taylor [55] were obtained with mixtures ofhydrogen and deuterium that had been adsorbedon various catalysts or had been diffused throughpalladium. Some of the observations of Gould,Bleakney, and Taylor plotted in figure 5 were notplotted by them in their published article.

Although the theoretical curve of figure 5 isthought to be more reliable than the experimentaldata, it is to be pointed out that the uncertaintiesin the zero-point energies of H2, HD, and D2 cangive rise to perceptible shifts in the curve. Thusa change in AE°0 of 3 cal mole"1, which is equivalentto about 1 cm"1 in 2(GO)HD~(GO)H2~(GO)D2,

changes Kexhy about 1.5 percent at 100° K. Itseems doubtful that AE°0 is known better than to avery few calories per mole, for while it is plausible,it is apparently not certain that De, the dissociationenergy above the minimum of the potential energycurve, is so nearly the same for H2, HD, and D2

[25]. The theoretical values of Urey and Bitten-berg [13] are, therefore, practically as reliable asthe newly calculated ones.

IV. PVT Data and Relations forHydrogen and Deuterium

In order to calculate the thermodynamic prop-erties of gaseous hydrogen at high densities (inprinciple at any densities other than very low)from values of the properties for the hypotheticalideal gaseous state, it is necessary to have informa-tion concerning the relations between pressure,volume, and temperature for each temperaturein question extending from very low to highdensities.

1. Hydrogen

The available PVT data for hydrogen fallbetween 14° and 700° K. They consist, in gen-eral, of measurements of volume of known amountsof gas at several different pressures along selectedisotherms. The quantities usually reported arevalues of PV or PV/P0V0 at the measured pres-

sures or densities. In this report this informationis presented in the form of tables in which integralvalues of the variables of state are spaced closelyenough to allow accurate interpolation.

The dependent variable Z appearing in thetables is PV/RT. Through the definition of R,this quantity has the value 1 at extremely lowdensities, and it is of the same order of magnitudeover a very extended range of densities. Theindependent variables chosen are T, the Kelvintemperature, and p, the Amagat density, which isdefined as the ratio of the observed density to thedensity at standard conditions (0°C and 1 atmos-phere). Density was chosen as an independentvariable of state in preference to pressure becausethis resulted in simpler representation of thePV/RT isotherms. The Amagat density is alsothe ratio of the volume Vo of the gas at standardconditions to its observed volume.

observed density- 3 - (4.1)density at standard conditions

The best value for Vo, the molar volume of hydro-gen at standard conditions, is 22.4279 liters or22428.5 cm3, according to the values of RT0

obtained by Cragoe [90] and the value of PV/RTfor hydrogen at standard conditions as given byCragoe and the present correlation. The densityof hydrogen at standard conditions is 0.089888gram liter -1.

Values of PV/RT, or Z for n-H2 are given intable 13 for different values of T and p. Cor-responding values of P and of the derivatives(dZ/dT)p, (d2Z/dT2)p and (dZ/dp)T needed for thecalculation of some of the more important thermalproperties of the real gas from ideal gas values (seesection V) are given as functions of the samevariables of state p and Tin tables 14, 15, 16, and17, respectively. The temperature intervals usedare of graduated size, being as small as 2 degreesat low temperatures and as much as 20 deg above0° C. The density intervals, except for entries atp= l , 2, 3, 6, and 10, are uniformly equal to 20Amagats from p=0 to p=500.


TAB

LE 1

3.

Val

ues

of Z

=P

V/R

T

for

hydr

ogen

at

inte

gral

val

ues

of T

, th

e ab

solu

te te

mpe

ratu

re,

and

p, t

he d

ensi

ty i

n A

mag

at

unit

s

Temperature

°K

16 18 20 22 24 26 28 30 32 34 36 38

.40 42 34 36 38 40 42 34 36 38

. .

40 42

...._

0.990904

. 992299

. 993373

. 994225

. 994917

. 995492

. 995977

. 996389

. 996742

. 997048

. 997315

. 997550

. 997758

. 997943

,=22

0

0.4937

5419

.5844

.6224

. 6565

,=52

0

0. 2775

. 3675

.44,89

. 5225

. 5890

2

0.981826

.984612

. 986758

. 988460

. 989846

.990994

.991962

. 992784

. 993490

. 994102

. 994636

. 995104

. 995520

. 995892

240

0.4619

. 5135

.5591

. 5998

.6364

540

0. 2878

.3809

.4657

.5421

.6110

3

0.972765

. 976940

. 980157

982708

. 984783

. 986505

. 987955

. 989187

990245

. 991162

.991962

. 992664

. 993287

. 993844

260

.04327

.4875

.5360

.5794

. 6183

560

0.3038

.3995

. 4882

. 5682

6

0.95401

. 96043

.96552

. 96965

. 97309

.97598

. 97844

98055

.98238

.98397

.98537

. 98662

.98773

280

0.4062

4639

.5151

.5610

.6021

580

0. 3258

4242

.5167

10

0 93430

94274

94962

.95532

96012

.96420

96771

97075

.97340

. 97573

.97779

.97964

300

0.3825

4428

.4966

.5446

.5880

600

0.3544

4557

.5520

20 0 90021

.91150

92100

92910

93606

94209

94735

. 95197

.95606

.95972

320

0.3613

4239

4799

.5302

.5758

620

0. 3913

4944

40 .8265

8452

8610

8747

8866

8969

. 90598

.91403

.92125

340

0.3425

4071

4652

. 517

!8

.565

7

640

0.43

8554

06

60 0 77

2479

58

8159

8334

8487

.862

0.8

739

.884

6

360

0 32

5939

2445

2550

73

.557

4

660

0.4971

5947

80 0.7334

7598

7826

.802

5.8201

.835

7

.849

7

380

0.31

1538

0044

21.4

990

.551

2

680

0. 567

8

100

0.6739

.706

2.7

342

.758

7.7

803

.799

5

.816

6

400

0.29

9237

00. 434

3.4

933

.547

6

700

0.6512

120

0.6551

.688

2.7

171

.742

6.7

652

. 785

4

420

0. 289

036

24.4

292

.490

5

.546

8

140

0. 606

7.6

446

.677

7.7

069

.732

8

.756

0

440

0. 280

8.3

573

.426

8.4

904

.548

8

160

0. 560

9.6

033

.640

4.6

732

.702

3

. 728

4

460

0. 274

8.3

547

.427

2.4

933

. 55

37

180

0. 517

6.5

644

.605

4.6

415

. 673

7

.702

6

480

0.2717

.355

0.4

305

. 499

2

.561

7

200

0. 5279

. 57

26.6

119

. 647

0

.678

6

500

0. 272

4.3

590

.437

4.5

085

.573

0

GO g

TAB

LE 1

3.

Val

ues

of Z

— P

V/R

T

for

hydr

ogen

at

inte

gral

val

ues

of T

, th

e ab

solu

te te

mpe

ratu

re,

and

p, t

he d

ensi

ty i

n A

mag

at

unit

s—C

ontin

ued

Temperature

QK

42 44 46 _

48 50 52 54 56 58 60

._

65 70 75 80 85 90 95 100

105

110

115

120

125

130

135

140

145...

._..

150

155

160

165

170

180

190

200

210

220

230

240. _

250

p =

l

0.997943

. 998110

. 998261

. 998399

. 998524

. 998638

. 998743

. 998839

. 998930

. 999013

. 999196

. 999348

. 999479

. 999590

. 999686

. 999770

. 999844

. 999909

. 999968

1.000021

1.000068

1. 000112

1.000152

1.000188

1.000221

1.000252

1.000280

1.000307

1.000332

1.000355

1. 000376

1.000396

1.000430

1.000462

1. 000490

1.000515

1. 000537

1.000556

1.000573

1.000589

2

0.995892

. 996226

. 996528

. 996802

. 997052

. 997280

. 997488

. 997682

. 997863

. 998030

.998394

. 998699

. 998960

. 999182

. 999374

. 999542

.999690

. 999821

. 999940

1.000046

1.000141

1.000228

1.000307

1.000379

1.000444

1.000505

1.000562

1.000615

1.000665

1.000711

1.000754

1.000793

1. 000862

1.000925

1.000981

1.001031

1.001075

1.001113

1.001147

1.001178

3

0.993844

. 994344

. 994798

. 995209

. 995583

. 995925

. 996238

.996528

. 996800

. 997049

.997595

. 998053

. 998444

. 998776

. 999065

. 999318

.999539

. 999735

.999911

1. 000070

1.000212

1.000343

1.000463

1.000572

1.000672

1.000763

1.000847

1.000926

1.001000

1.001069

1.001133

1.001192

1.001296

L. 001390

L. 001474

L 001548

L 001614

.001673

L. 001725

L. 001771

6

0.98773

. 98873

.98963

.990453

. 991199

. 991881

. 992507

. 993086

. 993628

. 994125

. 995214

. 996129

. 996910

. 997573

.998151

. 998656

. 999098

. 999489

. 999841

1. 000158

1.000442

1.000703

1.000943

1.001160

1.001356

1.001542

1.001710

1.001868

1.002016

1.002154

1.002282

1.002398

1.002607

1.002795

1.002962

1.003108

1.003240

1 003358

1.003463

1.003555

10

0.97964

. 98130

. 98280

. 98417

. 98541

. 98654

.98758

. 98854

. 98944

. 99027

.99208

. 99360

. 99490

. 99600

.99696

.99780

. 99854

. 99919

. 99978

1. 00031

1.00078

1. 00121

1.00161

1.00197

1.00230

1. 00261

1. 00289

1. 00315

1. 00340

1.00363

1. 00384

20

0.95972

.96302

. 96601

. 96872

. 97118

. 97342

. 97549

. 97742

.97920

.98084

.98443

.98746

.99004

.99224

. 99415

. 99583

. 99730

. 99860

. 99977

1.00082

1.00176

1. G0262

1.00341

1.00413

1.00479

1.00540

1.00596

1. 00648

1.00697

1.00743

1.00785

1.00403

1.00823

1.00438

1. 00469

1.00497

.1.00521

1.00543

1.00563

1. 00580

1.00596

L. 008

93

L. 009

55

L. 010

10

L. 010

58

L. 011

0201141

1.01175

] .01205

40

0.92125

. 92775

. 93364

. 93898

. 94384

. 94827

. 95235

. 95615

. 95966

. 96289

.96998

. 97597

. 98106

. 98543

.98923

.99255

.99548

. 99807

1.00038

1.00246

1.00434

1.00604

1.00760

1.00903

1.01035

1. 01155

1.01267

1. 01370

1.01467

1. 01558

1.01641

1. 01716

1.01856

1. 01978

1. 02085

1.02181

1.02267

1 02344

1.02411

1.02470

60

0.8846

.8942

. 90288

.91079

. 91798

. 92455

. 93060

. 93620

. 94138

. 94615

. 95669

. 96555

.97308

. 97958

. 98523

. 99017

99454

. 99841

1 00186

1.00495

1. 00775

L 01028

1.01260

.01473

1 01668

1. 01847

1.02014

1.02168

1. 02312

1.02447

1. 02571

L 02684

I. 02891

L 03072

.03231

1. 03372

1.03499

1 03611

1.03710

] L 03798

80

0. 8497

.8623

.8737

.8841

.8936

.90224

.91018

.91755

. 92436

. 93062

. 94454

.95619

. 96611

. 97470

. 98217

. 98871

. 99449

. 99962

1.00420

1.00830

1.01200

1. 01535

1.01842

1.02124

1. 02383

1. 02621

1.02842

1. 03047

1. 03237

1.03413

1.03577

1 03728

1. 04001

1.04238

1.04447

1.04632

1.04798

1 04945

1.05076

1. 05191

100

0.8166

.8321

. 8462

.8590

.8707

.8814

.8911

. 90020

.90859

.91631

.93351

. 94793

.96020

.97082

.98007

. 98819

. 99536

1. 00173

1.00741

1.01250

1.01710

1.02127

1.02509

1.02860

1.03183

1.03479

1.03753

1. 04007

1.04242

1.04460

1. 04663

1 04850

1.05186

1. 05478

1 05735

1 05964

1.06167

1 06348

1 06508

1.06649

120

0. 7854

.8037

.8204

.8355

.8493

.8619

.8735

.8842

.8941

.90323

.92363

. 94077

. 95537

. 96798

. 97899

. 98865

.99719

1.00477

1.01154

L. 01762

L. 02311

L 02809

L. 03265

1.03684

1.04069

1.04422

1.04748

L 05050

L 05330

L. 05590

L. 05831

nfioss

1.06451

1.06796

1.07100

1 07370

1.07610

1 07823

•1.08011

] . 08177

140

0.7560

.7770

.7961

.8135

.8293

.8438

.8572

.8695

.8808

.8914

. 91492

. 93473

.95163

. 96623

. 97895

. 99011

. 99999

1.00878

1.01665

1.02371

1.03009

1.03586

1.04114

1. 04599

1.05044

1.05453

1.05830

1.06179

1.06502

1.06803

1. 07082

1 07338

1.07797

1.08193

1.08541

1 08850

1.09125

1 09370

1.09587

1.09777

160

0. 7284

.7520

.7734

.7929

.8108

.8272

.8422

.8560

.8688

.8808

.90736

.92983

. 94904

. 96561

. 98002

. 99266

1.00386

1.01385

1.02280

1.03084

1.03808

1.04464

1.05062

1.05610

1.06113

1.06578

1. 07002

1.07396

1.07762

1.08102

1.08417

1.08710

1.09226

1.09671

1.1006

1.1041

1.1072

1 1099

1.1124

1.11451

180

0.7026

.7287

.7524

.7740

.7938

.8120

.8287

.8440

.8582

.8715

. 90102

. 92613

. 94764

. 96617

. 98225

.99635

1.00884

1.02004

1.03006

1.03905

1.04713

1.05445

1.06113

1.06725

1.07285

1.07797

1.08269

1.08707

1.09114

1.09492

1.09841

1.1016

1.1074

1.1124

1.1167

1.1205

1.1239

1 1270

1.1297

1.1320

200 C. 6786

.7071

.7330

.7567

.7783

.7982

.8165

.8334

.8490

.8636

.8960

.9237

.9475

.9680

.9857

1.0012

1.0150

1.0274

1.0385

1.0484

1.0573

1. 0653

1.0727

1.0794

1.0856

1.0912

1.0964

1.1012

1.1057

1.1098

1.1136

1.1172

1.1235

1.1289

1.1336

1.1378

1.1415

1 1448

1.1477

1.1502

260.

270_

26° C

s I §

300-

.

320.-

340. .

360.

. 100°

C-

—

400.

.

420.

.440..

460..

480..

500..

520..

540_.

560..

580..

1.000602

1.000614

1.000618

1.000626

1.000644

1.000646

1.000662

1.000.676

1.000688

1.000694

1.000697

1.000706

1.000713

1.000719

1.000724

1.000728

1.000732

1.000735

1.000737

1.000740

1.000741

1.000743

1.001206

1.001231

1.001238

1.001254

1.001290

1.001293

1.001326

1.001354

1.001377

1.001390

1.001396

1.001413

1.001427

1.001439

1.001449

1.001458

1.001465

1.001471

1.001476

1.001481

1.001484

1.001487

1.001812

1.001848

1.001859

1.001882

1.001936

1.001942

1.001991

1.002,032

1.002067

1.0Q2086

1.002096

1.002121

1.002142

1.002160

1.002175

1.002188

1.002199

1.002208

1.002216

1.002222

1.002228

1.002232

1.003637

1.003709

1.003731

1.003777

1.003886

1.003896

1.003994

1.004076

1.004145

1.004183

1.004203

1.004253

1.004294

1.004330

1.004360

1.004385

1.004408

1.004426

1.004441

1.004454

1.004465

1.004474

1.00609

1.006211

1.006247

1.006324

1.006508

1.006520

1. 006683

1.006819

1. 1. 1.007030

1.007112

1.007181

1.007240

1.007290

1.007332

1.007367

1.007397

1.007423

1.007444

1.007462

1. 007476

1.01232

1.01257

1.01264

1.01279

1.01315

1.01318

1.01350

1.01377

1.01400

1.01412

1.01418

1.01434

1.01448

1.01459

1.01469

1.01477

1.01484

1.01490

1.01495

1.01499

1.01502

1.01505

1.02523

1.02572

1.02586

1.02616

1.02685

1. 1.02755

1.028Q7

1.02851

1.02875

1.02S87

1.02918

1.02944

1.02966

1.02984

1.03000

1.03013

1.03023

1.03032

1.03040

1.03046

1.03050

1.03877

1.03948

1.03969

1.04012

1.04114

1.04124

1.04215

1.04291

1.04355

1.04390

1.04407

1.04452

1. 1.04520

1.04546

1.04568

1.04586

1.04601

1.04613

1.04623

1.04631

1.04637

1.05293

1.05386

1.05414

1.05471

1.05603

1.05615

1.05734

1.05832

1.05914

1.05960

1.05981

1.06038

1.06085

1.06124

1.06157

1.06184

1.06206

1.06225

1.06239

1.06251

1.06260

1.06266

1.06775

1.06889

1.06923

1.06992

1.07154

1.07169

1. 07313

1.07431

1.07529

1.07584

1.07610

1.07678

1.07734

1.07780

1.07818

1.07849

1.07875

1.07895

1.07911

1.07924

1.07933

1.07939

1.08325

1.08459

1.08499

1.08580

1.08768

1.08786

1.08953

1.09091

1.09204

1.09267

1.09297

1.09373

1.09436

1,09488

• 1.09531

1.09565

1.09593

1.09614

1.09631

1.09643

1. 09652

1.09657

1.09945

1.10098

1.10143

1.10235

1.10449

1.10469

1.10653

1.10812

1.10938

1.11009

1.11042

1.11126

1.11195

1.11251

1.11297

1.11333

1.11361

1.11383

1.11399

1.11411

1.11418

1.11421

1.11638

1.11809

1.11859

1.11961

1.12199

1.12220

1.12429

1.12598

1.12736

1.12812

1.12848

1.12938

1.13012

1.13071

1.13118

1.13155

1.13183

1.13204

1.13219

1.13228

1.13233

1.13233

1.1341

1.13594

1.13648

1.13760

1.14019

1.14043

1.14269

1.14451

1.14598

1.14679

1.14716

1.14812

1.14888

1.14948

1.14996

1.15032

1.15059

1.15078

1.15090

1.15096

1.15098

1.15095

1.1525

1.15455

1.15514

1.15635

1.15913

1.15938

1.16179

1.16373

1.16527

1.16611

1.16650

1.16748

1.16826

1.16886

1.16932

1.16966

1.16991

1.17006

1.17015

1.17017

1.17014

1.17006

8 CO

TAB

LE 1

3.

Val

ues

of Z

=P

V/R

T

for

hydr

ogen

at

inte

gral

val

ues

of T

, th

e ab

solu

te te

mpe

ratu

re,

and

p, t

he d

ensi

ty i

n A

mag

at

unit

s—C

ontin

ued

Temperature

°K

42 44 46 48 50 52 54 56 58 60 65 70 75 _

80 85 . .

90

95 100

105

110

115

120

125 ...

130

135

140

145

150

155

_

160

165

170

180

190

200

210

220

230

240

250

260

270

0°C

280

P-220

0.6565

.6874

.7154

.7410

.7644

.7860

. 8059

.8242

.8412

.8572

. 8925

.9228

.9488

.9711

.9904

1.0074

1.0224

1.0359

1. 0481

1.0589

1.0686

1.0773

1. 0854

1.0927

1.0994

1.1055

1.1111

1.1163

1.1212

1.1257

1.1299

1.1337

1.1405

1.1463

1.1514

1.1559

1.1599

1.1635

1.1666

1.1693

1.1718

1.17397

1.17460

1.17588

240

0 6364

.6695

.6996

.7270

. 7522

.7754

.7968

.8166

.8350

.8522

.8906

.9234

. 9515

.9756

.9965

1.0148

1.0311

1.0457

1. 0588

1.0705

1.0810

1.0905

1. 0991

1.1070

1.1142

1.1208

1.1269

1.1325

1.1377

1.1426

1.1471

1.1512

1.1585

1. 1647

1.1701

1.1749

1.1793

1.1831

1.1864

1.1893

1.1919

1.19420

1.19487

1.19623

260

0 6183

6535

.6856

.7149

.7417

.7665

.7894

.8106

8303

.8488

.8902

9254

.9556

.9815

1.0040

1.0236

1.0411

1 0567

1.0707

1 0833

1.0946

1.1048

1.1140

1.1224

1.1301

1.1372

1.1438

1.1498

1.1553

1.1605

1.1653

1.1697

1.1774

1.1840

1.1898

1.1949

1.1995

1. 2036

1. 2071

1. 2102

1. 2129

1. 21530

1. 21600

1.21741

280

0 6021

6394

6734

7046

7332

7594

7838

.8064

8273

8470

.8912

9289

9611

9888

1 0128

1 0339

1. 0525

1 0691

1.0840

1 0974

1 1094

1.1203

1.1302

1.1391

1.1473

1.1548

1.1618

1.1681

1.1740

1.1795

1.1846

1.1892

1.1974

1. 2044

1. 2105

1. 2159

1. 2207

1. 2250

1. 2287

1. 2319

1.2348

1. 23728

1. 23801

1. 23948

300

0 5880

6274

6633

6962

7265

7544

7802

8041

8262

8470

8938

9339

9681

9976

1 0232

1 0456

1 0653

1 0829

1.0987

1 1128

1 1256

1.1371

1.1475

1.1570

1.1656

1.1736

1.1810

1.1877

1.1939

1.1997

1. 2050

1. 2098

1. 2184

1. 2258

1. 2322

1. 2379

1. 2430

1. 2474

1. 2512

1. 2546

1. 2576

1. 26020

1.26094

1.26245

320

0 5758

6174

6553

.6901

.7220

7515

7787

.8037

8269

8488

.8982

9406

9768

1 0080

1 0350

1. 0587

1.0795

1 0981

1.1148

1 1297

1 1431

1.1552

1.1662

1.1762

1.1853

1.1936

1. 2014

1. 2085

1. 2150

1. 2211

1. 2267

1. 2316

1. 2406

1. 2483

1.2550

1. 2610

1.2663

1. 2709

1. 2749

1. 2784

1. 2814

1. 28407

1. 28483

1. 28637

340

0 5657

6094

6494

6861

.7198

7507

7791

.8053

8296

8525

.9043

9489

9872

1 0201

1 0485

1 0733

1.0953

1 1148

1.1323

1 1480

1 1620

1.1747

1.1862

1.1967

1 2063

1. 2150

1.2231

1. 2305

1. 2373

1. 2436

1. 2495

1. 2547

1 2640

1. 2720

1. 2790

1. 2852

1L. 2906

L. 2954

L. 2996

I. 3031

L. 3062

I. 30894

.30971

L. 31127

360

0 5574

6033

6454

6840

.7194

7518

7815

.8088

8342

8582

.9123

9591

9994

1 0339

1 0836

1 0896

1 1127

1 1332

1 1514

1 1678

1 1825

1 1957

1 2077

1. 2187

1 2287

1 2378

1.2462

1 2539

1. 2610

1.2675

1. 2737

1. 2790

1 2886

1. 2969

1. 3041

1. 3105

1.3161

1.3210

1. 3253

1.3290

1.3321

1.33486

1.33563

1.33719

380

0 551°

5992

6433

6837

7208

7547

7858

8143

8408

8658

9223

9712

1 0133

1 0495

1 0806

1 1077

1 1317

1 1531

1.1722

1 1893

1 2046

1.2184

1.2309

1. 2423

1 2527

1. 2621

1. 2707

1. 2787

1. 2861

1.2928

1.2992

1. 3046

1 3145

1 3230

1.3305

1. 3370

1.3428

1.3478

1.3522 •

1. 3559

1.3591

1.36185

1.36263

1.36418

400

0 5476

5976

6435

6857

7244

7598

7922

8219

8494

8754

9342

9852

1 0291

1 0669

) 0QQ4

L 1275

L 1525

1748

1947

9194

1 2283

1 2427

1 2557

1. 2675

1 2782

1 2879

1.2968

1.3051

1. 3127

1.3195

1.3260

1.3316

1 3418

1 3505

1.3581

1.3648

1.3707

1. 3758

1.3802

1.3840

1.3872

1. 38998

1.39074

.39227

420

0 5468

5987

6464

6902

7304

7672

8008

8316

8601

8871

9482

1 0012

1 0469

1 0861

1 1199

1 1493

1 1752

1 1983

1 2189

1 2373

1 2538

1 2687

1 2821

1. 2943

1 3053

1.3153

1.3245

1. 3330

1. 3408

1. 3478

1.3542

1. 3601

1 3705

1 3794

1. 3871

1.3939

1. 3999

1.4051

1.4096

1.4134

1.4166

1. 41927

1.42003

1.42152

440

0. 5488

6025

6518

6971

7386

7766

8114

.8433

8729

9008

. 9643

1 0194

1 0668

1 1075

1 1425

1 1730

1 1999

1 2238

1 2450

1 2640

1 2811

1 2965

1 3103

1. 3228

1 3341

1 3444

1.3539

1.3626

1. 3705

1.3776

1.3841

1. 3901

1 4007

1 4097

1 4175

1 4244

1.4304

1.4357

1. 440?

1.4440

1 4472

1. 44980

1. 45053

1.45196

460

0 5537

.6091

.6599

.7064

.7490

.7882

.8242

.8572

.8878

.9167

.9826

1 0397

1.0889

1.1311

1.1672

1.1986

1. 2264

1. 2511

1. 2732

1. 2929

1.3104

1.3262

1.3404

1. 3533

1. 3648

1. 3753

1.3850

1.3939

1.4019

1.4091

1.4157

1.4216

1. 4323

1. 4414

1.4493

1.4563

1.4624

1. 4676

1. 4721

1. 4759

1. 4790

1. 48160

1.48230

1.48366

480

0. 5617

.6187

.6707

.7183

.7620

.8022

.8392

.8733

.9049

.9348

1.0032

1.0623

1.1132

1,1568

1.1941

1. 2265

1. 2551

1. 2806

1.3034

1.3236

1.3416

1. 3579

1. 3725

1. 3856

1.3974

1. 4081

1. 4179

1. 4269

1.4351

1. 4424

1.4490

1.4548

1. 4655

1. 4747

1.4827

1.4897

1.4958

1. 5010

1. 5055

1.5092

1.5122

1. 51473

1. 51539

1. 51666

500

0. 5730

.6315

.6846

. 7331

.7777

.8188

.8568

. 8919

. 9245

.9552

1.0262

1. 0874

1.1400

1.1850

1.2234

1. 2568

1. 2863

1.3125

1.3358

1.3565

1.3750

1.3916

1.4065

1.4199

1. 4319

1.4428

1.4527

1.4618

1. 4701

1.4776

1.4841

1. 4899

1. 5005

1. 5097

1. 5177

1. 5247

1. 5307

1. 5359

1.5403

1.5440

1.5469

1. 54925

1. 54986

1.55103

1.17883

1.17910

1.18164

1.18365

1.18525

1.18612

1.18651

1.18751

1.18827

1.]

1. 1.18960

1.:

1.18991

1.18995

1.18992

1.18983

1.18970

1.19932

1.19960

1. 20224

1.20432

1,20595

1,20682

1. 20722

1.20820

1.20895

L 20950

L 20989

1.21015

1. 21029

1. 21034

1. 21032

1.21022

1.21007

1.!

1. 22063

1. 22092

1. 22364

1.22576

1. 22740

1. 22826

1. 22865

1.22960

1. 23031

1. 23080

1.23113

1. 23132

1.23140

1. 23138

1. 23127

1. 23110

1. 23088

1. 23060

1. 24279

1.24308

1. 24586

1.24798

1. 24961

1.25045

1.25083

1. 25173

1. 25236

1.25279

1. 25304

1. 25315

1. 25314

1. 25303

1. 25284

1. 25258

1. 25226

1. 25189

1.5

1.26613

1. 26892

1.27104

1.27262

1. 27342

1.27378

1.27460

1.27515

1. 27549

1. 27564

1. 27565

1. 27554

1. 27533

1.27503

1.27466

1.27424

1.27377

1. 1. 29009

1. 29287

1.29494

1.29645

1. 29720

1.29753

1. 29825

1. 1.29892

1. 29896

1. 29884

1.29861

1. 29828

1. 29786

1.29738

1. 29684

1. 29626

.31470

.31500

.31774

.. 31973

.32114

.. 32182

.32210

.32270

.32302

.32310

1.32301

1.32276

1.32238

1.32191

1.32136

1.32074

1.32008

1.31936

1.34060

1.34089

1.34356

1.34544

1.34672

1.34731

1.34754

1.34799

1.34815

1.34807

1.34782

1. 34741

1.34688

1.34625

1.34555

1.34478

1.34397

1.34311

1.36752

1.36781

1.37036

1.37211

1. 37323

1.37370

1.37387

1.37414

1.37411

1.37386

1.37342

1.37283

1.37212

1. 37

132

1.37

044

1.36

951

1.36

853

1.36

751

1.39552

1.39579

1.39820

1.39976

1.40069

1.40102

1.40112

1. 40118

1.40095

1.40048

1.39983

1. 39904

1.39813

1.39714

1.39607

1.39495

1.39379

1.39260

1.42463

1.42489

1.42710

1.42844

1.42914

1.42932

1. 42934

1.42915

1.42868

1.42797

.42709

.42607

. 42494

.42373

.. 42245

.. 42113

.41977

.. 41839

1.45490

1.45514

1.45711

1.45820

.45862

.. 45862

.45854

. 45809

,45734

. 45636

1.45522

1.45394

1.45257

1,45112

1.44961

1.44807

1. 44650

1.44490

1.48638

1.48659

1. .48917

.48897

.48878

.. 48801

.48568

. 48426

1. 48270

1.48105

1.47934

1.47758

1.47579

1.47398

1.47216

1.51911

1. 51930

1. 52065

1. 52109

1. 52084

1. 52040

1. 52009

1.51898

1. 51758

1.51598

1. 51422

1.51236

1. 51041

1. 50841

1. 50638

1.50432

1. 50226

1.50019

1.55316

1.55331

1. 55428

1.55431

1. 55366

1. 55296

1.55252

1.55101

1.54923

1. 54727

1.54516

1. 54296

1. 54068

1. 53837

1. 53603

1. 53369

1. 53135

1. 52902

Tem

pera

ture

° K

16 18 20 22 24 26 28 30 32 34. _

.36 38 40

.

42 34 36 38 40 42 34 36 38 40 42 42 44 46 48 50 52 54 56

..6

8..

..60 65 70

TA

BL

E

14.

P=

I

0.05

800

.065

35.0

7269

.080

02.0

8736

0946

9.

1020

3.

1093

6

. 11

669

. 12

402

.131

36.

1386

9.1

4602

.153

34

p=

22

0

13.

511

15.

702

17.8

7420

.039

22.1

93

P=

52

0

17.9

5025

.170

32 4

5339

. 76

2

47.0

63

P=

l

0.15

334

.160

67.1

6800

.175

33.

1826

6

. 18

999

. 19

732

.204

64.

2119

7.

2193

0

.237

62.

2559

3

2

0.11

495

.131

71.1

4441

.159

12.1

7383

1885

3.2

0323

. 21

793

.232

62.

2473

2.

2882

1.

2766

9. 2

9138

.306

06

240

13.7

9016

. 23

218

. 655

21.0

66

23.4

70

540

19.3

3227

.091

34 9

6242

.840

50.6

99

2

0.30

606

.320

74.3

3542

.350

10.3

6478

.379

46.3

9414

.408

81.4

2349

.438

17

.474

85.5

1154

Pre

ssur

e {i

n at

mos

pher

es)

at i

nteg

ral

valu

es o

f T

,

3

0.17

083

.193

01.

2151

6

. 23

729

. 25

941

2815

2.3

0362

.325

71

.358

00.3

6988

.391

95.4

1402

.436

08

.458

14

260

13.

994

16.

694

19.3

7522

.046

24.7

02

560

21.1

6229

.466

38 0

0946

. 56

5

3

0.45

814

.480

20.

5022

6.

5243

1.

5463

6

. 56

841

.590

46.

6125

1.6

3456

. 65

660

. 71

171

. 76

681

6

0.37

696

.421

66

. 466

28.

5108

555

538

. 59

988

.644

35

. 68

879

. 73

320

. 77

759

. 82

195

.866

32

. 91

065

280

14.1

4817

.108

20.0

5122

. 98

8

25. 9

05

580

23.5

0632

.405

41 6

64

6

0. 9

1065

.954

98.

9993

01.

0436

11.

0879

2

1.13

221

• L.

1765

0L.

220

79L.

265

08L.

309

35

1.42

002

.530

66

10 0. 6

8364

. 75

880

,833

8290

873

.983

551.

058

28

1.13

295

1.20

754

1. 2

8206

1.35

652

1.43

093

1. 5

0532

300

14.

274

17.4

9620

. 70

823

. 910

27.1

06

600

26.4

5136

.012

46 0

46

10 1. 5

0532

1. 5

7968

1. 6

5400

1. 7

2832

1. 8

0261

1.87

686

1. 9

5110

2.02

533

2.0$

957

2.17

380

2.35

925

2. 5

4462

20 1. 5

8080

1 73

410

1. 8

8695

2.03

952

2.19

178

2.34

377

2.49

552.

6470

2. 7

983

2. 9

494

320

14.3

8217

. 86

621

.350

24.

829

28.3

13

620

30.1

7940

.372

20 2.94

943.

1005

3. 2

515

3.40

243.

553

1

3. 7

038

3.85

444.

005

14.

1557

4.30

62

4. 6

821

5.05

78

40 3 14

483.

4633

3. 7

805

4.09

674.

4114

4. 7

252

5. 0

382

5.35

05

5.66

24

340

14.4

8518

. 23

021

. 990

25.

764

29.

555

640

34.

909

45.

569

40 5. 6

624

5. 9

739

6. 2

851

6. 5

959

6. 9

062

7. 2

162

7. 5

260

7. 8

359

8.14

558.

4548

9. 2

268

9. 9

979

the

abso

lute

tem

pera

ture

, an

d p,

the

den

s

60 4. 7

475

5. 2

407

5. 7

313

6. 2

201

6. 7

069

7.19

047.

673

4

8.15

57

360

14.

594

18.6

0622

.647

26.

726

30.8

34

660

40.

811

51. 6

96

60 8.15

578.

6368

9.11

709.

596

410

.076

10.

554

11.0

3111

. 509

11. 9

8612

.462

13. 6

5114

.837

80-

6.43

97

7.11

637.

787

98.

4557

9.12

129.

7839

10.4

45

380

14.

724

19.0

1923

.356

27.7

50

32.1

85

680

48.0

28

80 10.4

4511

.105

11. 7

6312

.421

13.0

77

13.

732

14.

385

15.0

3915

.692

16.3

43

17. 9

7019

. 59

0

100

7.39

66

8. 2

678

9.13

299.

992

810

. 84

811

. 70

0

12.5

48

400

14.

887

19.4

9324

.152

28. 8

76

33. 6

58

700

56.

703

100

12.

548

13.3

9514

. 24

115

.085

15.

928

16.

768

17.6

0518

.443

19.

280

20.1

14

22.2

0024

.277

120

9. 2

035

10.

273

11.3

3412

.389

13. 4

38

14.4

82

420

15.0

9920

.047

25.0

6130

.148

35. 2

89

120

14.4

8215

. 52

516

. 56

817

.607

18.6

43

19.6

7720

.709

21. 7

3922

. 76

723

. 79

3

26.3

5828

. 912

ity

in A

mag

at u

nits

140

9.94

4111

. 226

12.4

9613

. 75

915

.014

16.2

63

440

15.3

6920

.706

26.1

0831

.577

37.1

05

140

16.2

6317

. 51

118

. 75

720

.000

21. 2

38

22.

527

23.7

0924

.940

26.1

6727

.395

30.4

6133

. 51

4

160

10.5

0712

.007

13.4

9514

. 97

516

.444

17.9

08

460

15. 7

2421

.490

27.3

2033

.208

39.1

38

160

17.9

0819

.369

20.8

2622

. 27

923

.731

25.1

8026

. 62

228

.060

29.4

9730

. 93

6

34. 5

2438

.101

180

10.9

0812

.637

14.3

5316

.053

17.

746

19.4

33

480

16.2

2322

.443

28. 7

2835

.066

41.4

29

180

19.4

3321

.115

22.

792

24.4

6626

.138

27.8

0629

.470

31.1

2632

. 78

034

.435

38. 5

6942

.693

200 13

.133

15.0

8317

.014

18.9

37

20.8

55

500 16

. 94

223

.642

30.4

0537

. 20

8

44.0

24

200 20

.855

22.

766

24.6

7226

. 57

728

.475

30.3

7132

.262

34.1

5036

.031

37.9

15

42. 6

1547

.312

75..

80..

85...

90..

95..

100.

105_.

110.

115.

120.

125..

130..

135_.

140..

145..

150..

155..

160..

165..

170..

180-.

190..

200..

210..

220..

230..

240-.

250..

260--..

270...

0° C—

280.-..

25°

C.

300....

320

340

360

100° C_.

380

400

420.

440.

460.

480.

500-.

520..

540..

560-.

580..

600-.

. 27425

. 29257

.31088

.32920

.34751

.36583

.38414

.40245

. 42077

.43908

.45739

.47571

.49402

.51233

.53064

.54896

.56727

. 58558

.60389

. 62221

.65883

.69545

.73208

. 76870

. 80532

. 841944

. 878565

. 915186

. 951806

. 988428

1.000000

1.02505

1.09155

1.09829

1.17153

1. 24476

1.31800

1.36619

1.39124

1.46447

1. 53771

1. 61094

1.68417

1. 75740

1.83064

1.90387

1. 97710

2.05033

2.12356

2.19679

.54822

.58490

. 62157

. 65824

.69492

.73159

. 76826

.80493

.84159

.87826

.91493

. 95159

.98826

1.02492

1.06159

1.09825

1.13492

1.17158

1. 20824

1. 24491

1.31823

1.39155

1.46487

1. 53819

1.61151

1.68482

1. 75814

1. 83145

1.90476

1.97807

2.00124

2.05138

2.18451

2.19800

2.34461

2.49121

2.63782

2. 73428

2. 78441

2.93101

3.07761

3. 22420

3.37079

3. 51737

3.66396

3.81054

3.95712

4.10370

4. 25027

4.39684

.82190

. 87699

.93207

.98715

1.04222

1.09729

1.15235

1. 20742

1. 26248

1.31754

1.37261

1.42767

1.48272

1. 53778

1.59284

1.64789

1. 70295

1. 75800

1. 81305

1. 86810

1. 97820

2.08830

2.19839

2.30848

2. 41857

2. 52865

2. 63873

2. 74880

2. 85887

2.96894

3.00372

3.0790J

3. 27888

3 29913

3. 51924

3. 73935

3. 95945

4.10427

4.17954

4.39963

4. 61970

4.83978

5.05984

5.27990

5. 49996

5. 72001

5. 94006

6.16010

6.38014

6. 60017

1. 64128

1. 75186

1. 86243

1. 97298

2. 08352

2.19403

2.30454

2.41505

2. 52554

2. 63604

2. 74653

2. 85701

2. 96748

3.07795

3.18842

3. 29888

3.40935

3.51981

3. 63027

3. 74071

3. 96158

4.18245

4.40331

4.62415

4.84499

5.06581

5. 28661

5. 50739

5. 72816

5.94891

6.01867

6.16966

6. 57052

6. 61113

7.05258

7. 49396

7.93532

8. 22572

8.37665

8. 81797

9. 25925

9. 70051

10.14175

10. 5830

11.0242

11. 4653

11.9065

12.3476

12. 7888

13. 2299

2. 72995

2.91516

3 10035

3. 28549

3. 47059

3. 65563

3.84067

4.05570

4. 21066

4.39562

4. 5806

4. 7655

4.9505

5.1354

5.3203

5. 5052

5. 6901

5.8750

6.0599

6. 2447

6. 6143

6. 9839

7.3535

7. 7231

8.0926

8.46213

8. 83154

9. 20099

9.57026

9.93956

10.05625

10.3089

10.9795

11.0474

11.7858

12. 5240

13. 2623

13. 7480

14.0004

14. 7385

15.4764

16. 2144

16. 9522

17. 6900

18.4277

19.1654

19. 9031

20. 6407

21.3782

22.1157

5. 4332

5. 80 83

6.1832

6. 5580

6. 9326

7.3069

7. 6813

8.0555

8.4296

8. 8036

9.1777

9. 5516

9. 9255

10.2993

10. 6731

11.0469

11.4207

11. 7945

12.1681

12. 5416

13. 2885

14. 0354

14. 7822

15. 5286

16. 2752

17. 0215

17. 7676

18. 5134

19. 2591

20.0048

20.2403

20. 7502

22.1038

22. 2409

23.7311

25.2210

26. 7107

27. 6904

28.1996

29. 6885

31.1772

32.6654

34.1535

35. 6413

37.1289

38.6163

40.1036

41. 5905

43.0772

44. 5639

10. 768

11.537

12.305

13.073

13. 840

14. 606

15.372

16.137

16.903

17. 667

18.432

19.196

19. 961

20.725

21. 489

22. 252

23.016

23.780

24. 543

25. 305

26. 831

28. 355

29. 879

31.402

32. 9255

34. 4480

35. 9693

37.4896

39.0093

40. 5291

41.0090

42.0482

44.8054

45.0850

48.1202

51.1536

54.1858

56.1797

57. 2161

60. 2456

63. 2739

66.3011

69.3270

72.3524

75.3766

78.3993

81. 4217

84. 4439

87.4649

90. 4844

16.020

17. 203

18.383

19. 562

20. 740

21. 917

23.092

24. 266

25.440

26. 613

27.785

28.957

30.129

31.300

32. 471

33.641

34. 812

35. 982

37.151

38. 319

40. 655

42. 989

45.322

47. 653

49. 983

52.312

54. 638

56. 963

59.287

61.6092

62.3428

63.9304

68.1434

68.5706

73.2059

77. 8380

82.4672

85. 5106

87.0921

91. 7154

96.3353

100. 952

105. 568

110.181

114. 791

119.400

124.007

128.612

133. 215

137. 817

21. 208

22. 823

24. 435

26.044

27. 652

29. 258

30. 861

32.463

34. 063

35. 662

37. 260

38. 858

40.454

42.050

43. 646

45. 241

46.835

48.428

50.021

•51.612

54.792

57. 967

61.141

64. 31

1

67.481

70.. 647

73. 811

76. 97

0

80.1

2783

.282

0

84. 27

91

86. 43

62

92.1

574

92. 7366

99.0306

105.318

111.599

115. 729

117. 873

124.144

130. 409

136. 669

142. 926

149.178

155.426

161. 672

167. 912

174.151

180.386

186. 617

26.3

47

28.415

30.4

78

32. 53

8

34. 595

36.6

49

38. 70

0

40. 74

8

42. 793

44.8

37

46. 88

0

48. 92

2

50.963

53.002

55.040

57.078

59.114

61.148

63.182

65. 212

69. 270

73.321

77. 368

81. 413

85.453

89.489

93. 521

97. 546

101.568

105. 5872

106.857

109. 603

116.889

117.626

125. 637

133.636

141. 626

146.878

149. 607

157. 580

165. 545

173. 502

181. 453

189.396

197.336

205. 267

213.194

221.116

229.032

236. 943

31.458

33. 998

36. 534

39.064

41. 591

44.112

46.630

49.144

51. 655

54.164

56. 671

59.177

61. 681

64.182

66. 682

69.180

71.677

74.172

76.664

79.152

84.123

89.085

94.041

98. 991

103. 937

108. 876

113.808

118.733

123. 651

128. 5657

130.118

133. 476

142. 379

143.281

153.068

162. 841

172. 598

179. Oil

182.342

192.073

201. 793

211. 502

221. 203

230.892

240. 574

250.245

259.910

269. 566

279. 216

288. 857

36. 557

39. 592

42. 621

45. 642

48. 659

51. 670

54.677

57. 678

60. 676

63.668

66. 659

69. 649

72.635

75. 619

78. 599

81.501

84. 553

87. 528

90.499

93.464

99. 385

105. 292

111. 190

117.082

122.967

128. 845

134. 714

140. 570

146.417

152. 260

154.105

158.096

168. 676

169. 748

181.374

192.978

204. 562

212.176

216.129

227. 676

239.209

250. 726

262. 231

273. 721

285.197

296.664

308.118

319. 565

330. 998

342.421

41. 666

45. 220

48.763

52. 297

55. 825

59.348

62. 866

66.377

69. 882

73.381

76. 876

80.368

83. 856

87.343

90.823

94.300

97. 776

101.248

104. 716

108.181

115.089

121. 977

128. 85

135. 72

142. 59

149. 43

156. 28

163.101

169. 910

176. 716

178.864

183. 510

195.827

197.072

210. 602

224.101

237. 574

246. 425

251.022

264.444

277. 849

291. 232

304. 596

317.943

331. 273

344. 588

357. 888

371.173

384. 446

397. 703

46. 805

50.901

54.983

59.053

63.115

67.174

71. 226

75. 269

79.302

83.328

87.350

91.368

95.380

99.385

103.385

107.383

111. 378

115.369

119.353

123.327

131. 27

139.19

147. 08

154.96

162. 83

170. 70

178. 55

186.37

194.18

201. 979

204.440

209. 766

223.879

225.308

240. 805

256.263

271. 686

281. 816

287.074

302. 436

317. 768

333.074

348.359

363. 619

378.859

394.078

409. 278

424. 459

439. 625

454. 773

51. 998

56. 664

61.307

65. 934

70. 556

75.177

79. 788

84.385

88.969

93.540

98.114

102.676

107. 238

111.783

116.327

120. 865

125.404

129. 930

134.449

138. 971

147. 98

156. 95

165. 90

174.84

183. 76

192. 66

201,55

210. 41

219.26

228.098

230. 885

236. 915

252. 887

254. 502

272! 033

289. 518

306. 954

318. 404

324.349

341. 707

359.032

376.322

393. 582

410. 814

428.023

445. 201

462. 359

479.492

496.604

513. 693

5T

empe

ratu

re

42 .

_4

4..

.46

__

48.

..50

.

52

..54 56 58 66 6

5..

.7

0..

.75 80 8

5..

.90 95 10

0

105.

110

115.

.1

20

....

125.

.13

013

514

0..

145

150.

.15

516

0..

165

170,

.18

019

0..

20

0..

..

g 2

10

..§

220

Q

230

*~*

24

0..

2,

250

j?

260.

...§

270

3 0°

<7F

28

0

TA

BL

E

14.

Pre

ssu

re (

in a

tmo

sph

eres

)

p=

22

0

22.1

9324

.344

„26

^488

.->

28

,62

3.

:

30.

763

32.

897

35.0

2837

.150

39]

27j£f

o{ -

41.3

97

! 46

.694

51.9

9357

. 27

662

. 53

0

67.7

5972

.976

78.1

7783

.378

88.5

7993

. 753

98.9

1210

4.05

109.

20

114.

3411

9.46

124.

57

129.

6813

4. 7

813

9.88

144.

97

150.

0615

5.13

165.

2417

5.30

185.

35

1195

.38

205.

3921

5.39

225.

3623

5.29

245.

22

255.

128

258.

252

265.

008

240

23.4

7025

,866

28.2

573

0.6

41

33.0

24

35.

404

37.

780

40.1

5442

. 52

4

44.

897

50.

830

56.

756

62.6

61

68,

531

74.

374

80.1

9586

. 01

091

. 81

9

97.

618

103.

4010

9.16

114.

90

120.

64

126.

3613

2.08

137.

78

143.

48

149.

1615

4.84

160.

52

166.

1917

1.84

183.

1019

4.31

205.

48

216.

64

227.

8123

8.93

.

250.

0226

1.07

272.

1128

3.11

7

286.

592

294.

102

260

24.

702

27.

352

30.0

003

2.6

4$

35 2

76

37.

914

40.

549

43.1

8045

.809

48.4

44

55.

041

61. 6

1968

.175

74.

691

81.1

7887

. 63

194

. 08

110

0. 5

2

106.

9411

3.35

119.

74

126.

11

132.

46

138.

8014

5.12

151.

44

157.

76

164.

06

170.

3417

6. 6

2

182.

9018

9.15

201.

60

213.

99

226.

36

238.

6925

1.02

263.

3327

5. 5

828

7.80

299.

98

312.

130

315.

965

324.

252

280

25.

905

28.

820

31.7

3234

. 6

4$

^3

7.5

55

40.4

5243

.358

46.2

60;

49.1

5452

. 06

0

59.

342

66.

610

73.

842

81.0

35

88.1

8995

. 32

210

2. 4

310

9. 5

2

116.

60

123,

6613

0. 7

013

7. 7

2

144.

7215

1. 7

015

8. 6

716

5. 6

2

172.

57

179.

49

186.

4119

3. 3

3

200.

2320

7.10

220.

79

234.

4224

8.01

261.

5727

5.11

288.

63

302.

0831

5. 4

9

328.

8834

2.21

9

346.

429

355.

525

at i

nteg

ra

300

27.1

063

0.2

99

33.4

8936

. 67

839

. 86

9

43.0

5746

. 24

249

.423

52.

595

55.

779

63.

766

71.

752

79.

692

87.

596

95.

458

103.

29

111.

0811

8.86

126.

62

134.

3514

2.08

149.

77

157.

4316

5.09

172.

71

180.

34

187.

95

195.

54

203.

1121

0. 6

8

218.

23

225.

73

240.

71

255.

6327

0.49

285.

3230

0.14

314.

9032

9.59

344.

26

358.

8837

3.45

6

378.

048

387.

979

I va

lues

of

320

28

.31

3,

31.8

0435

. 29

138

. 78

142

. 26

4

45.

751

49.

230

52.

692

56.1

4959

. 62

4

68.3

5277

.084

85.

769

94.4

09

103.

0011

1.55

120.

0612

8.56

137.

0414

5.48

153.

9016

2.29

170.

67

179.

0118

7.34

195.

64

203.

95

212.

2322

0.48

228.

74

236.

9724

5.12

261.

4427

7.68

293.

86

310.

0232

6.15

342.

22

358.

22

374.

17

390.

0540

5.89

8

410.

892

421.

685

T,

the

abso

lute

tem

pera

ture

, an

d p,

the

den

sity

in

340

29.

555

33.

354

37.1

5940

. 96

644

. 76

9

48.

558

52.3

33

56.0

9759

. 85

463

. 62

6

73.1

1782

. 62

592

.100

101.

51

110.

86

120.

1612

9. 4

313

8.67

147.

8915

7.08

166.

2217

5.35

184.

4419

3.52

202.

57

211.

59

220.

61

229.

60

238.

56

247.

51

256.

4626

5.33

283.

0230

0.63

318.

20

335.

72

353.

1937

0. 6

238

7.98

405.

24

422.

4543

9.62

9

445.

026

456.

713

360

30.

834

34.

962

39

.10

243

. 24

347

.376

51.4

9055

. 58

359

. 65

563

. 72

667

. 82

0

78.1

0388

.425

98.

723

108.

94

119.

07

129.

1613

9. 2

214

9. 2

5;

159.

23

169.

1917

9.11

188.

98

198.

8320

8. 6

721

8. 4

722

8. 2

4

238.

00

247.

72

257.

4326

7.11

276.

8028

6.38

305.

50

324.

55

343.

52

362.

4738

1.36

400.

1741

8. 9

343

7. 6

0

456.

1747

4.69

7

480.

530

493.

138

380

32

.18

536

.654

41.1

4045

. 62

550

.105

54.5

6058

. 99

363

. 39

767

.798

72.

222

83.3

4694

. 51

610

5. 6

611

6. 7

3

127.

70

138.

60

149.

47

160.

31

171.

1218

1.88

192.

59

203.

27

213.

9122

4. 5

323

5.11

245.

65

256.

1626

6. 6

627

7.14

287.

57

298.

0330

8.34

328.

95

349.

4736

9.95

390.

3441

0. 7

143

0.97

451.

1847

1.26

491.

2751

1.20

0

517.

479

531.

041

400

33.6

5838

.480

43.3

19

48.1

67;

53.0

06

57.8

2062

.604

67.

357

72.0

9676

. 86

6

88.8

6410

0. 9

211

2. 9

512

4. 9

1

136.

7614

8. 5

016

0. 2

317

1. 9

2

183.

58

195.

1720

6. 7

221

8. 2

3

229.

70

241.

1425

2. 5

326

3. 8

7

275.

1828

6.49

297.

76

308.

96

320.

1933

1. 2

835

3.45

375.

5139

7. 5

0

419.

4344

1. 3

046

3.08

484.

76

506.

35

527.

82

549.

220

555.

952

570.

500

420

35.

289

40.4

79

45.6

9050

:907

56.1

17

61.3

0266

. 44

871

. 55

976

. 65

581

. 78

7

94.

706

107.

69

120.

65

133.

51

146.

27

158.

94

171.

55

184.

13

196.

66

209.

1422

1.56

233.

94

246.

26

258.

5527

0. 7

728

2.95

295.

1130

7. 2

431

9. 3

433

1.37

343.

3435

5. 2

937

9.07

402.

72

426.

29

449.

7947

3. 2

449

6. 5

951

9.84

542.

96

565.

9658

8. 8

34

596.

044

611.

610

Am

agat

uni

ts—

-Con

tinue

d t

440

37.1

0542

. 67

5

48

.26

653

1.86

4

59.4

49

65.0

0870

. 53

476

.022

81.

500

87

.00

5

100.

9011

4. 8

712

8. 8

014

2. 6

3

156.

33

169.

94

183.

5019

7. 0

0

210.

4422

3.82

237.

1625

0.45

263.

6627

6.82

289.

9330

2.99

316.

0232

9.02

341.

9635

4.82

367.

64

380.

4240

5.87

431.

1745

6.37

481.

52

506.

5853

1. 5

755

6.42

581.

13

605.

7263

0.14

3

637.

839

651.

455

460

39.1

3845

.104

51.0

8757

.064

63.0

27

68.

978

74.

903

80.

787

86.

659

92.

566

107.

49

122.

4813

7. 4

415

2. 2

9

166.

97

181.

55

196.

0821

0. 5

5

224.

9923

9.35

253.

61

267.

83

281.

9829

6. 0

831

0. 0

832

4. 0

4

337.

98

351.

88

265.

70

379.

43

393.

1240

6. 7

243

3.89

460.

9048

7. 8

2

514.

68

541.

4556

8.08

594.

59

620.

97

647.

1667

3. 2

36

681.

437

699.

141

480

41.

429

47.

806

54.1

80

60.5

4866

. 90

8

73.

256

79.

582

85.

883

92.1

6998

. 49

7

114.

51

130.

59

146.

62

162.

52

178.

24

193.

85

209.

39

224.

89

240.

3425

5.68

270.

9428

6.16

301.

28

316.

3333

1. 2

934

6.19

361.

0537

5.87

390.

6340

5. 2

9

419.

8643

4. 3

246

3. 2

549

2.05

520.

76

549.

3857

7.90

606.

27

634.

5266

2.58

690.

467x

8. 2

15

726.

937

745.

765

500

-n

44.

024

50.

829

57.

607

64.3

7171

.132

77.8

8784

.636

91.3

6798

.089

104.

84

122.

0213

9. 2

415

6.40

173.

42

190.

23

206.

92

223.

54

240.

10

256.

58

272.

9528

9. 2

630

5. 4

8

321.

61

337.

66

353.

62

369.

50

385.

3240

1.11

416.

83

432.

47

447.

8046

3.33

494.

07

524.

7255

5. 2

6

585.

72

616.

02

646.

21

676.

2470

6.11

735.

73

765.

191

774.

451

794.

443

25°

C.-_.

300

320

340

360

100° C__.

380

400

420

- -

440

460

480

500

520

540

. .

560

580

600

- -

282.903

284. 714

304.349

323.921

343.439

356. 255

362.904

382.326

401.700

421.037

440.338

459. 599

478.830

498.029

517. 201

536.344

555.457

574. 547

313.986

315.997

337.806

359.539

381. 204

395.424

402.806

424.350

445. 844

467. 287

488.685

510.042

531.355

552.633

573.878

595.084

616. 260

637. 405

346.195

348.415

372.470

396. 435

420.317

435.987

444.119

467.855

491.532

515.143

538. 703

562. 212

585. 675

609.093

632.463

655. 797

679.097

702.354

379. 594

382.026

408.406

434.669

460.839

478.007

486.916

512.912

538.829

564.681

590. 467

616.188

641. 862

667.479

693.046

718.565

744.038

769.467

414.248

416.903

445.677

474.323

502.848

521. 558

531. 268

559.589

587.822

615.978

644.053

672.061

700.003

727. 883

755. 701

783.462

811.176

838.838

450.228

453.112

484.361

515.458

546.415

566.717

577. 252

607.971

638. 586

669.113

699. 549

729.897

760.174

790.387

820. 514

850. 589

880. 600

910. 559

487. 607

490.727

524. 533

558.159

591. 623

613.565

624.944

658.134

691.208

724.167

757.032

789. 797

822.469

855. 064

887. 582

920.023

952.405

984. 709

526.460

529.824

566. 271

602. 505

638.553

662.185

674.438

710.172

745. 769

781.236

816.595

851.840

886.984

922.032

956.997

991.873

1,026. 68

1.061.40

566. 867

570.486

609.653

648. 584

687. 296

712. 665

725.817

764.168

802.359

840.414

878.333

916.128

953.806

991.380

1,028.85

1.066. 23

1,103.52

1,140. 72

608.920

612. 796

654. 778

696.478

737.937

765. 092

779.172

820. 216

861.086

901. 787

942.340

982. 756

1,023.04

1,063. 21

1,103. 25

1,143. 20

1,183.04

1, 222. 79

652. 702

656. 850

701. 727

746. 285

790.571

819. 575

834. 608

878.418

922.036

965.462

1,008. 72

1,051.83

1,094. 79

1,137. 61

1,180.31

1, 222.88

1.265.35

1,307. 71

698.312

702. 738

750. 602

798. Ill

845.302

876.202

892. 214

938. 883

985. 320

1,031. 55

1,077.59

1,123.45

1,169.16

1, 214. 71

1,260.12

1,305.40

1,350.56

1,395. 59

745.850

750. 559

801. 506

852.047

902. 234

935.090

952.108

1,001. 70

1,051. 04

1,100.15

1,149.05

1,197. 75

1, 246. 27

1, 294. 62

1,342.82

1,390.86

1,438. 77

1, 486. 55

795.416

800.425

854. 545

908. 217

961.484

996.343

1,014.40

1,067.01

1,119. 33

1,171.39

1, 223. 21

1, 274.83

1,326. 24

1,377.46

1,428. 51

1,479.40

1, 530.13

1, 580. 71

847.130

852. 440

909. 838

966. 721

1,023.16

1,060.083

1,079. 21

1,134.90

1,190. 28

1, 245.38

1,300.22

1,354.82

1,409.18

1, 463.35

1, 517.32

1.571.12

1, 624. 75

1, 678. 22

4* S

TABLE 15. Values of (dZ/dT)p at integral values of T, the

Temperature

0 K16 _ -18 .

20

222426 -28 - - .30

32 .343638

40

42

3436 _ _3840

42

42 .444648 _ _

50

52 ._

545658 . .

60

657075 . . .

80 _

85 -9095100 .

105 . -110115120 - - -

125 - _130135 :.-_ -_140

145150155160

°K-i8.04X10"4

6.074 76

3 823 142.622.211 90

1.641 431 251.100 982

.881

220

256X10~4

226200180

162

1

0 881X10"4

794719653596

.546502466434403

335281238

.205

.180159

.140

.124

.110

.099

.091

.083

.075069

.064059

.054

.051

.047.044

2

16.1X1O~4

12.19.50

7 646.285.244.433.80

3.282.852.502.201.96

1.76

240

274X10"4

242215193

174

2

1. 76X10"4

1.591.441.301.19

1.091.00

0.931.867.803

.669

.5«L

.473

.411

.360

.317

.279

.247

.220

.198

.182

.166

.151

.139

.128

.118

.108

.102

.095

.088

3

oK-\

24. IX10-*18.214.2

11.49.417.866.655.69

4.914.273.743.302.94

2.64

260

291X10"4

257229205

185

3

2. 64X10-42.382.161.961.78

1.631.511.401.301.20

1.000. 841.713.616

.540

.475

.418

.371

.331

.297

.272

.248

.226

.208

.191

. 177

.163

.153

.142

.133

6

36.2XHH28.4

22 818.815.713.311.3

9.798.517.466.595.87

5.26

280

306X10"4

271242217

196

6

5.26X10"4

4.754.303.913.56

3.263.012.792.592.39

2.001.681.421.23

1.080.949.836.742

.662

.593

.544

.495

.452

.416

.383

.354

.328

.305

.284

.265

10

47.1X1O~4

37 931.226.022.018.8

16.314.212.410.99.73

8.74

300

320X10"4

284254228

207

10

8. 74X10"4

7.897.156.495.92

5.435.004.644.303.97

3.322.792.392.07

1.801.581.391.24

1.100.991

.904

.824

.754

.692

.638

.588

.543

.508

.473

.442

20

°K-i

59. 8X10~4

51.143.737.5

32.328.124. 621.719 4

17.4

320

331X10"4

296266240

218

20

17. 4X10~4

15.714.212.911.8

10.89.999.248.537.84

6.595.564.734.09

3 573.142.772.46

2.201.981.801.64

1.501.381.271.17

1.081.010.94

.88


absolute temperature, and p,

40

100 5X10~4

85 973 6

63.455.148.342.738.1

34.2

340

340X10-4306276251

229

40

34. 2X10-430.928.025.423.2

21.319.718.216.815.5

13.011.09.408.13

7.096.235.504.89

4.383.953.573.25

2 982.742.522.32

2.152.001.861.74

60

124 8X10~4

108 0

93.581.371.263.056.2

50.4

360

349X10-*315287262

240

60

50. 4X10-*45.541.337.634.4

31.629 126.924.923.0

19.316.313.912.1

10.69.308.207.30

6.535.885.324.84

4.444.083.753.46

3.212.982.772.58

the density in

80

141 5X10~4

122.4106.593.482.773.7

66.1

380

358X10-«325297272

250

80

66.1X10"4

59.854.449.645.3

41.538.235.332.730.4

25.421.518.416.0

14.012.310.99.69

8.667.787.046.42

5.895.414.984.60

4.263.953.663.40

Amagat units

100

173.3X10~4

150.1130.8114.8101.690.5

81.3

400

370X10-*336308283

260

100

81. 3X10-*76.667.061.155.9

51.247.143.540.337.5

31.326.522.819.8

17.315.313.512.0

10.79.638.737.97

7.326.736.195.71

5.284.894.534.20

120

176. 8X10-<154.2135.4119.7106.6

95.9

420

384X10-*349320294

270

120

95.9X10-487.079.272.266.0

60.555.751.447.744.5

37.231.527.123.5

20.618.216.114.3

12.811.510.49.50

8.718.007.366.79

6.285. 815.384.99

140

202. 3X10-4176.6155.1137.2122.4

110. 2

440

401X10-4364332304

280

140

110. 2X10-499.990.982.975.8

69.563.959.054.851.3

43.036.531.427.2

23.821.018.616.6

14.913.412.111.0

10.19.248.517.85

7.256.716.215.76

160

°K-i

226.3X10-4197.7173.9154.0137.5

123.8

460

419X10-4380346316

289

160

123. 8X10-4112. 2102.193.285.4

78 372.066.561.857.9

48.741.535.630.8

26.923.721.118.9

17.015.313.812.5

11.510.49.588.84

3.187.577.026.51

180

249.0X10-4217.8191.7170.1152.1

137.1

480

437X10-4396360328

298

180

137.1X10-4124.2113.0103. 294.5

86 879.873.868.764.4

54.346.439.834.4

30.026.523.621.2

19.017.115.414.0

12.811.710.79.84

9.108.437.827.25

200

oK-\

237.1X10-4208.8185. 5166.1

149.8

500

455X10-4412373338

307

200

14918X10-4

135.7123.5113.0103.7

95.387.781.175.670.8

59.951.244.038.0

33.129.226.123.4

21.018.917.015.4

14.112.911.810.8

9.999.268.597.96


TABLE 15. Values of (dZ/dT)p at integral values of T, the absolute

Temperature

° K165 _170

180 - -190 _

200

210 -220230240 . . . _ _ _

250

260270

0°C

280 - --25°C

300 - -320340360

100° C380 . . . _

400

420 - - -- -440 . . - - - -

460 -480 - - - - - -500 - -

520 - - - -540

560 -580 - -600

42444648 _50 . .

52 .-_54 _56 _ - _ _58 .

60 - - -

65 _ _7075

80 -

8590 . - - - -95 _ _ . _ - _ -

100

105no _-_ -..115

120

I

4.1X10-63.83.32.92.6

2.32.11.81.61.4

1.31.200

1.1641.0890. 9169.9014.7517.6306.5315.4759.4496.3814

.3241

.2756

.2344

.1990

.1687

.1424

.1196

.0998

.0824

.0675

220

162X10"4

147134122112

10395.388.382.477.2

65.456.048.141.5

36.131.828.525.5

23.020.718.616.8

2

8. 3X10-67.66.65.85.2

4.64.13.63.22.9

2.62.398

2.3252.1751.8321.8011.5021.2591.0610. 9503

.8977

.7614

. 6468

.5500

.46763970

.3363

.2838

.2383

.1986

.1639

.1335

240

174X10-4158144131121

11110395.489.083.6

70.960.652.044.9

39.034.430.827.6

24.822.320.118.2

3

12.4X10-611.59.98.87.7

6.96.25.44.84.3

4.03.594

3.4843.2592.7442.6982.2491.8861.5901 4231.3441.140

0. 9683.8231.69965939

. 5029

.4243

.3561

.2967

.2447

.1991

260

185X10-4

168153140129

119

11010295.489.9

76.365.155.848.2

41.937.033.029.6

26.523.821.619.5

6

24.8X10-6

22. 919.917.615.8

13.912.410.79.58.6

7.97.167

6.9476.4995.4705.3774.4823.7573.1642 8322.6752.267

1.9251.6351.3891.1780. 9965

8399.7039. 5855.4819.3911

280

196X10-4178162149137

12611610810195.9

81.569.559.651.5

44.839.635.231.5

28.225.322.820.7

10

41.2X10-638.233.129.326.0

23.120.418.016.014.4

13.111. 90

11.5310. 799.0758.9217.4316.2265.2414.6894. 4273.750

3.1812.7002.2911. 9411.640

1.3801.1550. 9583.7867.6361

300

207X10-4188172158145

133123114107102

86.673.963.454.7

47.742.037.333.3

29.8 •26.824.221.9

20

82X10-676665851

4641363229

2623.57

22.8421.3517.9417.6414.6712.2710.329.2198.7007.355

6. 2275. 2744.4643.7713.176

2.6622.2161.8281.4891.193

320

218X10-4198182167153

140129120113107

91.478.367.157.9

50.444.439.435.1

31.428.225.423.0


temperature, and p, the density in Amagat units—Continued

40

162X10-6151131114101

9181716356

5146.21

44.7641.8135.0434.4328.5623.8219.9517.7916.7714.12

11.9010.038.4437.0865.922

4.9194.0503.2952.6362.059

340

229X10-*209192175161

148136126118112

96.182.770.861.0

53.046.641.436.9

33.029.626.624.1

60

240X10-6223194169150

1341191059383

7567.90

65.7461.3451.2650.3641.6334.6128.8825.6924.1820.27

17.0014.2511.929.9268.221

6.7535.4844.3833.4232.585

360

240X10-*220201184169

155143132124117

10186.674.564.1

55.648.843.338.7

34.530.927.825.1

80

316X10-293254222197

175156138122109

9888.63

85.7679.9266.5965,4053.8744.6137.0832.8830.9025. 78

21.5017.9014.8612.2710.05

8.1446.4995.0743.8342.754

380

250X10-*230211193177

162149138129122

10590.678.067.2

58.251.045.240.4

36.032.229.026.2

)?100

390X10361312273242

215192170151134

120108.4

104.897.5380.9979.5165.2453.8044.5239.3436.9130.62

25.3720.9717.2514.0911.39

9.0707.0765.3503.8522.549

400

260X10-*240220202185

169156144134126

10994.381.470.3

60.853.147.142.0

37.533. 530.227.3

120

463X10-6429370323285

253226200178158

141127.0

122.8114.194.4292.6675.7062.1351.1545.0442.1634.75

28.5823.4119.0615.3612.21

9.5107.1925.1903.4561.951

420

270X10"*249229210192

176162149140131

11498.184.773.2

63.255.248.943.6

38.934.831.328.3

140

534X10-6495427372327

290258229203180

161144.7

139.7129.7106.9104.885.2269.5856.9549.9446.6438.15

31.1025.2120.2516.0612.49

9.4396.8254.5742.6270.943

440

280X10"*257236216198

182167155145137

11810287.975.8

65.457.250.645.1

40.336.032.329 1

160

604X10-6559483419367

325289257228202

180161.2

155.6144.1118.2115.993.7476.0961.8753.9950.2940. 78

32.9026.3320.8116.1512.20

8.8335.9533.4781.345

—0. 496

460

289X10-*265243223204

188173160150142

12310691.078.5

67.659.152.346.6

41.637.233.329.9

180

672X10-6622536464405

358319283251222

198176.6

170.3157.5128.5125.9101.281.6265.8757.1653.0842.59

33.9326.7320.7015.6211.32

7.6654.5491. 880

—0. 414—2. 387

480

298X10-*272249228209

193179166156148

12711094.181.0

69.861.053.948.0

42.838.234.230.6

200

737X10-682587507442

391348308274242

215190.7

183.7169.6137.7134.8107.686.1368.9059.4054.9643.56

34.1726.3819.8814.429.81

5.9062.588

—0. 246—2. 673—4. 751

500

307X10-*279254233214

198183171161153

13211397.283.5

72.162.955.549.3

43.939.235.031.3


TABLE 15. Values of (dZ/dT)p at integral values of T, the absolute

Temperature 220 240 260 280 300 320

100° C125130135140 I

145 1150155 .160

165

170180190200 :

210220230240250

260 .2700° C.28025° C300320340360100° C380400

420440460480500

520540 _ . . . . .560580600

15.3 X10-*14.012.811.7

10.810.19.358.65

8.00

739X10-6634547478

423375332294260

230203.6196.0180.5145.6142.5112.989.5670.9260.6755.8843.62

33.5625.2418.32 ,12.527.644

3. 5270.041

-2.925-5.455—7.613

16.5X10-*15.113.812.7

11.710.910.19.32

8.60

792X10-6

678585512

453

401

354

312

275

243215.2206.9190.1152.3148.9117.091.8671.8760.9255.8042.75

32.0723.2715.979.8854. 786

0.496-3.122-6.186- 8 . 789

-11.00

17. 7X10"4

16.214.813.6

12.511.610.89.96

9.16

841X10-6719621544

482

426

375

330

290

255225.4216.4198.3157.6154.0119.892.9571.7060.0754.6640.88

29.6520.4212.816.479

1.201

-3. 220

-6.930

-10.06

-12.70

-14.93

18.8X10-4

17.2

15.8

14.5

13.3

12.3

11.4

10.5

9.67

887X10-6758655575

509

450

396

347

304

266234.1224.5205.1161. 5157.7121.292.7870.3358.0952.4037.96

26.2416.668.7762.260

-3.148

-7.657-11.42-14. 57-17.22-19.43

oK-\

19.9XKH18.216.715.3

14.113.012.011.1

10.3

20.9X10"4

19.217.616.1

14.813.712.611.7

10.7

933X10-6796687604

535473415362316

276241.3231.0210.3164.0159. 9121.391.2867.7054.9048.9733.94

21.8011.923.834

- 2 . 815-8.303

-12.85-16.62-19. 76-22.37-24. 54

980X10-6831718631

559493432376326

284246.8235.9213.9164.8160.5119.888.3663.7550.4444.2828.75

16.266.157

-2.069- 8 . 792

-14.30

-18.84-22. 58-25 . 66-28 . 20-30. 28


temperature,

340

21.9X10"4

20.118.416.9

15.514.313.212.2

11.2

102oX10-«863746656

580511447389336

290250.6239.0215.7164.0159.5116.883.9558.4044.6438.2922.33

9.574- 0 . 687- 8 . 984

-15 . 72- 2 1 . 20

- 2 5 . 67-29 . 32-32. 30-34. 72-36. 69

and p, the density in Amagat units—Continued

360

22.9X10"4

21.019.217.6

16.1. 14. 813.712.7

11.6

1050X10-6

892771678

598527460399344

295252.6240.3215.8161.4156.7112.177.9651.5537.4030.9014.60

1.666- 8 . 670

-16.97- 2 3 . 64-29. 03

- 3 3 . 38-36. 89-39. 72- 4 1 . 99- 4 3 . 79

380

23.9X10"4

21.819.918.2

16.715.314.113.0

11.9

1O8OX1O-6

918792696

613539470406348

297252 6239.7213.9157.0152.0105.670. 3043.1428.6722.03

5. 500

- 7 . 529- 1 7 . 86- 2 6 . 08- 3 2 . 63- 3 7 . 85

- 4 2 . 0 3- 4 5 . 34- 4 7 . 96- 5 0 . 0 3- 5 1 . 63

400

24.8X10"4

22.520.518.8

17.215.814.513.3

12.1

III0XIO-6940810711

625548476410350

297250.5237.0210.0150. 5145.497.2260.8533.0618.3411.61

- 5 . 058

-18.08-28 . 31-36. 37-42. 72-47. 72

- 5 1 . 65-54. 71-57. 08-58 . 89-60. 24

420

25.6X10-4

23.221.119.3

17.716.214.813.5

12.3

113oX10-6958825722

634554480412350

294246.1232.0203.8142.0136.686.8549.5221.216.314

- 0 . 457-17.16

-30. 08-40.11-47. 92-53 . 99-58 . 68

-62. 30-65 . 04-67.10-68 . 60-69. 65

440

26.3X10-4

23.821.719.8

18.116.615.113.7

12.5

1140X10-8971837730

640558482412348

290239.5224.7195.3131.1125.674.3436.197.491

- 7 . 497-14. 28-30. 88

-43 . 59- 5 3 . 33-60.79-66. 49-70. 80

-74. 03-76. 39-75 . 08-79. 22-79 . 92

460

27.0X10"4

24.422.120.2

18.516.915.313.9

12.6

1140X10-6980846736

644560482410343

283230.2214.9184.3117.9112.259.5720.73

- 8 . 216- 2 3 . 21-29.95-46. 34

-58 . 70-68.04-75 . 06-80. 30-84.14

-86. 91-88 . 82-90.06-90. 78- 9 1 . 08

480

27. 6XIO-4

24.922.520.5

18.817.115.514.0

12.6

1140X10-6986852741

646561481407337

273218.3202. 4170. 7102.196.2542.393.011

-26.03-40.93-47. 58- 6 3 . 61

-75 . 52-84. 33-90.81-95.48-98 . 76

-101.0-102. 4-103.1-103.3-103.2

500

28.1X10"4

25.322.920.8

19.017.315.614.0

12.6

1140X10-6

988 •854744

647560479402329

262203.5187.0154.283.5977.5822. 64

-17.11-46.08-60.78-67. 30-82.82

-94.14-102.3-108.1-112.1-114.7

-116. 3-117.1-117.3-116.9-116.2


TABLE 16. Values of (-d2Z/dT*)p at integral values of T,

Temperature

16 .1820

2224.262830.

3234.363840

42

3436384042

42__44..46..48..50..

52. .54_.56_.58__60__

65. .70..75. .80..

85 . .90..95_.100-

105.110.115.120.

125.130.135.140.

145.150.155.160.

165.170.

11.4X10-57.95.5

4.03.02.31.81.4

1.20.96.79.66.55

.47

220

167X10-51381149682

0.47X10-5.40.35.31.27

.23

.20

.18

.16

.15

.12

.10

.078

.063

.051

.042

.036

.030

.025

.020

.017

.015

.013

.012

.011

.0095

.0082

.0071

.0063

.0058

.0054

.0050

22.6X10-515.711.0

7.95.94.63.62.9

2.42.01.71.41.1

0.92

240

178X10-514612110287

0. 92X10-8.79.69.61.53

.46

.40

.35

.32

.30

.24

.19

.15

.12

.10

.084

.070

.059

.049

.041

.035

.030

.026

.023

.021

.019

.016

.014

.013

.012

.012

.011

33. 7X10-523.516.5

11.78.86.85.44.3

3.52.92.42.01.7

1.4

260

186X10-515412810892

1.4X10-51.21.00.91.80

.60

.53

.48

.45

.37

.30

.23

.18

.15

.13

.11

.091

.075

.062

.052

.045

.039

.035

.032

.028

.024

.021

.019

.018

.017

.016

47.0X10-533.0

23.517.413.510.78.6

7.05.84.84.03.3

2.8

280

190X10-516013511497

2. 8X10-52.42.11.91.6

1.41.21.10.96

.90

.74

.60

.48

.31

.25

.21

.18

.15

.12

.10

.071

.062

.055

.048

.043

.039

.036

.033

.031

10

54.0X10-5

38.629.222.717.814.3

11.79.78.06.65.5

4.6

300

190X10-5.162138118101

4. 6X10-54.03.53.12.7

2.32.01.81.71.5

1.20.96

.76

.61

.49

.41

.35

.29

.24

.20

.17

.15

.13

.12

.11

.081

.070

.062

.057

.054

.050

48. 5X10-«41.034.328.4

23.319.015.512.810.7

9.2

320

188X10-5161138119103

9. 2X10-58.06.96.05.2

4.54.03.63.33.0

2.41.91.51.2

0.97.80.67.57

.48

.40

.34

.30

.26

.23

.21

.19

.16

.14

.13

.12

.12

.11


the absolute temperature, and p, the density in Amagat units

40

79. 8X10~5

67.155.8

45.937.530.725.421.2

18.0

340

183X10-5158137119104

40

18.0X10-515.513.511.810.3

9.08.07.26.55.8

4.63. 62.92.3

1.91.61.31.1

0.94.80.68.58

.51

.46

.41

.37

.33

.29

.26

.25

.24

.22

60

99. 0X10-581.9

67.255.145.437.631.3

26.4

360

178X10-5154134118105

60

26. 4X10-522.719.717.215.2

13.411.910.69.58.6

6.75.34.23.4

2.82.42.01.7

1.41.21.00.87

.77

.69

.61

.54

.48

.44

.40

.37

.35

.32

80

106.2X10-5

87.271.959.649.441.0

34.3

380

177X10-5152132117106

80

34. 3X10-529.325.422.419.9

17.615.613.912.411.3

8.86.95.54.5

3.73.12.62.2

1.91.61.31.1

1.00.90

.80

.72

.65

.60

.55

.51

.47

.43

100

128. 0X10-5

105.787.872.960.650.3

42.0

400

181X10-5154133118108

100

oK-242. 0X10-535.630.827.324.3

21.619.217.115.313.8

10.88.56.75.4

4.53.83.32.8

2.42.01.61.4

1.21.11.00.91

.82

.75

.69

.63

.58

.53

120

123.0X10-5102.885.671.159.1

49.4

420

189X10-5160138122112

120

°K-249. 4X10-541.836.232.028.5

25.422.720.318.216.3

12.710.08.06.5

5.34.53.93,3

2.82 42.01.7

1.51.31.21.1

0.97.89.82.75

.70

.64

1

140

139. 6X10-5116.997.581.167.5

56.5

440

199X10-5170147130118

140

56. 5X10-547.941. 536.632.5

29.025.923.120.718.5

14.511.59.37.6

6.15.24.43.7

3.22.82.42.0

1.71.51.31.2

1.11.00.93

.87

.81.74

160

155. 6 X 10-5130.1108.590.475.3

63.1

460

210X10-5182159141127

160

63.1X10-553.846.941.236.4

32.428.825.622.820.4

16.113.010.68.6

7.05.84.84.1

3.63.22.82.4

2.11.81.51.4

1.31.21.11.0

0.91.84

180

oR-2

171.0 X 10-5142.9118.998.882.4

69.6

480

220X10-5193171153137

180.

°K~2

69. 6X10-559.852.145.740.2

35.631.628.024.822.0

17.514.411.89.7

7.96.45.34.6

4.13, 63.12.6

2.32.01.81.6

1.41.31.2*1.1

1.00.93

200

°K-*

155. 2 X 10-5128.0106.689.3

75.9

500

229X10-*204183165148

200

°K~2

75.9X10-565.656.949.743.7

38.734.430.526.923.8

19.015.713.010.7

8.87.25.95.0

4.43.93 42.9

2.52.22.01.8

1.61.41.31.2

1.1

1.0


TABLE 16. Values of ( — d2Z/dT2)p at integral values of T, the

Temperature

°K170180190200210220230240250260.270

0° C.280 _...

25° C.300320340360 .

100° C.380 -

400420440460480 :

500520540560 .580600 .

42..44..46-48__50..

52..54..56..58..60-

65..70-.75..80..

85..90..95..100.

105.110.115.120.

125.130.135140.

P = l

0. 50X10-7.44.38.33.29.26.23.20.17.14.1185.1140.10501.08510.08336. 06706.05459.04490.03969.03727

. 03119

.02629

.02231

. 01905

. 01635

. 01410, 01221. 01062. 00928.00812.10714

p=220

82X10-571615347

4237332926

21171412

9.77.86.45.5

4.84.33.73.2

2.82.52.21.9

1.1X10-70.88.76.66.58.51.45.40.34.28.2370.2280.2100.1701.1666.1341.1091

, .08970. 07932.07448

. 06233

. 05254

.04458

. 03806

. 03266

.02817

.02440

.02122

.01853

. 01623

.01426

240

87X10-575655750

4439353128

23191613

108.36.96.0

5.34.64.03.5

3.12.72.42.1

1. 6X10-71.31.11.00.87.78

.52

.43

.3554

.3419

.3148

.2551

.2498

.2010

.1636

.1345

.1189

.1116

.09342

.07875

.06682

.05703

.04894

. 04221

. 03660

.03180

. 02776

. 02431

.02136

260

92X10-579696154

4741363230

25211714

118.97.56.5

5.75.04.33.8

3.32.92.62.3

3.1X10-72.62.22.0

.42.00.86.7101.6830.6289.5094.4990.4013.3266.2685.2373.2228

.1864

.1571

.1333

.1137

.08414

.07286

. 06336

.05530

.04842

.04254

280

97X10-584736558

5144383431

26221815

129.58.17.0

6.15.34.64.0

3.53.12.72.4

10

5.0X10-74.33.73.22.72.42.22.01.71.41.1821.1371.0470.8476.8302.6674.5430.4463.3945.3702

.3097

.2609

.2213

.1888

. 1620

.1396

.1209

.1051

. 09169

. 08027

.07049

300

101X10-587766861

5446403632

27231916

13108.67.5

6.55.64.94.3

3.73.22.92.6

20

11X10-79.3

' 7.56.25.44.74.23.73.22.82.3562.2712.0861.6911.6631.3281.079

.7837

.7350

. 6143

.5173

. 4385

.3739

.3205

. 2761

.2389

.2076

.1810

. 1583

.1390

320

103X10-590807264

5749423734

28242016

13119.18.0

6.96.05.24.5

3.83.33.02.7


absolute temperature^ and p, the density in Amagat units—Continued

40

°K-2

22X10"7

191512109.68.57.56.55.54.6844. 5234.1423.3583.2762.6282.1321.7491.5461.448

1.2081.0160.8602.7325.6271

.5395

.4662

.4045

. 3522

.3077

.2696

340

104X10-5

93847668

6153453936

29252117

14119.68.4

7.36.45.54.7

4.03.53.22.9

60

oK-2

32X10-7

272218151413119.18.06.9866.75bb.1715.0014.8723.9003.1602.5872.2862.138

1.7061.4961.2651.0750.9193

.7897

.6814

.5903

.5132

. 4476

.3916

360

105X10"3

95878073

6557484036

30262218

1512108.8

7.76.75.84.9

4.23.73.43.1

80

oR-2

43X10-73529242018171513109.2648.9708.1756.6206.4405.1454.1603.4003.0042.806

2.2021.9571.6511.4021.196

1.0260.8839.7645.6635.5777. 5045

380

106X10-598918477

69615142 .37

31272319

1613109.1

8.07.06.05.2

4.44.03. 63.2

100

53X10-'44362925222018161311.5211.1710.158.2167.9816.3625.1354.1893.6973.450

2.8652.3972.0191.7111.458

1.2481.0730.9265

.8026

.6975

.6078

400

108X10-5101958881

7364544539

32282420

1714119.5

8.47.36.35.5

4.84.23.73.3

120

64X10-753433529262320181613.7613.3512.119.7869.4967.5536.0824.9514.3664.069

3.3732.8162.3682.0031.703

1.4551.2491.0760.9299

.8063

.7011

420

112X10-5104989285

7666564741

33292522

1814119.8

8.77.66.65.8

5.14.43.93.4

140

oK-2

74X10-761504134292623201815.8215. 5114.0511.3310.988.716V. 002

' 5.6875.0084.664

3.8573.2142.6962.2761.931

1.6461.4101.2121.0450. 9038.7838

440

118X10-51091029587

7767564742

34302623

19151210

9.18.06.96.0

5.34.64.03.6

160

°K-*84X10-770584739332926232018.19

17.6615.9612.8512.459.8517.8936.3955.6245.232

4.3173.5863.0042.5292.141

1.8211.5551.3331.1470. 9892.8554

460

127X10-51151069787

7767574844

36312723

19151211

9.4 ,8.37.36.3

5.54.84.23.7

180

0^-2

93X10-7

79655344373229262320.3819. 7917.8614.3513.8910.968.7567.0756.2115.773

4.7513.9393.2892.7622.332

1.9781.6851.4401.2351.0620. 9152

480

137X10-51231119987

7767574945

38332924

20161311

108.67.66.6

5.85.04.43.8

200

100x10-787725948413632292622. 5621.9119. 7415.8115.3012.039.5887.7256.7686.285

5.1584.2653.5502.9732.503

2.1161.7971.5311.3091.1210. 9626

500

148X10-513211610188

7665564946

39343025

20161311

108.97.96.9

6.05.24.53.9


TABLE 16. Values of (—d2Z/dT2)p at integral values of T, the

Temperature

°K145150155160165

170180190200

210220230240

250260270—

0° C_280

25° C_300

320340360

100° C.380400

420440460. . . . .480500

520540560580600

p=220

1.7X10-5

1.51.41.31.3

110X10-7967863

52454036

322924.73

24.0121.60

17.2516.68

13.0810.398.343

7.2936.7675. 536

4.5643.7873.1612.6532.236

1.8921.6061.3671.1660.997

240

°K-21.8X10"5

1.61.51.41.4

120X10-71008367

56

4440

353126.

26.23.

18.18.

14.11.8.

89

0944

6604

0916929

•7.7857.5.

4.

218885

8343.9983.325

2.780

2.334

1.1.1.1.1.

9676o3409196018

°K-22.0X10-5

1.81.61.51.5

130X10-71108770

5953

383329.03

28.15

25.

20.19.

15.11.

.26

0437

0889

9.480

8.7.6.

5.4.3.2.2.

242635202

076181464884411

2.022

1.1.1.1.

701434211023

280

2.1X10-51.91.81.71.6

140X10-71209173

63565146

403531.17

30.2027.07

21.3820.67

16.0212.589.994

8.6628.0176.485

5.2854.3353.5752.9622.464

2.0561.7201.4411.2081.014

300

2.3X10-8

2.11.9

1.81.6

150X10-71209678

66595348

433733.30

32.2328.85

22.6921.94

lb.9313.2410.47

9.0438.3606.733

5.461

4.458

3.658

3.014

2. 493

2.067

1.718

1.429

1.188

320

2.4X10-'2.22.01.91.8

160X10-71309981

69625651

464035.42

34.2430.62

23.9623.17

17.8013.9110.90

9.3838.b646.942

5.6014.5473.7093.0382.496

2.0551.6931.3961.1490.944

TABLE 17. Values of (dZ/dp)T at integral values of T, the

Temperature

°K161820

2224262830

3234363840

42

P=0

—9,105X10-6

— 7 , 709

-6,633

-5 , 781

-5,087- 4 , 512

-4,027

-3,615

- 3 , 262

-2,955

-2,688

-2,453

-2,245

-2,059

1

—9,087X10-6

-7,694

-6,621

-5,770-5,077

- 4 , 503

-4,019

-3,608

-3,255

-2,949

-2,682

-2,448

- 2 , 240

-2,054

2

—9,070X10-6

—7,679

-6,608

- 5 , 759

-5,067-4,494

-4,011

-3,601

-3 , 249

-2,943

-2,676

- 2 , 443

- 2 , 235

-2,050

3

—9,052X10-6—7,664

—6, 595

5, 748

-5,058-4,485

-4,003

-3 , 594

-3 , 242

-2,937

-2,671

-2,438

- 2 , 230

-2,045

6

—7,620X10-6

-6 , 557

-5 , 714

-5,029

-4,460

-3,980- 3 , 572

-3,223

-2,919-2,654

-2,422

- 2 , 216

-2,032

10

-6,506X10-6

-5,670-4,990

-4,425

-3,949

-3 , 544

-3,197

-2,895

-2,632

-2,402

-2,197

-2,014

20

-4,892X10-6

-4,338

-3,871

-3,474

-3,132

-2,836

- 2 , 577

-2,350-2,149

-1,968


absolute temperature, and p, the density in Amagat units—Continued

. 340

°K-*2.6X10"8

2.32.12.02.0

170X10-714010083

72656055

494337.53

36.2332.36

25.2024.37

18.6314.4211. 29

9.6788.9257.110

5.704

4.6013.7283.0322.471

2.0161.6461.3421.0900.883

360

2.8X10"5

2.52.22.12.1

180X10-714010085

75686358

524639.63

38.1934.08

26.3925.53

19.4114.9411.63

9.9269.1397.236

5.7654.6183.7132.9932.416

1.9511.5731.2641.0100.801

380

°X-2

2.9X10-5

£62.32.22.2

180X10-714011088

78716560

544841.71

40.1335.77

27.5426.64

20.1415.4111.92

10.129.3057.314

5.7844.5943.6602.9762.331

1. 8571.4741.1620.908.700

400

oR-2

3.0X10-62.72.52.42.3

190X10-715011092

81736762

565043.78

42.0537.44

28.6327.71

20.8215.8312.16

10.279.4177.344

5.7564.5273.5672.8122.212

1.7321.3461.0350.782.576

420

3.1X10-52.92.72.52.4

190X10-715012096

83756964

585245.83

43.9339.07

29.6828.74

21.4416.1812.33

10.369.4737.319

5.6784.4153.4332.6642.057

1. 5741.1890.880.630.428

440

oR-2

3.3X10-5

3.02.82.62.5

200X10-715012098

85777166

605447.86

45.7840.66

30.6729.70

21.9916.4712.44

10.399.4687.238

5.5474.2533.2532.4751.864

1.3821.0000.695.452.257

460

3.4X10-53.23.02.72.5

200X10-715012099

87807468

625649.87

47.6042.22

31.6030.61

22.4816.6912.49

10.369.3977.094

5.3594.0392.0382. 2421.631

1.1530.776.480.245.059

480

oR-2

3.4X10-53.23.12.92.6

200X10-7150120100

88817569

645851.85

49.3843.73

32.4731.^5

22.9016.8312.46

10.259.2566.885

5.1103.7692.7461.961

1.354

0.884.517.231.007

- .167

500

3.5X10-«3.33.23.02.7

210X10"7

160120100

89817570

656053.80

51.1245.19

33.2732.22

23.2316.8912.34

10.089.0396.605

4.7953.4382.4111.630

1.032

0.573.220

- .052- . 262- .422

absolute temperature, and p, the density in Amagat units

40

-4,165X10"6

-3,715- 3 , 332

-3,003-2,717-2,467-2,248-2,053

-1,878

60

-3,559X10-6

-3,191

- 2 , 874- 2 , 598-2,357-2.145-1,957

-1.788

80

-3,049X10-6

-2,745- 2 , 479-2,247-2,043-1,861

-1.698

100

-2,908X10-6

- 2 , 615-2,360-2,137-1.940-1,765

-1.608

120

-2,486X10-6-2,242-2,027-1,838-i ,669

-1.517

140

-2,357X10-6-2,123-1,917-1,735-1,573

-1,427

160

- 2 , 228X10-6-2,004-1,807-1,633-1,477

-1,337

180

-2,098X10-6-1,885-1,697-1,530-1,381

-1,245

200

-l,768X10-6

-1,588-1,428-1,283

-1,151


TABLE 17. Values of {dZ/dp,)T at integral values of T, the absolute

Temperature P=220 240 260 280 300 320 340

34 _36 _

40.

42.

34 _

38..40_

42_

42 .44 .46-48.50.

52..54.56..58..60..

65 . .70.75-80-

8 5 _ .9 0 . -95_ _100..

105.110_115.120.

125_130.135_140_

145_150_155_160.

165_.170_.180_.190 _200_.

210_220_230 _240 _250_

260_270.

°K-1,650X10-6- 1 , 4 7 7- 1 , 3 2 2- 1 , 1 8 1

- 1 , 0 5 4

p=520

387X10-8538681812

940

P = 0

-2,059X10-6-1,892- 1 , 740-1,603-1,478

-1,364-1,259-1,162-1,072

- 8 0 7- 6 5 4- 5 2 4- 4 1 2

- 3 1 5- 2 3 1- 1 5 7

- 9 0

- 3 4

+1967

111

151187221252

281307331353

373393429460489

512534553571587

602613.9

- 1 , 526X10-6-1,360-1,210-1,077

- 9 5 6

-1,392X10-6-1,239-1,100

- 9 7 3

- 8 5 7

-1,254X10-6-1,117

- 9 8 9- 8 7 0

- 7 5 8

-1,121X10-6-1 ,001

- 8 8 2- 7 6 8

—658

-998X10-6- 8 9 2- 7 8 2- 6 7 0

- 5 5 8

540 560 580 600 620

654X10-6795961

1,135

941X10-61,0771,260

1,255X10-61,4001,587

1,627X10-61,750

2,090X10~«2,120

-2,054X10-6-1 ,887- 1 , 736- 1 , 599-1,474

-1,360- 1 , 256-1,159-1,069

- 9 8 5

- 8 0 4- 6 5 1- 5 2 2- 4 1 0

- 3 1 3- 2 2 9- 1 5 5

- 3 2+21

69113

153189223254

309333355

375395431462490

514536555573589

604

615. 3

2,050X10-6

,883

,732

, 595

,470

,357

,252

-1,156

-1,066

-982

-801

-649

-519

-407

-310

-227

-153

-86

-30

+23

71

115

155

191

225

256

285

311

335

357

377

397

433

463

491

516

538

557

574

605

616.8

-2,045X10-6

-1,879

-1,728

-1, 591

-1,467

-1,353

-1,249

-1,152

-1,063

-979

-799

-646

-516

-405

-224

-150

-84

-28

+25

73

117

157

193

227

258

312

378

398

434

465

493

517

539

558

576

592

618.2

10

-2,032X10-6

-1,866

-1,716

-1,580

-1,456

-1,342

-1,239

-1,143

-1,053

-970

-790

-638

-509

-301

-218

-144

-77

-21

+32

79

123

139

199

232

263

292

318

342

404

439

470

522

544

580

596

610

622.5

-2,014X10-6

-1,849

-1,700

-1, 564

-1,441

-1,328

-1,225

-1,129

-1,040

-779

-628

-499

-292

-209

-135

-12

+40

87

131

171

206

240

271

297

325

349

371

391

411

446

477

504

529

550

569

586

602

616628.3

-884X10-6

-788

-684

-573

-461

640

2,638X10-6

2, 504

20

-1,968X10-6

-1,806

-1,659

-1,526

-1,404

-1,293

-1,191

-1,096

-1,009

-751

-601

-474

-364

-186

-113

+961

108

151

190

226

259

289

317

343

367

389

409

429

464

494

521

545

566

585

602

617

631642.8


temperature, <

360

-774X10-6

-680

-577

-467

-356

660

3, 227X10-6

40

-1,878X10-6

-1,721

-1.578

-1,448

-1.330

— 1,222-1,123

-1,031

-946

-867

-694

-548

-424

-317

-223

-141

-69-5

+52103149191

230265297326

354380404426

446465499529555

578599618634648

661672.6

%nd p, the density in Amagat units—Continued

380

-667X10-6-564-456-347

-241

680

3,848X10-6

60

-1,788X10-6-1,636-1,497-1,371-1,256

-1.151-1,054-965-883-807

-637-495-374-269

-177-96-25+38

95145191232

270305337366

394419442464

484503536565590

613633651666680

692703. 4

400

-562X10-6-442-324-213

-111

80

-1,698X10-6-1,550-1,416-1,294-1,182

-1,080-986-899-820-746

-580-441-322-219

-129-50+2082

138188233274

311346379408

435459482503

523541574602626

648667684699713

725735.2

420

-460X10-6-318-188-73

+29

100

-1,608X10-6-1,465-1,335-1,216-1,108

-1,009-918-834-757-685

-523-386-269-168

-80-2+67128

183233277318

355389421450

476500523544

563581613640663

684703719734747

758768.1

440

-357X10-6-194-51+69

173

120

-1,517X10-6-1,379-1,254-1,139-1,034

-938-849-768-693-623

-465-330-215-115

-29+48115176

231280324364

401434465493

519543565585

604622653679701

721739755770782

792802.0

460

-234X10-6-62+89218

321

140

-1,427X10" •-1,294-1,173-1,062-961

-868-781-702-629-561

-407-273-159-60

+25100167227

281330373413

449481511538

563586608628

646664694719741

760777792806818

828837.2

480

-68X10-6

+99247374

475

160

-1,337X10-6-1,209-1,092-985-887

-797-713-636-565-497

-346-214-100-2

+82156222281

335383426465

500531560586

610632653672

690707736761782

800816831844855

865873.6

500

+148X10-6302437550

639

180

-1,245X10-6-1,122-1,011-908-813

-726-645-570-499-431

-280-150-37+59

142215279338

392439481518

552583611637

660681701719

736752780804824

842857871884894

903911.4

200

-1,151X10-6-1,033-926-827-736

-652-573-499-428-361

-210-81+30124

204275339397

449495536573

606636663688

711731750768

785800826849868

885900913925935

943950.4


TABLE 17. Values of (dZ/dp)T at integral values of T, the

Temperature P = 0 10 20

0° C.280

25° C_300

320340360

100° C.380_400

420440460480500

520540560580600

617.7X10-6625.4

643.5645.2

661.7675.5687.1

693.7696.9705.2

712.3718.3723.4727.7731.4

734.5737.1739.3741. 2742. 7

619.1X10-6626.8

644.9646.6

663.0676.8688.4

695.0698.1706.4

713.4719.4724.5728.8732.5

735.6738.2740.3742.2743.6

620. 5X10-6628.2

646.3647.9

664.4678.1

696.2699.4707.6

714.6720.5725.6729.9733.5

736.6739.2741.4743.1744.6

621. 9X10-6629.6

647.6649.3

665.7679.4

697.4700.6708.8

715.8721.7726.7731.0734.6

737.6740.2742.4744.1745.6

626.2X10-6633.8

651.8653.4

19.713. 2

701.1704. 2712.4

719.3725.1730.1734.3

737.8

740.8743.3745.4747.1748.6

631.9X10-6639.5

657.3658.9

675.0688.4699.7

706.1709.2717.2

724.0729.7734.6738.7742.1

745.0747.5749.5751.2752.5

646.4X10-6

653.8

671.2672.8

688.5701.6712.5

718.7721.6729.3

735.8741.3745.9749.8753.0

755.8758.0759.8761.3762.5

42_44.46-.48-.50-.

52-.54_.56-.58-.60..

65-.70..75..

85..90. _95..100-

105-110_115.120-

125-130-135.140-

145.150.155..160..

165..170.

Temperature P=220 240 280

-1,054X10-6

-941

-837

-742

-655

-573

-496

-423

-353

-287

-137

-9

+99

191

270

340

402

457

507

552

592

660

716

740

762783

819

835

-956X10-6

-847

-747

-654

-489

-413

-342

-274

-211

-62

+64169

260

406

467

520

568

611

650

685

716

744

770

794

816

836

854

871

-857X10-6

-751

-653

-562

-477

-398

-324

-255

-190

-131

+15

137

241

331

475

534

586

632673711745

775

802

827

850

871

891

908

924

951

-758XHH

-653

-555

-464

-380

-301

-229

-163

-102

-46

+94213

316

403

480

546

604

655

739

775

807

836

930

948

965

993

1,005

300

-658X10-6

-553

-454

-362

-277

-200

-131

-68

-10

+44

177

293

393

480

555

620

676

725

768807841

872

900

925

948

970

990

1,008

1,024

1,038

1,051

1,062

320

-558X10-6

-452

-351

-258

-174

-31+30+86138

264

376

475

560

633

696

751

798

840

878

911

941

992

1,014

1,034

1,053

1,070

1,085

1,099

1,1111,121


absolute temperature, and p, the density in Amagat units—Continued

40

676.1X10"6

683.2

699.8701.3

716.2728.4738.6

744.4747.1754.2

760.1765.1769.2772. 6775.4

777.6779.5780.9782.1782.9

340

-461X10-6

-352-250-157

- 7 3

+2+68128183234

355464561644

715776829875

916953985

1,014

1,0411,0641,0851,103

1,1201,1361,1501,162

1,1731,183

60

706. 7X10-6713. 5

729.2730.7

744.7756.2765. 6

770.9773.4779.9

785.2789.5793.1796.0798.3

800.2801. 6802.6803.4803.9

360

-356X10-6-249-149

- 5 7+27

102168228282332

450556651733

802860911956

9961,0321,0641,092

1,1171,1401,1591,176

1,1911, 2051,2181,229

1,2401,248

80

738.4X10-6

744.8

759.7761.0

774.2784.8793.5

798.4800.6806.4

811.0814.8817.8820.2822.0

823.4824.4825.0825.3825.4

380

-241X10-6-139

- 4 2+48

130

204270329382431

547652745824

892948997

1,041

1,0801,1151,1461,173

1,1971,2181,2361,252

1,2661,2781,2901,300

1,3091,317

100

771.0X10-6

777.1

791.2792.4

804.7814.5822.3

826.6828.6833.7

837.7840.8843.3845.1846.4

847.3847.8848.0847.9847.6

400

-111X10-6- 1 5+76

162240

310373431484532

647751842919

9841, 0391,0861,129

1,1671,2011,2311,257

1,2801,2991,3161,331

1.3451,3561,3661,374

1,3811,388

120

804. 9X10-6

810.6

823.7824.9

836.2845.1852.1

855.9857.6861.9

865.2867.7869.5870.8871.6

871.9871.9871.7871.2870.5

420

+29X10-6+120

203281353

418478534587635

752855944

1,018

1,0811,1341,1801,221

1,2581,2911,3201,345

1,3661,3841,3991,413

1,4261,4361,4441,451

1,4571,462

140

839.8X10-6

845.2

857.4858.5

868.8876.7882.9

886.1887. 6891.1

893.7895.5896.6897.3897.5

897.3896.8896.1895.1894.0

440

173X10-6259335404468

528585640692740

861963

1,0501,122

1,1831,2341,2791,319

1,3551,3871,4141,437

1,4571,4731,4871,500

1,5111,5201,5261,531

1,5361,539

160

876.1X10-6881.1

892.3893.2

902.5909.5914.7

917.3918.5921.2

923.0924.1924.6924.6924. 2

923.5922.4921.2919.8918.2

460

321X10-6403471531588

643697750800848

9731.0751,1601,231

1,2911,3401,3841,422

1, 4561,4861,5111,533

1,5521,5671,5801,591

1,6001,6071,6121,616

1,6191, 620

180

913.6X10-6

918.2

928.4929.2

937.4943. 3947.6

949.6950.5952.4

953.4953.7953.5952.8951.8

950.4948.9947.1945.2943.1

480

475X10-6553612664715

765815864913961

1,0891,1911,2741,345

1,4051,4541,4961,531

1,5611,5871,6111,633

1,6521,6671,6781,686

1,6931,6991,7041,707

1,7071,706

200

952.5X10-6956.7

965.8966.5

973.5978.4981.6

983.0983.6984.5

984.7984.2983.3981.9

980.2

978.2976.1973.8971.3968.7

500

639X10-6707759804849

894939984

1,0311,079

1,2101, 3121,3941,465

1, 5261,5761,6161,646

1,6701,6931,7151,736

1,7541, 7681,7791,785

1,7901,7951,7991,801

1,8001,797


421

TABLE 17. Values of (dZ/dp) T at integral values of T, the absolute

Temperature

180 - -190

200

210220230 - -- -240250

260270 - -

0° C

28025° C.

300 - - - - - - --

320 _ _ _ _ _ . , _

340 -360

100° C

380400

420440 __.

460480500

520540 - -560580600 - --

p=220

874X10-6

895913

929943956967

976

984991.0992.9996.6

1,004. 51, 005.2

1,011.01,014. 71,016.91,017.61,017.81,017. 8

1,017.11,015.81,014.01,011.91,009. 5

1,006.91,004. 21,001.2

998.2995.1

240

922X10~6

943961

976989

1,0011,0111,020

1,0271,033.11,034. 71, 038.01,044. 71,045. 3

1,049.81,052.31,053.41,053. 41.053. 2

L, 052.2

L, 050. 6L, 048.4L, 045.8, 042.9. 039.8

1,036. 51,033.11,029.61.026.0] , 022. 3

260

972X10"6

9921,009

1,0241,0371,0481.0571,065

1,0721,076.81,078. 21, 081.01,086. 41, 086.8

1,090.11,091. 31,091. 21.090.41, 089.91,087.8

1,085. 21,082.11,078. 61, 074. 91,071. 0

1,067.01,062. 91,058. 71,054. 61, 050. 3

280

1 025X10-6

1,0431,060

1,0741,0861,0961,1041,111

1,1171,122.11,123. 31,125. 61,129. 71,130. 0

1,131.81,131.71,130. 31,128.81,127. 91,124. 7

1,121.01,116. 91,112. 61,108. 01,103. 3

1,098. 51,093. 71,088. 81,084.01,079. 2

300

l,081X10-6

1,0981,113

1,126• 1,137

1,1461,1541,160

1,1651,169. 31,170.21,172.01,174. 71,174.8

1,175.11,173.61,170. 91,168 51,167. 21,162. 9

1,158.01,152. 91,147 6i, 142.11,136.6

1,131. 01,125. 41,119.91,114.41,108.9

320

1,140X10-6

1,1561,169

1 1801,1901,1981 2051, 211

1 2151,218 31 219 01 220 21,221 41, 221.4

1, 220.11, 217.11, 212 91 209 71,207 91, 202.4

1,196.41,190. 21 183 81,177 41,171.0

1,164. 61,158.21,151 91,145. 71,139. 5


temperature, and p, the density in Amagat units—Continued

340

l,201X10-6

1,2151,227

1,2371,2461,2531,2591,264

1,2671, 269. 31, 269. 71, 270. 31, 269.91,269. 7

1,266.81, 262.21, 256.61, 252.41, 250. 21. 243.3

1, 236.11,228. 71, 221. 31, 213. 91, 206. 5

1,199. 21,192.01,184. 91,177.9

1,171.1

360

l,264X10-6

1,2761,287

1,2961,3041,3101,3151,319

1,3211, 322.41, 322. 51, 322.41, 320. 31, 320. 0

1, 315. 31, 309.01, 301.81, 296. 71, 293. 91,285.6

1, 277.21, 268. 61, 260.11,251.61, 243.2

1, 234. 91,226.81, 218. 91,211.21. 203. 6

380

l,330X10-6

1,3411,350

1,3581,3651,3701,3731,376

1,3781, 377. 61, 377. 41, 376.61, 372. 81. 372. 3

1, 365. 71, 357. 71, 348. 81, 342. 61. 339. 31, 329. 6

1,319.71,309. 91, 300.11, 290. 51, 281.1

1, 271.81,262.81, 254. 01, 245. 51, 237. 2

400

1, 399X10-6

1,4081,416

1,4221,4281,4321,434[,436

L, 436L, 435. 2L, 434.6L, 433.1L, 427. 31, 426.6

1, 418.11,408.21, 397. 51, 390. 21, 386.41,375.1

1, 363.81, 352.61, 341. 61, 330.81, 320.2

1,310.01. 300.01, 290. 31, 280. 9. 271. 7

420

1,471X10-61,4781,484

1,4891,4931,4961,4971,498

1,4971,495.11, 494.21, 491. 91,484.11,483. 2

1, 472. 51, 460. 61,448.01, 439. 61, 435. 21, 422.3

1, 409.41,396.81, 384. 51, 372. 41, 360. 7

1, 349.41, 338. 41, 327. 71,317.41, 307.4

440

1, 545X10-6

1,5501,550

1,5591,5611.5621,5631,563

1,5611, 557.61, 556. 31, 553.21, 543. 31, 542.2

1, 529. 21, 515.11, 500. 61, 490. 91,485. 81,471. 2

1, 456. 71, 442. 61,428. 9L, 415. 5L, 402. 6

L, 390.1L, 378.0L, 366. 3L, 355. 0, 344.1

460

1,623X10-61,6261,629

1,6321,6331, 6331,6331,631

1,6281, 622.81, 621.11, 617.11, 604. 81, 603. 5

1, 588.11, 571. 81, 555.11, 544.11, 538.41, 521.9

1, 505.81,490. 01, 474.81,460.11, 445.8

1, 432.11, 418. 91, 406. 21,393. 91, 382. 0

480

l,706X10-6

1,7071,708

1,7081,7081,7081,7061,702

1,697L, 690.8, 688.6

L, 683. 7L, 669. 0L, 667. 4

L, 649. 5L, 630.81,611.8L, 599. 41, 593. 01, 574.6

1, 556.61, 539.21, 522. 41, 506. 21, 490. 6

1, 475. 61,461.21, 447. 31, 434. 01, 421. 2

400

1, 794X10-61,7921,790

1, 7881,7871,7861,7831,778

1,7711, 761. 81, 759.11, 753.11,735.81, 734.0

1,713.41, 692. 01, 670. 71, 656. 81.649. 71, 629.2

1, 609. 31,590.11,571.61, 553. 91. 536. 9

1, 520. 51, 504.81,489.81, 475.41,461.5


Many thermodynamic equations involve de-rivatives in which P, V, and T are the variables ofstate. Applications of the tables of this paper inwhich the variables are Z, p, and T to calculationsof properties involving derivatives in which thevariables are P, V, and T may be facilitated bymeans of equations relating the P, V, T and theZ, p, T derivatives. The following are adequatefor many ordinary uses:

T(dP\ __T(dP\ _T(dSP \dTJy~P \dTJp~P \dV)T

.T/dZ\~i+Z \dT)p

_ T/dV\ __Tfdp\ _T/dS\ _V\dTjP~ P \dTjP~V\dPjr

1 +

(4.2)

T(dZ\Z\dTj

Z\dp/T(4.4)

The Joule-Thomson coefficient pi may be utilizedto illustrate the use of these formulas. Thus forpurposes of calculations with the tables of thispaper, the familiar equation

dT\ VYT(dVl

is put in the form

=(dT\ _ VofX~\dPjH~pCP - —1 , (4.6)

where Vo is the molar volume of hydrogen atstandard conditions and Cp is the molar heatcapacity at the given conditions of T and P orT and p.

In correlating the PVT data for hydrogen thefunction

T V PVl g ( 4 J )

was used, where To is the Kelvin temperature ofthe ice point. Reported temperatures were re-duced wherever possible to a thermodynamic scalehaving the ice point temperature 273.16°. Allavailable data were considered in this work butonly those appearing most reliable were used andthese were weighted according to their apparent

precision. The data used [59, 61, 63, 65, 66, 67,70 to 74, 76, 79, 81, 85, 88, 91, 177] are plotted infigure 6 with the exception of a few observationsat temperatures below 29° K and at densitieslower than p=lO, which were omitted because inthese regions of low precision the scattering is sogreat that the points would be confusing.

A lower boundary to the a versus p gas-liquiddiagram in figure 6 is furnished by the vapor-liquid saturation line and the freezing curve.These are represented in figure 6 by dashed lines.The saturation line for the vapor rises steeplyonto the diagram at low densities and with de-creasing slope approaches tangency to the criticalisotherm at the critical point which is indicatedby an asterisk. The saturation line for liquidhydrogen is a nearly straight and horizontal linefrom a density somewhat greater than the criticalto the triple point. The freezing curve, whichrepresents the values of a for liquid when for agiven temperature the pressure is great enoughto cause the liquid to freeze, rises nearly verticallyfrom the triple point and bends towards higherdensities.

The saturation curve on the vapor side wasobtained with the help of the vapor pressureequation (eq 7.2) and the PVT representationgiven by eq 4.14 and table 19. On the liquid sideit was obtained from the same vapor pressureequation and the volumes of the liquid at satur-ation pressure, given in table 31 and discussed insection VIII. The freezing curve was obtainedfrom the melting point-pressure relations givenin table 30 combined with extrapolations basedon the higher density observations of Bartholomefor the isotherms of the liquid which are given intable 32.

The isothermal curves of figure 6 represent finaltable values. The curves are not necessarily thebest fit for the experimental data for each indi-vidual isotherm inasmuch as the curves and tablevalues are the result of correlating all the dataand include the temperature dependence which,while it does not affect the relative position ofpoints on one isotherm, may shift the wholeisotherm somewhat. Isotherms that dependedupon only a few individual observations and cov-ered only a small range of densities were given lessweight than others. For a given isotherm, dataat higher densities, corresponding to larger devi-ations from the ideal gas law, were usually given


e Bartholomeo Bartlett• Holborn and Otto© Kohnstamm and Walsfrao Leiden• Michels, Nijhoff and Gerver• Wiebe and Gaddy

(472.067

^473.16°

68.46°65.26°60.35°55.77°

S U D -

coolediquid

FIGURE 6. Plot of PVT data for H2 in the fluid states.

greater weight than data at low densities. In fact of penetration of the containers by hydrogen. Atin some instances the low density data were given very low temperatures the deviations from thezero weight. Data at the highest temperatures ideal gas law have not been measured very pre-do not appear to be very reliable, probably because cisely because the pressure range over which


measurements can be made is limited by condensa-tion.

Cragoe has shown that for densities up to p=500the 0° C isotherm is fitted to within experimentalaccuracy by the equation c=b+cp. Figure 6shows that, although this linear relation betweena and p fails at low temperatures, it is valid withinexperimental error over a considerable range oftemperatures above 200° K. This relation wasmade the basis for the correlation of the PVT dataabove 0° C. The different method used for cor-relating the data below 0° C is described under (b).

(a) Region Above 0° C

Above 0° C, equations of the form a=b-\-cpwere fitted to the PVT data plotted in figure 6,and b and c, the intercept and slope of an iso-thermal line, were determined as functions of T.The quantity Z=PV/RT thus obtained as afunction of T and p,

PV/RT=exp 2.30259

exp (4.8)

was used for the calculation of the tables of Z,P, (dZ/dp)T, (dZ/dT)p, and (d2Z/dT2)p.

Before fitting functions of T to b and c, smallcorrections were applied to some of the data. Aconstant error in T and constant factor errorsalong an isotherm in P, V, and the number ofmoles of gas, cause deviations from the true iso-therm that are very nearly proportional to 1/p.Such hyperbolic deviations from a straight lineare most easily detected in data extending fromlow to high densities. A change in V by 0.2 per-cent is sufficient to considerably straighten the573.16° K (300° C) isotherm of Wiebe andGaddy, and raise the line drawn through theiradjusted data so that it intersects the a axis offigure 6 only 0.7 unit below the table line for573.16° K and crosses the table line at p=550.Wiebe and Gaddy call attention in their paper toan estimated error of 0.05 to 0.10 percent in thevolume of their high pressure steel pipette at200° and 300° C. It would seem that some partof the 0.2-percent adjustment, which straightensthe 300° C isotherm of Wiebe and Gaddy, mightbe attributed to small temperature and pressureerrors and to some loss of hydrogen in the steel.

Hyperbolic adjustments proportional to 1/p of

Bartlett's higher temperature data straighten theisotherms and improve their agreement with thelines representing the tables. A comparison of theobservations of Michels, Nijhoff, and Gerver [79]at different temperatures for nearly constantvalues of p, revealed apparent small hyperbolictrends of the data for the separate isothermssuperposed on one larger though small randompattern of scattering common to all their iso-therms. Using their 0° C isotherm as a referenceline, their other data were adjusted to remove thehyperbolic deviations. The points of figure 6represent reported data adjusted only to theKelvin scale having 273.16° at the ice point.

Least square determinations were made of thestraight lines fitting the adjusted a versus p iso-thermal data for the different observers separately.From these, values of intercept 6 and slope cwere obtained for the different observers at eachtemperature of measurement. Holborn's dataabove 0° C, however, were used only for obtainingintercepts, the slopes of adjacent isotherms ofother observers being used with his data.

Expanding the exponential of eq 4.8,

]p4+.. . (4.9)

shows that B (T) is the second virial coefficientand that a correlation of intercepts b of a-isothermsis essentially a correlation of values of the secondvirial coefficients of hydrogen. Formulas express-ing the dependence of the second virial coefficienton temperature have been derived theoreticallyon the assumption of simple laws of intermolec-ular forces. One of the most satisfactory form-ulas is based on a law of intermolecular force ofthe form \nr~n—\mr~m and is due to Lennard-Jones. For n=^l?> and m = 7, the Lennard-Jonesformula for B is

B=B1T-1/±+B2T-3/*+B3T-5'4+..., (4.10)

where all the coefficients Bt of this infinite seriesare determined by \n and \m. Following essenti-ally a procedure used successfully by F. G. Keyes[89], we used only the first three terms of thisseries and selected values for Blf B2, and B3

which resulted in the best fit of a three constantequation with the intercepts of the a-isotherms.Our formula,

5=0.0055478 T"1 0.036877T"3/4—0.22004T"574,(4.11)


intended for use above 0° C, passes through theintercept of the.—50° C isotherm determined bythe correlation below 0° C.

The slopes of the a-isotherms were representedby a two term empirical formula without theoreti-cal justification, except that it involves powers ofT which make C go to zero as T grows very large.

(7=0.004788 T-3/2-0.04053T-2. (4.12)

The exponents of T were chosen so as to simplifythe temperature function coefficients in the powerseries in p of eq 4.9.

The tables from 270° to 600° K have beencomputed on the basis of these formulas, and in

0.06 percent for the 100° C isotherm, and for theother isotherms it is of this approximate magnitudeor smaller. At low densities the deviation for the0° C isotherm does not appear to be systematic.On the other hand, it will be seen that there is asystematic deviation at densities greater than 500with the experimental values for a less than thoseobtained by linear extrapolation from the interme-diate densities. This trend is supported by the highpressure data of Kohnstamm and Walstra [61, 81],also shown in the figure. If the representation ofthe a isotherm by an equation is extended beyondp=500, it will be necessary to include a smallquadratic term in the expression for a.

CATA OF MICHEU3 a GOUDEKETH2 . D2 o

! DATA OF KOHNSTAMM & WALSTRAH2 AT 20 °C

100 200 300 400 500 600 700 800 900 1000

FIGURE 7. A plot of part of the PVT data for H2 and D2 from 0° C to 150° C.

this temperature range the various derivativestabulated have been calculated analytically.

It was not until considerably after the prepara-tion of the tables on hydrogen that we were able toexamine the data of Michels and Goudeket pub-lished in Physica 1941 [91]. Values of a for thesedata on H2 are shown as solid circles in figure 7with the tables represented by the solid straightlines. The agreement for H2 is not complete butseems fairly satisfactory at moderate densities. Atlow densities there are discrepancies, roughlyhyperbolic, which have the appearance of thehyperbolic deviations resulting from small sys-tematic errors discussed earlier in this section. Ifthe hyperbolic deviation is attributed to a sys-tematic error in the volume, the error amounts to

(b) Region Below 0° C

At low temperatures the a versus p isothermsare curved; making it difficult to decide how theisotherm should be drawn at low densities wherethe data were meager and the precision was low.

Another function, T^2V/V0 ( l - ^ Y plotted

against p = V0/V as abscissa gave lines which ap-peared to be straight at low densities for tem-peratures below 56°K, though there is consider-able curvature at high densities. In figure 8,

l -^y+0 .0006 P =i /> is plotted against

p, the term 0.0006p being added to make isothermsnearly horizontal at low densities and thus in-crease the scale of the plot. The sensitivity to


small changes of PV/RT at P=200 and T=55° Kis 18 times greater in figure 8 than in figure 6 and14 times greater at p=200 and T=3S° K. Thecurves of figure 8 were drawn to fit the data foreach particular isothern considered independently,and though the curves do not represent the tablesexactly they agree closely with them. Below31°K the data were not sufficient and preciseenough to determine consistent isothermal curveswhen the isotherms were considered independently.The data lower than 29°K were not plotted be-cause the double valued nature of xp causes thedata below 29° K to fall in the same region on thediagram as is covered by the data above 29° K.

At first it appeared that the critical isotherm infigure 8 could be represented by a straight linefrom p equal to zero to p greater than the criticaldensity. However, the conditions that (dP/dV)T

and (d2P/dV2)T be zero at the critical point imposeupon the slope and curvature of the isotherm atthe critical point the conditions

f T(r a vT

PeVt o1 1+0.0006,(4.13)

In addition, values for the critical temperature andpressure should satisfy the vapor-pressureequation.

Only a single determination has been made ofthe critical temperature and pressure of hydrogen[62]. The critical isotherm was located some-where between the 2 measured isotherms at 32,94°7

and 33.29° K, and was at the time (1917) con-sidered to be 33.19° K with a certainty of about0.1°, though in 1925 it was stated in a footnote toLeiden Communication 172a that Tc should beabout 0.1° lower. The critical pressure inferredfrom the P versus V isotherms in 1917 was 12.80atm. Later in 1917 [142] the vapor pressureequation of H2 above the boiling point was deter-mined and the value 12.75 atm deduced for Pc

using Tc=33.18° K (on basis of «T0-273.09). Twodeterminations [62] were made of the critical den-sity based on the extrapolation of the rectilineardiameter. These gave pc=345. The values re-ported in later Leiden Communications havenot in all cases been the latest determined values.The most recently reported Leiden values [69] are

7 Unless otherwise stated, temperatures are expressed on the Kelvin Scalewith TQ=273.16°.

0.6O

2.0.55ooob

>j>°0.50

0.45

rfi<-29.2l°K<jj>. 6-31.22°

^36.79°

41.70°

,•33.19°

100 200 300 400 500 600 700

FIGURE 8. Plot of PVT data for H2 at low temperatures.


Tc=33.19°lf (on basis of T0=273A6), Pc=12.751atmand l/pc=0.02909 orPc=344. The lower criti-cal temperature 33.1° K inferred from LeidenCommunication 172a is supported by the agree-ment of the vapor pressure 12.81 atm, calculatedfrom vapor pressure equation (eq 7.2) with thecritical pressure determined in 1917 from theP versus V isotherms.

Difficulties are encountered in obtaining agree-ment with the experimental PVT data (fig. 8)

vapor pressure equation (7.2). These critical con-stants are listed in table 18.

TABLE 18. Critical constants of hydrogen

°K33.19

Pc

atm12.98

Pc=yc

335

vc

cmzmole-1

66.95

PcVc

RTe

0. 3191

It seemed reasonable to assume that the iso-

0.70

0.65

0.60

0.55

0.50

0.45

OCD

O

o~'~? 0

00

0

° } LEIDEN

~°) SCHAFER0 J

0 0

v-. DEUTEV,(A)

0

0

HYDRO

RIUM

00

00

\

\

5

- 5

—1

10 20 30 n 40 50T °K

FIGURE 9. Intercepts and slopes from figure 8.

6 0

on the basis of Tc=33.1° and Pc= 12.81 atm,however, unless the critical density is inferred tobe about 320, in Amagat units, instead of thereported values 345 or 344. This difference incritical density seemed too large on the basis ofthe probable precision of the density measurementsThe adjustment has instead been so made andthe critical isotherm in figure 8 so drawn that Tc=33.19°, Pc= 12,98 atm, and pc=335. This valueof Pc is consistent with the PVT data and with


therms of figure 8 are straight lines up to p=200.This assumption was used in correlating theobserved data below the critical temperaturewhere the data were scarce and the precision low.In figure 9 the intercepts A and the slopes C ofthe isotherms of figure 8 are plotted as functionsof the temperature. The curve for the slopewas extrapolated smoothly to lower temperaturesas slopes could not be obtained from the databelow 33° K.

429

Also shown in figure 9 are values for A calculatedfrom second virial coefficients determined experi-mentally by Schafer [85]. Schafer reported theresults of his PVT measurements as virial coeffi-cients B'(T)=d(PV/RT)/dP at constant tempera-ture and at P=0. The values of A= — (RT5/2/V0)B'(T) obtained from Schafer's results agreewell with those obtained from data of the LeidenLaboratory as shown by figure 9. Schafer observedno consistent difference between the second virialcoefficients of para hydrogen, normal hydrogen,and a one to one mixture of ortho and paravarieties.

The equation for the straight part of the \[/-isotherms of figure 8 may be written

^3/2

where C=C — 0.0006, C being the slopes plottedin figure 9 of the ^-isotherms in figure 8. Valuesof A and C and their derivatives are given forhydrogen in table 19. The values of PV/RT fromp=0 to p=200 and from T=U° to T=56° K in

TABLE 20. Pressure, density, and PV/RT for saturatedH2 vapor

TABLE 19. Hydrogen values of A and C (and derivatives)in the equation for isotherms

[Applicable at Amagat densities less than 2001

° K1416182022

2426283032

3436384042

4446485052

5456

A

° KW0. 5754.5827.5887.5933.5965

.5981

. 5981

.5966

.5940

.5904

.5858

.5805

.5746

.5679

.5604

.5521

.5429

.5330

.5225

.5114

.4996

.4871

C

o ^ 3 / 2

-5,62lX10~7

- 5 , 636- 5 , 653-5,672-5,693

- 5 , 716- 5 , 743- 5 , 774-5,809-5,848

-5,892- 5 , 943-6,003-6,071-6,146

-6, 229-6, 320-6, 420-6, 529- 5 , 646

-6, 770-6, 900

dA/dT

° KM0.00388.00330. 00264. 00192.00116

. 00040-.00032-.00097-.00154-.00202

-.00243-.00280-.00317-.00356-.00397

-.00438-.00476-.00509-.00540-.00572

-. 00608- 00650

dC/dT

° KM-75XHT 8

-82-90-100

-112

-127-145

-165

-187

-213

-245

-282

-320-358

-396

-436

-478

-522

-565

-603

-636-664

T

° K141 6 __•_ _

182022

242628303233.19

P

atma 0728.2018.4551.88911.5645

2. 54533 89865 6958.01010 93312.98

P

Amagats1.4453.5627 32113.31122. 235

35. 01753 0278 55116. 33180 94335

PV/RT

0. 98415. 96768.94396.91283.87420

.827837729870776.6273251554. 3191

table 13 were calculated using eq 4.14 with table19. Table 20, giving the pressure, density, andvalue of PV/RT for saturated H2 vapor, was pre-pared similarly using the vapor pressure equationfor TI-H2 (eq 7.2). For certain uses eq 4.14 withtable 19 may be more convenient than the tablesof PV/RT and its derivatives.

For temperatures below 56° K and densitiesgreater than p=200 where \p could not be rep-resented by a simple function of p, a table wasmade of values of \p for each p and T entry in theZ-table. The ^-values of this table were obtainedfrom figure 8 by graphical interpolation. Largeplots of i/'-isochores, 20 Amagat units apart, on\// versus T graphs were made of values of \f/ readfrom figure 8. Values of \f/ at 2-degree intervalswere read from the isochores. A Z(p,T) table wascalculated from the yp(p,T) table.

From 56° to 273° K, the ^-function rather thanthe yp-iunction was used because above 56° K the(r-isotherms approach linear functions of thedensity. The method of graphical interpolationused below 56° K was used above, also, to obtaina table of er-values for the p and T entries of theZ-table. The accuracy of graphical interpolationwas improved by using more sensitive plots thanfigure 6 of modified <x-functions obtained by addingto a simple functions of T and p, which broughtthe isotherms and isochores closer together so thatthey could be easily plotted to a large scale.Values of <r were obtained at densities as high asp=5Q0, although between 70° and 200° K meas-urements were not available at densities this high.This region was filled in by extrapolation of a-curves to higher densities along isotherms and byinterpolation along isochores between the upper


and lower temperature regions where there weredata to determine the trend. From the o{pyT)table a Z(p,T) table was obtained by calculation.

The Z(p,T) table obtained through graphicalinterpolation of the \[/ and a isotherms as has justbeen described was smoothed along isotherms andalong isochores by inspection of second differences.In general the Z-tables are smooth to one unit inthe last digit.

The tables of (dZ/dT)p and (dZ/dp)T below 0°Cwere for the most part calculated from thesmoothed Z table by the method of Rutledge[179] for the calculation of derivatives fromsmooth sets of tabular values of data.8 In theregion below 56° K and p=200, where the \f/versus p isotherms are straight lines, the followingequations, obtained by differentiating eq 4.14,were used with table 19 to calculate the deriva-tives

(l-Z) p_ dA p^dCT3/2 JT T3/2 jrp v"*.T3/2 dT

(4.16)

Where the derivatives could be obtained both bythe method of Rutledge and by eq 4.15 and 4.16,the agreement was very satisfactory. Th e (dZ/dp) T

and (dZ/dT)p tables were also smoothed alongisotherms and isochores by inspection of seconddifferences.

The (d2Z/dT2)p table below 0° C was obtainedthroughout by the method of Rutledge from thesmoothed {dZ/dT)p table and was also smoothed.The equation for (d2Z/dT2)p corresponding to eq4.15 for the first derivative was considered tooinvolved for easy computation.

In general, the tables of derivatives are smoothto the last digit recorded.

(c) Reliability of Tables of PVT Data

By inspecting figures 6 to 8 it is possible toarrive at some general conclusions regarding thedeviations of the observed data from the Z(p,T)table. It may be noted that, except at lowdensities, the deviations of the observationalvalues of a from the curves representing the tableare of about the same magnitude at differentdensities along a given isotherm up to p=500.9

This means that deviations of (PV/RT) — l8Assuming that differences of higher order than the fourth are negligible.9 For still greater densities larger deviations occur as shown by figure 7.

along an isotherm are approximately proportionalto the density. At low densities the deviations arelarge because the sensitivity of the a and \f/ plotsapproaches infinity as p approaches zero. It isdifficult to make an estimate of the probable errorin PV/RT based on the deviations because, asis seen, the greatest deviations are the systematicdifferences between the results of different ob-servers and are not accidental errors as should bethe case if error theory were to apply. The userof the tables can make an estimate of the meandifference between the observed and tabulatedvalues of PV/RT, in any particular region oftemperature and density by noting the deviationsshown on the graph and from these calculatingthe corresponding deviations in PV/RT. Fortemperatures below 60° K it would be best to usefigure 8 for this purpose as it is plotted to a largerscale than is figure 6.

In constructing the tables for the intermediatetemperature regions where analytical equationsof state were not used, just enough digits wereretained so that changes made in smoothingwould be confined to the last digit. As a con-siderable amount of smoothing resulted from thegraphical methods used, many of the irregularitiesin the measured values were not apparent in theunsmoothed tables.

It is believed that throughout the table thevalues were carried out to at least as manysignificant figures as were at all justified by thedata, and that the last digit recorded should beconsidered very uncertain. In that part of thetable between 77° and 200° K which was filled inby interpolation and extrapolation the last twodigits should be considered uncertain, the lastrecorded digit being retained to achieve continuitywith the rest of the table.

The tables are thought to be most reliable fortemperatures between 273° and 373° K (0° and100° C), because at these temperatures theexperimental difficulties encountered are not asgreat as at higher and lower temperatures. Also,as is shown by figure 6 the results of severaldifferent investigators are in agreement at thesetemperatures. Above 373° K the experimentaldata are not as self-consistent as at temperaturesimmediately below. As the values of PV/RTgiven in the tables for these higher temperaturesare derived largely from an extrapolation based onthe temperature region between 273° and 373° K,


an estimate of reliability of the high temperatureportion of the tables involves both the applica-bility of the correlating function, eq 4.8, and theprecision of the experimental data. Consideringthe differences between the isothermal linesdetermined by different sets of experimental dataof different observers and the same observer atdifferent temperatures, it seems probable that theextrapolation is more reliable than the experi-mental data at temperatures above 473° K.

It is doubted that PV/RT is known to betterthan 0.2 percent for densities as high as 100Amagats near 33° K, the critical temperature.

Below the critical temperature, the data are notvery satisfactory. In addition to the difficultiesof making measurements at low temperatures,there exists the circumstance that below thecritical temperature the range of vapor densitiesthat can be covered is limited by the density ofsaturated vapor. At low densities the deviations(1—Z) from the ideal gas law are small and hencedifficult to measure precisely.

There is another method of obtaining values ofsecond virial coefficients which may be advanta-geous for the low temperature region. It involvesthe determination of the velocity of sound, whichhas been carried out for gaseous hydrogen atliquid-hydrogen temperatures and various pres-sures by van Itterbeek and Keesom [77], using aresonance method. The change of the velocity ofsound with pressure at very low pressures is re-lated to the value of the second virial coefficientand to its first and second derivatives. Becauseof this relationship, it is possible to determine thesecond virial coefficient from the velocity of soundif the second virial coefficient is already known inan adjacent range of temperature. Van Itterbeekand Keesom concluded that the agreement be-tween their own measurements and the PVT datawas "rather good", although for both types ofdata the scattering was quite appreciable.

In calculating the tables of derivatives by themethod of Rutledge, the criterion for retainingsignificant figures in the recorded values was thesame as that previously mentioned, namely,enough places were carried so that the changesresulting from the smoothing were in general con-fined to the last digit. As in the case of the tablesof PV/RT, it is believed that the tabulated valuesof the derivatives are given to as many significantfigures as are justified by the data.

2. Deuterium

The interesting features of the PVT data fordeuterium are most evident when deuterium iscompared with hydrogen. The difference betweenthe second virial coefficients of H2 and D2 has beeninvestigated theoretically [86, 87], though a com-plete treatment of the problem has not been made.

Assuming the same intermolecular forces forH2 and D2, classical mechanics and statisticslead to the same equation of state for H2 and D2.The quantum theory of virial coefficients leadsto effective volumes of molecules and to secondvirial coefficients that are larger than the classicalvalues, the differences being small at ordinarytemperatures but becoming large at low tem-peratures.10

In table 21 are given ratios between quantummechanical and classical values of second virialcoefficients, for gases whose molecules are rigidnonattracting spheres. They may also be con-sidered as ratios between apparent molecularvolumes for the two treatments. These ratios arebased on formulas derived by Uhlenbeck and Beth[84]. Columns 2 and 3 are for gases with molecularweights 2 and 4, respectively. The value of theratio depends, among other things, upon thediameters of the rigid spheres. Here the size ofthe spheres was taken to be the same for the two

TABLE 21. Ratio between quantum mechanical and classicalsecond virial coefficients for nonattracting rigid sphericalmolecules a of molecular weight M

T

° K60030010025.5

B quantumB classical

1.211.301.522.74.6

B quantumB classical l0T11

1.151.211.372.02.6

» With diameters calculated from the van der Waals' b for hydrogen.

10 The application of quantum mechanics instead of ordinary mechanicshas as one effect for rigid spherical molecules the removal of the classicaldiscontinuity in the calculated distribution of molecules for pair separationscorresponding to contact between the spheres. As smaller separations areprevented by the impenetrability of the spheres, the continuity is establishedby a reduction of the molecular density for separations greater than thatcorresponding to contact. The effect is large for separations of sphere surfacesup to a considerable fraction of the de Broglie wavelength (for whichh(-y/2mkT is a representative value) and depends through this uponthe temperature. This reduction of molecular density beyond the mini-mum separation could be represented roughly in a classical description asan increase of the volume from which 1 molecule causes the centers of othermolecules to be excluded. In classical theory the second virial coefficientfor nonattracting rigid spheres is proportional to the excluded volume.


gases and to be equal to the size calculated fromthe van der Waals b for H2.

Although it would scarcely be expected that theresults of calculations for rigid nonattractingspheres would apply to real H2 and D2 molecules,it would seem likely that qualitative indicationswould be correct, at least at higher temperatureswhere the excluded volume predominates over theintermolecular attractive forces in determiningthe magnitude of the second virial coefficient.This is borne out by experiment, the difference insecond virial coefficients (B^—B^^, being posi-tive, though smaller than would be indicated bytable 21 for rigid spheres by a factor of about 2.6at 300° K. Uhlenbeck and Beth derived an ap-proximate quantum mechanical representationfor the second virial coefficient applicable at high

80

60

3*40

ih

20

n

/

[1

I \1

• — I05 BHz

(BH2-BD2)

-200 <

-400

-600

200 400 600T °K

FIGURE 10. Second virial coefficient for H2 and the differencebetween second virial coefficients for H2 and D2.

temperatures for molecules with radially sym-metrical force fields. Their formulas were appliedto hydrogen and deuterium by de Boer andMichels [87J upon the assumption that the inter-molecular forces were the same for H2 and D2.They obtained differences between the virial co-efficients for H2 and D2 represented by the uppertemperature portion of one of the curves of figure10. In a later paper by Michels and Goudeket[92] attention was called to the fact that the inter-molecular forces of hydrogen and deuterium dodiffer a little because the mean internuclearseparations of H2 and D2 molecules are differentas a result of the different zero point vibrationsof their nuclei.

The effect of the intermolecular attractiveforces overbalances the effect of the excludedvolume or the repulsive forces of the moleculesin determining the magnitude of the second virialcoefficient at low temperatures, and makes thecoefficient negative. Nevertheless, at low tem-peratures, as at high temperatures, the differencein second virial coefficients B^2—BT>2 is positive,partly for the reason already discussed in the caseof high temperatures, namely the larger apparentquantum-mechanical volume of H2 molecules, andpartly for another reason. There is a closer spacingof the discrete negative energy states and smallerzero point energy for pairs of D2 molecules thanfor pairs of H2 molecules because of the mass dif-ference, so that by reason of the Boltzmann fac-tor, exp [—energy/& TJ, there is a greater degreeof association or clustering together of D2 mole-cules than of H2 molecules. Without a considera-tion of the Boltzmann factors for these negativeenergy levels the effect of the difference of masswould be less clear, as the quantum treatment forthe continuum would require that the spacing ofthe levels there be smaller for D2 than for H2 inessentially the same ratio as in the case of the .discrete negative energy levels. With these orsimilar ideas in mind, Schafer [86] derived aformula for the difference in second virial co-efficients for H2 and D2 at low temperatures,which involved a constant whose magnitude he sochose as to obtain a fit with his experimentalvalues for the difference in the second virialcoefficients.

Figure 9 shows values of A in the equation ofstate (eq 4.14) calculated from the second virialcoefficients of deuterium for the temperaturerange 23° to 45° determined experimentally bySchafer [85].

A=-T*'2(dZ/dp)T,p^=-T^B1, (4.17)

where Bi is the second virial coefficient in theequation of state PV=RT (l+BlP+B2p

2+. . .).The dashed line curve in figure 9 was obtained byadding to the A's for H2 the differences betweenthe A's calculated from the differences betweenthe second virial coefficients of H2 and D2 whichSchafer determined partly theoretically and partlyempirically. Schafer's measurements were madeon deuterium at low densities and hence do notgive information on higher virial coefficients.Approximate values of PV for deuterium at low


temperatures may be found by using values of Afrom figure 9 in eq 4.14, and either neglecting theC term or preferably using the correspondingvalue of C for H2.

Values of the function <r={TVIT{y^ log10

(PV/RT) calculated from the data of Michels andGoudeket [92] for D2 are shown as open circles infigure 7. The dashed straight lines for deuteriumare obtainable from the equation

where

and

PV/RT=exp[B(T)P + C(T)P2], (4.18)

= 0.0055298T-1/4-0.036040T-3/4-0.25878T-5/4

C(T) = 0.00580 r~ 3 / 2 - 0.0565 T~2.

The constants in the formula for B have been sochosen that the difference between D2 and H2

intercepts on the a—axis is in close agreementwith the theoretical result of de Boer and Michels[87] from 250° to 450° K.

In figure 10, a curve marked 105 (BH2— BD2) showsthe trend of differences between second virialcoefficients based on the theoretical calculationsabove 150° K and on the results of Schafer below50° K with an interpolation between. It may beinferred that the differences between the PVTdata for H2 and D2 decrease rather rapidly withincrease of temperature. For comparison, thecurve marked 105 BH2, in figure 10, shows on adifferent scale the magnitude of the correspondingsecond virial coefficient for H2 at the same temper-atures.

If it is assumed that the a or (TV/T0VQ) log(PV/RT) isotherms for D2 and H2 are parallel,values of PV/RT for D2 may be obtained fromthose tabulated for H2 by (1) calculating the<?H2 or a for H2, from the values of PV/RT, T andp, (2) subtracting the difference (V#2—(TD)P=Q toget aD2, and then (3) calculating the correspond-ing value of PV/RT for D2. A plot of thedifference 105 (<rH2—<rD2)p=o which may be usedfor this purpose is shown in figure 11. An alter-native method based on the assumption that onlythe second term of the series expansion eq 4.9for PV/RT is to be changed is as follows. 105

(BH2—BD2), obtained from figure 11 by multiply-ing 105 (<rH2—<rD2)P=o by 2.302585 To/T or obtaineddirectly from figure 10, is multiplied by 10~5 andthe product subtracted from PV/RT for H2 togive PV/RT for D2. This alternative method issimpler than the other method and may be asreliable.

I.

3

1

200 400 600T °K

FIGURE 11. Difference between intercepts of a versus pisotherms for H2 and D2.

V. Calculation of Thermal Propertiesof the Real Gas

The calculation of thermodynamic properties ofa real gas from values of these properties for theideal gas rests upon the principle that the differ-ence between values of a thermodynamic functionat different densities for the same temperaturemay be determined from data of state for the gasat the given temperature.

The entropy and free energy of a gas are depend-ent upon the pressure, even in the ideal state, andin tables 4 to 8 they are given for the hydrogensin the ideal gas state at a pressure of 1 standardatm. On the other hand, the internal energy,enthalpy, and specific heat in the ideal gas stateare independent of density at constant temperature.

Equations 5.1 to 5.8 show how, using the data ofstate expressed in the form, Z=Z(p, T), thethermodynamic properties of the real gas at atemperature T and an Amagat density p may becalculated from properties for the ideal gas state


at a pressure of 1 atm, given for the hydrogens intables 4 to 8.

- ("[T(dZ!dT),/P]dpJo

(5.1)This can be expressed in a slightly different formby using the identity

f' [ ( Z - 1)/P]dp+ f" [T(dZ/dT)Jp]dP=Jo Jo

(5.2)

P[(Z-Jo

In [/ (fugacity of real gas)/P]= \ (ZJo

lnZ+(Z-l).

(5.3)

ln p+

(5.4)

T (real gas) -^-T (ideal)

(5.5

/p\dp, (5.6)

(w»)Pt

- (P[T2(d*Z/dT2)p/p]dP. (5.7)Jo

T (real gas) \^P)T (ideal)

R ~ R

2 rp[r0Z/rfn/p]rfp- fP[Jo Jo

{[Z+ T(dZ/dT)p]2/[Z+p(dZ/dP)T]} - 1 .

(5.8)

In order to facilitate the calculation of thethermodynamic properties of hydrogen in the realgas state, tables 22 and 23 were computed.11

Lagrangian four point formulas [181] were usedfor the tabular integrations.

Table 22 is intended for use in the calculation of11 For the calculation of these tables the authors are indebted to Messrs

Roger E. Clapp, Kingsley Elder, Jr., and Robert Mann, who worked asstudent assistants at the National Bureau of Standards during the summerof 1941.

entropies. The values in the second column,headed (Sp^i—Sf^/R, are for the differencebetween entropies of hydrogen in the ideal gasstate at 1-atm pressure and at unit Amagatdensity, divided by R.

y = 1, T (ideal) f~0^0

R

-0.000618 + ln T/To (5.9)

The row at the bottom of the table, headed(S°p=1—S°)/R, is for the difference between entro-pies in the ideal gas states at Amagat densities oneand p, divided by R.

P = l , 2'(ideal)

RLe5»=lnp (5.10)

The other rows and columns of table 22 headed(S°—S)/R give the differences between the entro-pies in the ideal and real gas states at the sametemperature and density, divided by R.

S P,T (ideal) , r f r - ) = r> [(Z_ l)/p]dp +

(P[T(dZ/dT)p/P]dp (5.11)Jo

In order, then, to get S/R for the real gashydrogen at a temperature T and Amagat densityp, one subtracts from S°/R, obtained from S°given in table 8, the sum of three numbers for theappropriate values of T and p to be obtained fromtable 22: one comes from the second column,headed (So

v=1—S°p==1)/R; another from the bot-tom row of the table, headed (S°p=si—S°)/R;and the third from the rows and columns of thetable headed (S°-S)/R.

Table 23 is for the difference between theenthalpy of hydrogen in the ideal and real gasstates at temperature T and Amagat density p,divided by RT.

TT°£ 1 T

TTP £ 1

RTP,T (real) = [\T{dZldTyp\dp-{Z-\).

Jo(5.12)

Hence to obtain H/RT for hydrogen in the realgas state, one subtracts the appropriate value of(H°—HP)/RT in table 23 from the value ofH°/RT obtained from H° given in table 8 for thedeal eras state.


TAB

LE 2

2.

Ent

ropy

di

ffer

ence

s di

vide

d by

R,

for

norm

al H

2

—-—

-, E

ntro

py o

f id

eal

gas

min

us e

ntro

py o

f re

al g

as a

t sa

me

Tan

d p,

div

ided

by

R.

-, E

ntro

py o

f id

eal

gas

at p

ress

ure

of 1

atm

osph

ere

min

us e

ntro

py o

f id

eal

gas

at d

ensi

ty o

f 1

Am

agat

, di

vide

d by

/?.

— ,

Ent

ropy

of

idea

l ga

s at

den

sity

of

1 A

mag

at m

inus

ent

ropy

of

idea

l ga

s at

den

sity

of

p A

mag

ats,

div

ided

by

R.

°K

16..

18..

20..

22..

24..

26..

28_.

30..

32..

31..

36..

38..

40-.

42..

44-.

46..

48..

50-.

52.

54.

56.

58.

60.

65.

70.

75.

80.

85.

90.

95 100

105

110

115

120

-2. 83809

-2. 72030

-2. 61494

-2. 51963

-2.43262

-2. 35258

-2. 27847

-2. 20948

-2.14494

-2.08432

-2.02716

-1. 97309

-1.92180

-1.87301

-1.82649

-1.78204

-1.73948

-1.

-1.65943

-1.62169

-1.5

8532

-1.55023

-1.51633

-1.43629

-1.3

6218

-1.29319

-1.22865

-1.1

6802

-1.1

1087

-1.05680

-1.00E51

-0.9

5672

-.91

020

-.86

574

-. 82318

S°-S

p = l

imag

at

0.00373

. 00323

.00288

. 00264

.00245

.00230

.00218

.00208

. 00198

. 00190

. 00182

. 00175

. 00169

.00165

.00160

.00156

. 00153

. 00150

.00148

.00146

. 00144

.00142

. 00140

. 00135

.00131

.00127

.00124

.00121

.00119

.00117

.00115

.00113

.00112

.00111

.00110

2

0.00745

.00645

.00576

.00529

. 00489

.00459

.00435

.00415

.00396

.00379

.00363

.00349

.00338

.00329

.00320

.00313

.00306

.00300

.00295

.00291

.00287

.00284

. 00280

. 00270

. 00261

. 00254

. 00248

. 00242

. 00238

. 00234

. 00229

. 00226

.00224

. 00222

. 00220

3

0.01117

.00968

. 00864

.00792

.00733

. 00690

. 00653

. 00623

. 00595

. 00569

. 00545

.00524

.00508

.00494

.00480

.00469

. 00460

. 00451

.00443

.00436

.00431

.00426

. 00420

. 00405

. 00392

. 00381

. 00371

.00364

.00357

.00351

. 00344

. 00339

.00335

. 00333

.00331

6

0.01933

. 01724

.01583

.01463

. 01378

.01304

.01245

.01189

.01137

. 01088

.01047

.01014

.00988

. 00960

.00938

.00919

.00902

.00886

.00873

.00862

. 00851

.00840

. 00810

. 00784

. 00762

.00743

. 00728

.00714

. 00702

.00689

.00679

.00671

. 00665

. 00661

10

0.0287

. 0263

.0245

. 0230

.0218

.0208

.0198

.0190

.0181

.0174

.0169

.0165

.0160

.0156

. 0153

.0150

.0148

.0146

.0144

.0142

. 0140

.01352

.01309

.01271

.01240

.01214

.01192

.01171

.01149

.01133

.01120

.01111

.01104

20

0. 048

8. 045

8. 0436

.0416

.0396

. 0379

.0362

.0347

.0337

. 0329

.0320

.0313

. 0306

.0301

.0295

. 0291

.0287

.0284

.0280

.0271

.0263

.0254

.0248

.0244

.0239

.0234

.0230

.0227

.0224

.0222

.0221

40

0.0915

.0867

.0827

.0787

.0753

.0719

.0692

.0672

.0655

.0640

.0626

. 0613

.0601

.0591

.0582

.0575

.0568

.0560

.0540

.0525

.0510

.0499

.0490

.0481

.0472

.0464

. 045

8.0452

.0448

.0445

60

0.129

. 123

.117

.112

.107

.103

.100

.098

.096

.094

.092

.090

.089

.088

.087

.086

.084

.0812

.0790

.0768

.0752

.0739

.0726

.0712

. 0700

.0691

.0683

.0677

.0672

80

0.163

.155

.148

.142

.137

.133

.130

.128

.125

.122

.120

.118

.117

.116

.114

.112

.109

.106

.103

.101

.099

.097

.096

.094

.093

.092

.091

.090

100

0.202

.193

.184

.176

.170

.165

.161

.158

.156

.153

.150

.148

.146

.144

.142

.140

.136

.132

.129

.126

.124

.122

.120

.118

.117

.115

.114

.113

120

0.230

.220

.211

.203

.198

.193

.190

.186

.183

.180

.177

.175

.173

.171

.168

.164

.159

.155

.152

.150

.148

.145

.143

.141

.139

.138

.137

140

0.266

.255

.245

.236

.229

.225

.221

.217

.213

.210

.207

.204

.202

.200

.197

.192

.186

.182

.179

.176

.173

.170

.168

.165

.163

.162

.161

160

0.302

.290

.278

.268

.261

.256

.252

.248

.244

.240

.237

.234

.231

.228

.225

.219

.213

.209

.205

.202

.199

.196

.193

.190

.188

.186

.185

180

0.337

.324

.311

.300

.293

.287

.283

.278

..274

.270

.266

.262 259

.257

.254

.248

.241

.236

.232

.228

.225

.221

.218

.215

.213

.211

.209

200

0.358

.344

.332

.324

.317

.313

.308

. 303

.299

.295

.291

.288

.285

.283

.276

.269

.263

.259

.254

.251

.247

.244

.241

.238

.236

. 234

125

130

135

140

145

150

155

160

165

170

180

190

200

. -

210

220

230

240

250

260

270

280

300

320

340

360

...

380

400

420

440

460

480

500

520

540

560

580

600

R

-.78

236

-.74

314

-.70

540

-.66

903

-.63

394

-.6C

004

-.56

725

-. 53550

-.50

473

-.47

488

-.41

772

-. 36365

-.31

236

-.26

357

-.21

705

-. 17260

-. 13004

-.08

922

-.04

999

-.01

225

+.02

4114

+. 093106

.157645

. 218270

. 275428

. 329495

. 38079

. 42958

. 47610

. 52055

. 56311

. 60393

. 64315

. 68089

. 71726

. 75235

.78625

. 00109

.00109 ,

. 00109

. 00108

. 00107

. 00107

. 00106

. 00106

. 00105

. 00104

.00103

.00102

.00100

.00099

. 00098

.00097

.00096

. 00095

. 00095

.000938

. 000931

. 000916

.000902

. 000890

. 000879

. 000868

. 00086

.00085

.00084

.00083

. 00082

.00082

.00081

.00080

. 00080

.00079

. 00078

0

. 00219

. 00218

.00217

.00215

.G0214

.00213

. 00212

.00211

.00210

.00209

. 0C207

.00204

.00202

. 00200

.00197

.00195

. 00193

. 00191

. 00190

. 001877

. 001861

. 001833

. 001805

. 001781

. 001759

.001737

. 00172

. 00170

. 00168

.00166

. 00165

.00163

. 00162

. 00160

. 00159

. 00158

. 00157

.69315

. 00329

. 00327

.00325

. 00323

. 00321

. 00320

. 00318

.00316

.00315

.00313

.00310

. 00307

. 00303

.00299

. 00295

.00292

.00289

. 00286

. 00284

. 002816

. 002793

. 002750

.002708

.002672

.002639

.002605

.00257

. 00255

.00252

.00250

.00247

.00245

.00243

.00241

.00239

.00237

.00235

1. 09861

.00658

.00654

.00G50

. 00647

.00643

.00639

. 00636

.00632

. 00629

. 00626

. 00620

.00613

.00606

.00599

.00592

.00585

.00578

.00573

.00568

.005636

.005589

,005504

.005420

.005347

.005280

.005213

.00515

.00510

.00504

.00499

.00494

. 00490

.00486

. 00482

. 00478

. 00474

. 00470

1. 79176

.01098

.01092

. 01086

. 01080

. 01074

. 01068

. 01062

.01056

. 01051

.01046

. 01033

. 01021

. 01007

.01004

.00994

. 00976

.00964

. 00952

. 00948

. 009402

. 009323

. 009180

. 009042

. 008919

.008807

. 008696

. 00859

. 00850

. 00841

. 00833

.00825

. 00817

. 00810

. 00803

. 00797

. 00790

.00784

2.30259

.0220

.0219

.0218

.0216

.0215

.0214

.0213

.0212

.0211

.0210

. 0208

.0205

.0203

.0201

.0198

.0196

.0194

.0192

.0191

.01885

. 01869

. 01840

.01812

.01787

.01765

.01742

.0172

.0170

.0169

.0167

.0165

.0164

.0162

.0161

.0160

.0158

.0157

2.9957

.0442

.0440

.0438

.0435

.0433

.0430

.0428

.0426

.0424

.0422

.0418

.0413

.0408

.0403

.0398

.0394

.0389

.0385

.0382

. 03786

. 03754

. 03694

. 03639

.03589

.03543

.03498

.0346

.0342

.0338

. 0335

.0332

.0329

. 0326

.0323

.0320

.0318

.0315

3. 6889

.0667

.0664

.0662

.0656

.0653

.0649

.0646

. 0642

.0639

.0636

.0629

.0622

.0615

.0608

.0600

.0593

.0587

.0581

.0576

.05705

.05656

. 05565

. 05481

.05404

. 05334

.05266

.0520

.0515

.0509

.0504

.0499

.0495

.0490

. 0486

.0482

.0478

.0474

4.0943

.090

.089

.089

.088

.088

.087

.087

.087

.086

.085

.0843

.0834

.0824

.0814

.0805

.0795

.0786

.0778

.0770

.07641

.07574

.07451

.07338

. 07234

.07140

.07048

.0697

.0689

.0681

.0675

.0668

.0662

.0656

.0650

.0645

.0640

.0635

4.3820

.113

.112

.112

.111

.110

.110

.109

.109

.108

.107

.1059

. 1047

.1035

.1023

.1011

.0999

.0988

.0978

.0968

.09595

.09510

. 09354

.09210

. 09079

.08959

. 08844

.0874

.0864

.0855

.0846

.0838

.0830

.0823

.0815

.0809

.0802

.0796

4.6052

.136

.136

.135

.134

.133

.133

.132

.131

.130

.129

.127

.126

.125

.124

.122

.120

.119

.1.18

.117

.1157

.1146

.1127

.1110

.1094

.1079

.1065

.1053

.1041

.1030

. 1019

.1009

.1000

.0990

.0982

.0974

.0966

.0958

4. 7875

.160

.159

.158

.157

.157

.156

.155

.154

.153

.152

.149

.147

.146

.145

.143

.141

.139

.138

,137

.1356

. 1344

.1321

. 1300

. 1281

.1264

.1248

.1233

.1219

.1205

. 1193

.1181

.1170

.1160

.1149

.1140

.1131

.1122

4. 9416

.184

.183

.182

.181

.180

.179

.178

.177

.176

.174

.172

.169

.167

.166

.165

.162

.160

.158

.157

.1557

.1543

.1516

.1492

.1470

.1450

.1431

.1414

. 1398

.1383

.1368

.1355

.1342

.1330

. 1318

.1307

.1297

.1286

5.0752

.208

.207

.206

.205

.204

.203

.202

.201

.199

.197

.194

.191

.189

.188

.186

.184

.181

.179

.177

.1760

. 1743

.1713

.1686

.1661

.1638

.1617

.1597

.1578

.1561

.1545

.1530

. 1515

.1501

.1488

. 1476

.1464

.1452

5.1930

.233

.232

.230

.229

.228

.227

.225

.224

.222

.220

.217

.214

.211

.210

.208

.205

.203

.200

.198

.1965

.1946

.1912

. 1881

.1853

.1827

.1803

.1781

.1760

.1741

.1723

.1706

. 1689

. 1874

.1659

.1645

.1632

.1619

5. 2983

CO

Tem

per-

atur

e

16

..1

8..

.2

0..

.

22 24

.26 28 30 32 34 36

_38 40 42

_ ._

44 46 48-

.50 52

_

54__

56 58 60 65 70 75 80 85 90 _

95 100-

.

105

_11

0-

.11

5_

.._

120

125

130

135

14

0..

.14

515

0

CO

C

O

°(P

=1)

°(

p=l)

R -2.0

84

3-2

.02

72

-1.9

73

1-1

.92

18

-1.8

73

0-1

. 82

65-1

. 78

20-1

.73

95

-1.6

98

7

-1.6

59

4-1

.62

17

— 1

585

3-1

.55

02

-1.5

16

3

-1.4

36

3-1

.36

22

-1.2

93

2-1

.22

86

-1.1

68

0-1

.11

09

-1.0

56

8-1

. 00

55-0

.95

67

-.9

10

2-

. 86

57

-.8

23

2-.

78

24

-.

7431

-.

7054

-.

6690

-.

6339

-.

6000

P=

22

0

0.39

1.3

76.3

64.3

55

.348

.343

.338

.333

.328

.325

.320

.317

.314

.312

.305

.298

.291

.286

.281

.278

.274

.270

.267

.264

.262

.259

.258

.257

.255

.25

3.2

52.2

51

240

0.42

4.4

08.3

95.3

86

.379

.374

.36

8.3

63

.358

.354

.350

.346

.343

.341

.334

.326

.320

.314

.308

.304

.301

.297

.294

.290

.287

.28

5.2

84.2

82

.280

.278

.277

.275

TA

BL

E

22

260

0.45

7.4

41.4

27.4

17

.409

.404

.399

.393

.388

.383

.379

.37

5.3

73.3

71

.363

.356

.348

.342

.336

.332

.328

.324

.320

.317

.313

.311

.309

.307

.305

.303

.301

.300

Ent

ropy

dif

fere

nces

div

ided

by

R,

for

norm

al

s°—

sR

280

0.48

9.4

73.4

58.4

48

.439

.434

.428

.422

.417

.413

.408

.405

.402

.401

.393

. 38

5

.377

.370

.364

.359

.355

.351

.347

.344

.340

.337

.335

.333

.331

.329

.327

.325

300

0.52

0.5

04.4

89.4

78

.470

.465

.459

.45

3.4

47

.44

3.4

38

.434

. 432

.431

.424

.415

.40

7.3

99.3

92.3

87

.383

.379

.37

5.3

71

.367

.364

.361

.359

.357

.354

.352

.35

1

320

0.55

1.5

35

.519

.509

.500

.495

.489

.483

.478

.472

.467

.464

.462

.461

.454

.445

.436

.428

.421

.416

.411

.407

.402

.39

8'

.394

.390

.388

.386

.383

.380

.378

.376

340

0.58

1.5

65.5

50.5

39

.531

.526

.520

.514

.508

.503

.497

.494

.492

.492

.485

.476

.467

.458

.450

.445

.440

.43

5.4

30.4

26.4

21

.41

8.4

15

.412

.409

.407

.404

.402

360

0.61

0.5

94.5

80.5

70

.563

.557

.55

1.5

45

.539

. 53

4.5

28.5

24.5

23.5

23

.516

.507

.497

.488

.480

.474

.469

.464

.459

.454

.449

.445

.442

.439

.436

. 433

.431

.429

H2—

Con

tinu

ed

380

0.6

38

.623

.610

.601

.594

.589

.583

.577

.57

1

.565

.559

.554

.554

.554

.548

.539

.529

.519

.510

.503

.498

.493

.487

.482

.477

.473

.470

.467

.463

.460

.457

.455

400

0.66

6.6

52.6

40.6

32

.626

.622

.616

.609

.603

.596

.590

.586

.585

.586

.580

.57

1

.560

.550

.540

.534

.528

.522

.517

.51

1.5

05

.50

1.4

98.4

95

.491

.488

.485

.482

420

0.69

4.6

82.6

71.6

64

.65

8.6

55

.649

.642

.636

.629

.622

.617

.616

.618

.613

.604

.593

.582

.571

.564

.558

.552

.546

.540

.534

.530

.526

.523

.519

.515

.512

.509

440

0.7

23

.712

.701

.696

.69

1'

.68

8.6

82.6

76.6

69

.661

.654

.649

.648

.650

.646

.637

.62

5.6

14.6

03.5

95

.589

.58

3.5

76

.570

.564

.559

.555

.552

.547

.544

.540

.537

460

0.7

53

.742

.733

.728

.724

.721

.716

.710

.702

.694

.687 68

2.6

81.6

83

.679

.67

0

.659

.647

.635

.627

.620

.614

.60

7.6

00.5

94

.589

.584

.580

.576

.572

.568

.56

5

180

0.78

4.7

74.7

66.7

62

.758

.75

5.7

50.7

43

.736

.728

.720 71

5.7

15

.717

.714

.70

5

.692

.680

.667

.659

.652

.64

5.6

38

.63

1.6

24

.619

.614

.610

.60

5.6

01.5

97

.593

500

0.81

6.8

07.8

00.7

96

.792

.789

.784

.777

.769

.762

.754 74

9.7

49.7

51

.749

.739

.727

.714

.700

.691

.684

.677

.670

.662

.65

5

.649

.644

.640

.634

.630

.626

.622

I §

155

..160

165

170

180

190

200

210

220

230

240

250

260

270.

..28

0--.

300.

320

340

360

380.

400

420

440

460

480.

500

520

540

560

580

-

600

—. 5673

-. 5355

-.50

47-.

4749

-.41

77

-.36

37-.

3124

-.26

36-.

2170

-. 1726

-.13

00-. 0892

-.05

00-.

0123

+. 0241

.0931

.1576

.2183

.2754

.3295

.3808

.4296

.4761

.5206

.5631

.6039

.6432

.6809

.7173

. 7524

.7863

C P-,

)-S°

R

.249

.248

.246

.244

.240

.237

.234

.232

.230

.227

.224

.221

.219

.2172

.2151

.2113

.2078

.2047

.2018

.1991

.197

.194

.192

.190

.188

.186

.185

.183

.182

.180

.179

5.3936

.274

.272

.270

.267

.263

.259

.256

.254

.252

.249

.246

.243

.240

.2381

.2358

.2316

.2277

.2242

.2211

.2181

.215

.213

.210

.208

.206

.204

.202

.200

.199

.197

.195

5. 4806

.298

.297

.294

.291

.287

.282

. 279

.277

.274

.271

.268

.264

.262

.2593

.2567

.2520

.2478

.2440

.2405

.2372

.234

.231

.229

.226

.224

.222

.220

.218

.216

.214

.212

5. 5607

.323

.321

.319

.316

.310

.306

.302

.300

.297

.293

.290

.286

.283

.2806

.2778

.2727

.2680

.2638

.2600

.2565

.253

.250

.247

.245

.242

.240

.238

.235

.233

.231

.230

5. 6348

.349

.346

.344

.340

.334

.329

.326

.323

.320

.316

.3127

.308

. 305

.3022

.2991

. 2935

.2885

.2839

.2797

.2759

.272

.269

.266

.263

.260

.258

.255

.253

.251

.249

.247

5. 7038

.374

'.372

.369

.365

.359

.353

.349

.347

.343

.339

.335

.331

.327

.3240

.3206

. 3145

.3091

.3041

.2996

.2955

.292

.288

.285

.282

.279

.276

.273

.271

.269

.266

.264

5. 7683

.400

.397

.394

.390

.383

.378

.373

.370

.367

.362

.358

.353

.350

.3460

.3424

. 3358

.3299

.3245

.3197

.3152

.311

.307

.304

.300

.297

.294

.291

.289

.286

.284

.282

5. 8289

.426

.423

.420

.415

.408

.402

.398

.394

.391

.386

.381

.376

.372

.3683

.3644

.3572

.3508

.3451

.3399

. 3351

.331

.327

.323

.319

.316

.313

.310

.307

.304

.302

.299

5. 8861

.453

.450

.446

.441

.433

.427

.422

.419

.415

.410

.405

.399

.395

.3908

.3866

.3789

.3720

.3658

.3603

. 3551

.350

.346

.342

.338

.335

.331

.328

.325

.322

.319

.317

5. 9402

.479

.476

.472

.467

.458

.452

.447

.443

.439

.434

.428

.423

.418

.4136

.4090

. 4008

.3934

.3868

.3808

.3753

.370

.366

.361

.357

.353

.350

.346

.343

.340

.337

.335

5. 9915

.506

.503

.498

.493

.484

.477

.472

.468

.464

.458

.452

.447

.441

.4366

.4317

.4228

.4149

.4079

.4015

.3957

.390

.385

.381

.376

.372

.369

.365

.362

.358

.355

.352

6.0403

.534

.530

.525

.519

.510

.503

.498

.494

.489

.483

.477

.471

.465

.4598

.4546

.4451

.4367

.4292

. 4224

.4162

.410

.405

.400

.396

.392

.388

.384

.380

.377

.374

.370

6.0868

.561

.557

.552

.546

.537

.529

.523

.519

.514

.508

.501

.495

.489

.4833

.4777

.4676

.4587

.4507

.4434

.4369

.431

.425

.420

.415

.411

.407

.403

.399

.395

.392

.389

6.1312

.589

.585

.579

.573

.563

.555

.550

.545

.540

.533

.526

.519

.513

.5071

. 5011

.4904

.4808

.4723

.4647

.4577

.451

.445

.440

.435

.430

.426

.421

.418

.414

.410

.407

6.1738

.617

.613

.607

.600

.590

.582

.576

.571

.566

.559

.552

.544

.538

.5311

.5248

.5133

.5032

.4942

.4861

.4787

.472

.466

.460

.455

.450

.445

.441

.436

.432

.429

.425

6.2146

TAB

LE 2

3.

Ent

halp

y of

ide

al g

as m

inus

en

thal

py o

f re

al g

as a

t th

e sa

me

T a

nd p

, div

ided

by

RT

, fo

r no

rmal

H2

Tem

pera

ture

16-

18-

20-

22..

24-

26-

28-

30-

32-

34_.

36-

38-

40-

44_.

46-.

48-.

50-

52-.

54-.

56..

58-.

60_.

65..

70-.

75-.

80..

85_

90-

100-

105-

3 11

5-Q

12

0-.

*"*

125.

2,

130-

.

°K

135.

140-

145-

150.

II

°-H

RT

= lA

mag

at

0.02193

.01869

.01614

.01420

.01263

.01132

.01023

.00931

.00851

. 00781

.00719

. 00665

.00617

. 00576

. 00538

.00504

.00473

00445

.00419

.00396

. 00376

. 00356

.00337

. 00296

.00261

.00231

. 00205

. 00184

. 00165

. 00148

. 00133

.00119

.00107

. 00097

. 00088

. 00079

.00071

. 00064

. 00057

. 00051

.00045

0.04381

.03725

.03226

.02838

.02524

. 02260

.02045

.01860

. 01701

.01556

.01436

.01329

.01234

.01151

.01077

.01008

.00946

.00837

.00792

.00750

.00710

.00674

.00592

. 00522

. 00461

.00411

. 00368

.00330

.00296

.00265

.00238

. 00215

.00194

.00175

. 00158

.00143

. 00128

.00115

. 00102

.00090

0. 06568

. 05583

.04835

.04253

.03783

. 03388

. 03063

. 02786

. 02549

.02337

.02153

.01991

.01850

.01727

.01615

.01513

.01420

.01324

.01257

.01188

.01126

.01066

. 01010

.00887

.00782

. 00692

.00616

. 00551

. 00495

. 00444

.00357

.00322

.00290

.00262

.00236

. 00213

.00193

. 00172

. 00154

. 00136

6

0.11144

.09650

. 08489

. 07551

. 06765

. 06116

. 05564

. 05089

. 04666

. 04298

. 03976

. 03692

. 03445

. 03223

.03020

. 02834

.02665

.02509

.02371

.02247

.02127

.02017

.01771

.01542

. 01382

.01230

.01101

. 00987

.00884

.00793

.00712

. 006

42

. 00579

.00523

. 00472

. 00425

. 00382

.00341

. 00304

.00269

10

0.1604

. 1411

. 1255

.1125

. 1017

.0925

. 0846

.0776

.0714

.0661

.0614

. 0573

.0536

. 050

2.0471

.0443

.0417

.0393

. 0374

.0354

. 0335

.02943

. 02595

. 02297

. 02043

. 01828

.01639

.01469

.01316

.01182

.01065

.00960

.00867

.00781

.00704

.00632

.00563

.00502

.00444

20

0. 2494

.2237

.2023

. 1839

.1681

.1541

. 1420

.1313

.1220

.1140

. 1067

.1000

.0937

.0881

.0830

.0784

.0743

.0704

.0666

.0584

. 0515

.0456

.0406

.0363

.0325

. 0291

.0261

.0234

.0210

.0190

.0171

.0154

,0139

.0124

.0110

.0098

.0087

40

0. 4420

.3994

.3637

.3318

.3042

.2801

.2592

.2409

.2249

.2104

.1971

.1849

.1739

.1638

.1548

.1466

. 1389

.1315

. 1153

. 1017

.0900

.0799

.0714

. 0639

.0571

.0511

.0458

. 041

2

.0370

. 0333

.0299

.0268

.0239

.0212

.0186

.0165

60

0.591

. 538

.491

.451

.414

.383

.357

.333

.312

.292

.274

.257

.243

.229

.217

.206

. 194

.1706

.1503

. 1330

.1182

.1055

.0942

.0842

.0752

. 0673

.0603

.0540

.0485

.0434

.0388

.0345

.0305

.0269

.0235

80

0.708

.646

.593

.545

.504

.469

.438

.410

.384

.360

.338

.319

.302

.286

.271

.256

.2242

.1974

.1747

.1552

.1384

. 1235

.1101

.0983

.0878

. 0785

.0701

.0628

.0561

. 0499

.0442

.0389

.0340

.0295

100

0.872

.797

.731

.673

.622

.578

.539

.505

.473

.444

.417

.394

.373

.353

.333

.315

.276

.243

.215

.191

.170

.152

.135

.120

.107

.095

.085

.076

.068

.060

.053

.046

.040

.035

120

0. 943

.865

.796

.736

.684

.638

.598

.560

.526

.494

.466

.441

.417

. 394

.373

.327

.287

.254

.225

.201

.179

.159

.141

.125

.111

.099

.088

.078

.069

.061

.053

.045

.039

140

1.085

0.995

.916

.847

.787

.734

.688

.645

.605

.569

.537

.507

.479

.453

.429

.376

.330

.291

.258

.230

.204

.181

.161

.143

.126

.112

.099

.088

.077

.087

.058

.050

.042

160

1.222

1.121

1.032

0.955

.887

.828

.775

.727

.682

.641

.605

.571

.540

.511

.483

.423

.371

. 327

.290

.257

.229

.203

.179

.159

.140

.124

.109

.097

.084

.073

.063

.053

.044

180

1.355

1. 243

1.145

1.059

0.984

.918

.860

.806

.757

.711

. .671

.632

.598

.566

.536

.469

.411

.362

.320

.283

.252

.223

.197

.174

.153

.135

.118

.104

.090

.078

.066

.055

.045

200

1.361

1.253

1.160

1.078

1.005

0.942

.883

.829

.779

.734

.692

.654

.619

.586

.513

.449

.395

.349

.308

.273

.241.

.213

.187

.165

.144

.126

.110

.095

.081

.068

.056

. 015

155.

160.

165.

170.

180.

190.

200 :

210.

220.

230.

240.

250.

260.

270.

280.

300.

320.

340.

360.

380.

400.

420.

440.

460.

480.

500.

520.

540.

560.

580.

600

.00040

.00035

.00031

.00026

. 00017

.00010

. 00003

—.00

003

-.00

008

-. 00013

-. 00018

-. 00023

-.00

026

-.00

0290

-.00

0321

-.00

0375

-.00

0422

-.00

0462

-.00

0496

-.00

0526

-.00

055

-.00

058

-.00

060

-.00

062

-.00

063

-.00

065

-.00

066

-.00

067

-.00

068

-.00

069

-.00

070

. 00081

. 00071

. 00061

.00052

.00034

.00019

.00006

-. 00005

-. 00016

-.00

027

-.00

036

-. 00046

-.00

053

-.00

0584

-. 000645

-.00

0752

-.00

0845

-.00

0926

-.00

0994

-.00

1054

-.00

111

-.00

116

-.00

120

-.00

123

-. 00127

-.00

130

-.00

132

-.00

135

-.00

137

-.00

139

-.00

141

00121

00106

00092

00076

00050

00028

00008

00009

00024

00041

00056

00068

00078

000877

000969

001131

001271

001390

001493

001584

00166

00174

00180

00185

00190

00195

00199

00202

00205

00208

00211

.00239

.00209

.00180

.00152

.00100

.00053

. 00015

-. 00020

-.00

051

-.00

084

-.00

113

-.00

138

-.00

159

-.00

1770

-.00

1953

-.00

2276

-.00

2556

-.00

2794

-.00

3000

-.00

3181

-.00

334

-.00

348

-.00

360

-.00

372

-.00

382

-.00

390

-.00

398

-.00

406

-.00

412

-.00

418

-.00

424

.00393

.00344

. 00295

.00248

. 00161

.00085

.00021

-.00

037

-.00

091

-.00

143

-. 00192

-.00

233

-.00

268

-.00

2985

-.00

3290

-.00

3827

-.00

4291

-.00

4688

-.00

5031

-.00

5332

-. 00560

-.00

583

-.00

604

-.00

622

-. 00639

-.00

653

-.00

667

-.00

679

-.00

689

-.00

699

-.00

708

.0076

.0067

.0057

.0047

.0030

.0015

.0002

-.00

09-.

0020

-.00

30-.

0040

-.00

48-.

0055

-.00

615

-.00

675

-.00

782

-.00

375

-.00

954

-.01

022

-.01

081

-.01

13

-.01

18-.

0122

-.01

26-.

0129

-.01

32

-.01

35-.

0137

-.01

39-.

0141

-.01

43

.0144

.0125

.0105

.0087

.0052

.0022

-.00

03-.

0026

-.00

47-.

0068

-.00

88

-. 0104

-.01

18-.

0129

9-.

0142

1-.

0163

4

-.01

816

-.01

973

-.02

108

-.02

226

-.02

33

-.02

42-.

0250

-.02

57-.

0264

-.02

70

-.02

75-.

0279

-.02

84-.

0287

-.02

91

.0204

.0174

.0145

.0117

.0066

.0021

-.00

17-.

0051

-.00

82-.

0113

-.01

42

-.01

66-.

0187

-.02

058

-.02

238

-.02

556

-.02

826

-.03

058

-.03

259

-.03

433

-.03

59

-.03

72-.

0384

-.03

95-.

0404

-.04

13

-.04

20-.

0427

-.04

33-.

0439

-.04

44

.0254

.0214

.0176

.0139

.0070

.0011

-.00

59-.

0084

-.01

26-.

0166

-.02

04

-.02

36-.

0265

-.02

891

-.03

131

-.03

549

-.03

907

-.04

213

-.04

478

-.04

707

-. 0491

-.05

09-.

0524

-.05

38-. 0550

-.05

62

-.05

71-.

0580

-.05

88-.

0596

-.06

02

.029

.024

.020

.015

.0065

-.00

08

-. 0070

-.01

26-.

0178

-.02

28-.

0274

-.03

14-.

0349

-.03

802

-.04

099

-.04

620

-.05

062

-.05

440

-.05

766

-.06

049

-.06

30

-.06

52-.

0671

-.06

88-. 0703

-.07

16

-.07

28-.

0739

-.07

49-.

0758

-.07

66

.033

.027

.021

.015

.005

-. 004

-.01

1-.

018

-.02

4-.

030

-. 035

-.04

0-.

044

-.04

79-.

0515

-.05

77

-.06

29-.

0674

-.07

13-.

0746

-.07

76

-.08

01-.

0824

-.08

44-. 0862

-.08

77

-.08

92-.

0904

-.09

16-.

0926

-.09

35

. 035

.028

.021

.014

.003

-.00

8

-.01

6-.

024

-.03

1-.

038

-.04

4

-.05

0-.

054

-.05

87-.

0G28

-.06

99

-.07

60-.

0312

-.08

56-. 0395

-.09

28

-.09

58-.

0984

-.10

06-. 1027

-.10

45

-.10

61-.

1075

-.10

88-.

1100

-.11

10

.036

.028

.020

.012

-.00

1-.

012

-.02

2-.

031

-.03

9-.

047

-.05

4

-.06

0-.

065

-. 0703

-. 0749

-. 0830

-.08

99-.

0957

-.10

07-.

1C51

-.10

88

—.

1121

-.1150

-.1176

-.1199

-.1218

-.

1236

-.1252

-.1267

-.

1280

-.1291

.036

.027

.018

.010

-.005

-.018

-.029

-.039

-.047

-.056

-.064

-.071

-. 077

-.08

28-.

0879

-. 0970

-. 1046

-.11

11-. 1166

-. 1214

-. 1256

-. 1292

-. 1324

-.13

52-.

1377

-. 1399

-.14

19-.

1436

-.14

52-. 1466

-.14

78

.035

.025

.015

.006

-.01

1-.

025

-.03

7-.

048

-.05

7-.

067

-.07

6

-.08

3-.

090

-.09

62-.

1018

-.11

18

-.12

02-.

1273

-.13

34-.

1386

-.14

32

-.14

71-.

1506

-. 1536

-.15

63-.

1587

-.16

08-. 1627

-.16

43-. 1658

-. 1671

CO

TAB

LE 2

3.

Ent

halp

y of

ide

al g

as m

inus

ent

halp

y of

rea

l ga

s at

the

sam

e T

and

p,

divi

ded

by R

T,

for

norm

al H

%—

Con

tinue

d

Tem

pera

ture

16..

18..

20..

22..

24..

26..

°K

H°—

HR

T

p=

22

0

240

260

280

300

320

340

360

380

400

420

440

460

480

500

32..

34..

36..

38..

40..

42..

44..

46_.

50.

52.

54.

56_

58_.

60..

65..

70_.

75..

80..

85.

90..

95

...

100.

.10

5..

110.

.

115.

120.

125.

130.

135.

MO

-

1.50

..

1.47

51.

359

1.25

71.

168

1.09

01.

021

0.95

7.8

98.8

44

.796

.750

.708

.670

.63

5.5

55.4

86

.427

.376

.331

.29

3

.259

.228

.200

.17

5

.152

.132

.11

5

.068

.056

.044

1.58

21.

461

1.35

11.

256

1.17

11.

097

1.02

80.

965

.907

.855

.806

.761

.720

.682

.596

.521

.457

.401

.35

3.3

12

.274

.241

.211

.184

.159

.137

.11

8.1

00

.084

.06

8.0

54.0

41

1.69

21.

559

1.44

21.

340

1.250

1.170

1.097

1.029

0.968

.911

.859

.811

.767

.727

.634

.554

.425

.373

.253

.221

.191

.165

.141

.121

.101

.066

.051

.037

1.793

1. 653

1.529

1.421

1.325

1.241

1.163

1.091

1.025

0.96

6.9

10.8

59.8

12.7

70.6

71.5

86

.511

.44

7.3

91

.344

.301

.26

3.2

29

.19

7

.144

.12

1.1

01

.08

1.0

63

.04

7.0

32

1.89

01.

743

1.61

21.

499

1.398

1.308

1.226

1.150

1.081

1.017

0.958

.904

.855

.811

.706

.615

.536

.468

.408

.357

.312

.272

.235

.202

.171

.144

.121

.099

.078

.059

.041

.026

1.983

1.830

1.693

1.574

1.467

1.373

1.286

1.206

1.133

1.06

61.

003

0.94

7.8

96.8

49.7

39.6

43

.559

.487

.42

3

.322

.279

.240

.204

.17

2.1

44.1

18

.09

5

.07

3.0

53

.03

5.0

18

2.07

11.

912

1.77

01.

646

1. 534

1.436

1.344

1.260

1.183

1.11

21.

046

0.98

7.9

34.8

86.7

70

.580

.50

3.4

36

.329

.284

.24

2.2

05

.17

2.1

41

.11

5.0

90

.06

7.0

46.0

26

.00

9

2.15

41.

991

1.84

41.

715

1.59

91.

496

1.40

01.

311

1.23

1

1.15

61.

087

1.02

50.

969

.919

.799

.69

2

.599

.518

.44

8

.33

5.2

87.2

44.2

04

.13

7.1

09

.08

3

.05

9.0

37

.01

6.0

02

2.23

42.

065

1.91

41.

781

1.66

11.

553

1.45

31.

361

1.27

6

1.198

1.125

1.060

1.002

0. 950

.825

.714

.616

.531

.457

.394

.243

.202

.164

.131

.102

.075

.049

.026

.005

-.015

2.310

2.136

1.981

1.843

1.71

91.

608

1.50

4.4

07.3

19

.. 23

7.1

61

.09

3.0

33

0.97

9.8

49.7

33

.63

1.5

42

.464

.340

.288

.240

.19

7

.15

8.1

23

.09

3.0

64

.014

-.00

9-.

029

2.38

22.

204

2.04

41.

902

1.77

41.

659

1.55

11.

451

1.35

9

1.273

1.195

1.123

1.061

1.005

0.870

.750

.644

.551

.469

.400

.340

.28

5.2

36.1

91

.150

.114

.08

2.0

52

.024

-.0

01

-.0

24

-.0

45

2.45

32.

269

2.10

41.

958

1.82

61.

707

1.59

61.

492

1.39

6

1.307

1. 225

1.151

1.087

1.028

0.889

.764

.654

.557

.47

2.4

00

.33

7.2

80.2

29.1

82

.140

.10

2.0

68

.00

9-.

01

7-.

04

2-.

06

4

2.52

12.

331

2.16

12.

010

1.87

41.

752

1.63

71.

530

1.43

0

1.33

81.

253

1.17

71.

110

1.04

90.

906

.776

.66

2.5

61

.47

2

.33

2.2

74.2

20.1

71

.05

3.0

21

-.00

8-.

036

-.06

1-.

084

2.58

62.

390

2.21

52.

059

1.91

91.

792

1.67

41.

564

1.46

1

1.36

61.

279

1.20

01.

130

1.06

90.

920

.786

.667

.562

.470

.393

.325

.264

.208

.158

.112

.072

.036

.003

-.028

-.056

-.082

-. 106

2.647

2.446

2.265

2.104

1.959

1.829

1.707

1.594

1.489

1.391

1.301

1.219

1.148

1.085

0.931

.793

.561

.466

.385

.315

.252

.195

.142

.095

.053

.016

-.018

-.050

-.079

-.106

-.130

r H rH rH CN CN CN C O C O C O C O C O

f r r r r r r r r r rN fi to a H

r r r r rCO iO to N 00

i i i i i i i i i i I I I I I

r r r r r r r r r r r8 §5 8 S 8 S ,rH CO Tjf tC N 00^ Tfl T * HH Tf< T * • rJH Tf »O iO »O

r r r r rlO *O *O *O *O

I I I I I I I I I I I I I I I

"3 tO tO tO O OO N CO N Ho M •* ffi o « 10 N a H CO CO CO CO i

i 00 »O CMI rH 1>- ©I © rH CO

I I I I I I I I I I

CM

r r r f r I I I I I I I I I I

r r r r r rto orH COCO CO

»—t co i—t"* »O t>.CO CO CO

8 S Si I§ gT« < * T* T*< HH

r r r r ri i i i i i i i i i i i i i

r r r r r r r r r r rCO CO CO CO CO CO CO CO

r r r r r r r r

r—c CO IO CO !>•CN CN CN CN CN

i' r r r rr r i i i i i

3 S :

r r r r r r r r r rI CO H N I

CO CO CO < CO CO CO CO CO

\ \ \ \ \i i i r r r r r I I I I I

) *O © rH OS CO C^ t-I H •* C N a H N

I I I I I I I I I I I I I i I ICO CO CO CO CO

r r r r rCO CO CO CO CO

r i* r r r

) O i—I CO) t^ t^ 1^) CO CO CO

I I I I I

r r r r r r r r r r rCM (N CO W COCO CO CO CO CO

^ i« N

i i I I I I I I I I I I I I I I I I

iO 00 O Cl "^ *-O !>• 00

r r r r r f r rr(< to a o NH H H N NCO CO CO CO CO

I I I I i I I I I I I I I I I I I I I I !

i d 0C O 00 tl OS CO iI TH tO OS © CM CO "5 'i © © © rH rH rH rH •

CM CO O t ^ r - l

i CV| CO TH " 3 tO § S S i ^i i i i i i i i i i i i i i i. i i i i i i i i i i i i i i

io to

r rI Tt CN O TH O I O) O C

1^ CS O t>» CO

r r r r r ra co N H •** io io to toCM CM CM CM CM

f r r r rtO O H N *(O (D N N N<M CM CN IM CM

r r r r ri i i i i r r

© © © © ©

1O CO t o 00 CN rH OJ

S r - l - H r H i H rH . CM CN CM CN

r r r r r r r r r rS 1 1 I CN CN

1" \r r r r r r r i i i i i i i

© © © © O © © © © r H r H

r r i

rH HH OS "<*l CO 00 CN£Q ^ CM OS i—' CM rH 00 *O •C N C O T t H H H C O N 00 00 O)

I o m N O!I <M (M (N CM

I I I I I I I I I I I I I I I I I I I I I I I I

* (N O5 »Q HiC O (O N OC

I I I I I I I I I I i i r r r r

l r H r H t > . C O (O N X © II r H O r H C O • * IO O N (i © © © © © © © © <

tO T)< 6 tC N © Th H tO HO J O r H r H C M CO T( ( IO >O tD

r r r r r r r r r rS r H r H r H r H 0 0 0 0 0 0 0 0 0 0

r r r r r r r r r ri i i i i i i

i © N •* to 00 8 M '


Values of F/RT, E/RT, and ln(//P) may beobtained rather simply from values of S/R andH/RT and the Z-table in accordance with thefollowing equations:

F/RT=(H/RT)-(S/R)

P~ RT

E/RT=(H/RT)-Z.

(5.13)

(5.15)

This may be carried out using the (d2Z/dT2)p

table (table 16), and a method of tabularintegration. Table 23 may be used to obtain

(P[T(dZ/dT)p/p]dP, since from eq 5.12 it followsJothat

j; [T(dZ/dT)p/p]dP =

—^2^ (from table 23) + (Z-1 ) . (5.16)

The value of [F°p, Tu<i*i)-Fp§Titni)/RT]may be obtained by subtracting (S°—S)/R, givenin table 22, from (H°—H)/RT, given in table 23.

The calculation of the heat capacities of the realgas involves the evaluation of

£[T2(d2Z/dT2)p/p]dP.

In the temperature and density ranges where Zmay be represented by an analytic expression/2

these two integrals may be evaluated by usingseries expansions for Z and its derivatives in theintegrands. The difference between the specificheats at constant pressure for the real and idealgas states may be calculated using the equation

d (dP\ l(dP\ ( d fRT

H°-HRT (5.17)

The derivatives in eq 5.17 may be calculated fromtables 14 and 23, using a method of tabular differ-entiation. Except for the first term, the deriva-tives in eq 5.17a are given in tables 15 and 17.

1014

1.012

1.010

1.008

1006

1.004

1002

IDOO/

/

/

//

/

/

/

FIGURE 12. Effect of density on specific heat of H2 at 50° C.

Figure 12 shows the dependence of the specificheat at constant pressure for hydrogen at 50° Cupon the Amagat density p. The curve representsthe results of the evaluation of formula 5.8, using

the PVT correlation of this paper. The plottedpoints are observations by Workman [49]. Noother direct experimental data on the effect ofpressure upon the specific heat at constant pressureare available for hydrogen.

An indirect indication of the effect of pressureon the specific heat of hydrogen is found in thework of van Itterbeek [78], who used the results ofvan Itterbeek and Keesom [77] on the effect ofpressure on the velocity of sound in hydrogen atliquid hydrogen temperatures. The results ofvan Itterbeek at a pressure of one-tenth of anatmosphere indicate that the increase of Cp withpressure above the zero-pressure value agreeswith the PVT prediction within 3 percent at17.5° K and at 19.0° K, but is lower by morethan 30 percent at 20.5° K. At pressures abovey2 atm at 20.5° K, this difference in heat capacityhas become approximately 0.1 cal deg"1 mole"1,but this discrepancy is reduced by roughly 50percent if the data of van Itterbeek and Keesomare evaluated with values of Cp—Cv based onthe PVT tables of this paper.

12 Up to p=500 at temperatures above 0° C, the equation Z=has been used. This is eq 4.8 and eq 4.9 is its series expansion. The symbolsstand for functions oi T, which are given by eq 4.11 and 4.12.

From P=0 to p=200 and T=14° to 56° K Z can be expressed by Z=\—(^4/T3/2)/O-(C/T3/2)p2, which is equivalent to eq 4.14. The symbols A andC stand for functions oi T, whose values are tabulated in table 19.


The specific heat of hydrogen at constantvolume has been determined by Eucken [169]for various combinations of temperature anddensity in the ranges 35° to 110° K and 60 to150 Amagats.

Joule-Thomson coefficients of hydrogen may beof interest. These may be calculated from eq 4.6.For this calculation there are required: the valueof Cp which may be calculated using eq 5.8 or5.17. Values of Z, (dZ/dT)p, and (dZ/dp)T aregiven explicitly in tables 13, 15, and 17. Byusing values of Cp for H2 at 50° C derived fromfigure 12, the following values of \x for 50° C wereobtained by calculation: at p=2Q, n=— 0.0350deg atm"1; p=40, /x= —0.0364; p=60,»= —0.0378;p=80, fx=— 0.0390, and p=100, M=— 0.0402.By extrapolation, one obtains for ju at p=0 thevalue —0.0335.

There are no accurate measured Joule-Thomsondata for hydrogen for 50° C with which thesecalculated values of JJL may be compared.

Results of measurements on Joule-Thomsoneffects in hydrogen and deuterium at liquid airand room temperatures have been publishedrecently by Johnston and coworkers [57, 58],with curves showing calculated values for hydro-gen based on the tables of this paper.* Consider-ing that the Joule-Thomson coefficients are notobtained with great simplicity from the PVTdata and depend sensitively on the trends of therepresentation, the agreement is considered fairlysatisfactory.

The location of the inversion curve for theJoule-Thomson effect in hydrogen on a p-T graphmay be determined from tables 15 and 17 by find-ing values of p and T for which T(dZ/dT)p=p(dZ/'dp)T, in accordance with eq. 4.6.

An expression for ix in terms of derivatives of theenthalpy, H, is

(dH/dP)TH\ (dP\ _/dH\/dP\dp)T\dTjp \dTJP\dp)7

(5.18)

In accordance with this equation the inversioncurve may be determined by inspection of the(H°—H)/RT table (table 23), since M = 0 where

/d(Ho-H)/Rdp T-\

= 0. (5.19)

*The tables of this paper were completed before the papers by Johnstonand coworkers [57, 58] on the Joule-Thomson coefficients of H2 and D2 appeared.Our correlation of PVT data would doubtless have been better if these Joule-Thomson data had been available at the time the correlation was made.

The heavy curve in figure 13 is the inversioncurve of hydrogen as given by the correlation ofthis paper. In locating it, values of P were deter-mined with the help of table 14. For temperaturesbelow 75° K some extrapolation beyond the limitof the tables was necessary. In this extrapolatedregion the a versus p diagram, figure 6, was workedwith, and the relation for the inversion curve onthis diagram was used to get the extrapolated partof the inversion curve directly from the a versus pdiagram.

In a Joule-Thomson expansion of hydrogen atconstant temperature from a high to a very low

80 120 160 200P, Atmospheres

FIGURE 13. Curves related to the Joule-Thorns oncooling of H2.

density, approaching zero density, there is achange in enthalpy equal to (H°—H). In figure13 the curves that cross the inversion curve hori-zontally are curves of constant H°—H. As H° is afunction of temperature, these constant (H°—H)curves are not isenthalpics.

The horizontal crossing of the inversion curveby the (H°—H) curve is related to the fact that /x,which is zero along the inversion curve, is equalto (dH/dP)T/CP, which means that along the inver-sion curve (dH/dP)T is zero. The enthalpy change(H°—H) is equal, very nearly, to the amount ofrefrigeration, per mole of gas, available for theliquefaction of hydrogen in a Hampson or Lindelow pressure type of hydrogen liquefier in which a


continuous flow of gaseous hydrogen is allowed toexpand from a high to a low pressure withoutdoing work against an external force system.The fraction x of the high pressure hydrogen flowthat might, theoretically, be liquefied is

X=-H'-H H'-H

(5.20)

where H and H' are the enthalpies of the com-pressed and expanded hydrogen at the tempera-ture at which the compressed hydrogen leaves theprecooler and enters the last stage interchangerbefore expansion; Lv is the heat of vaporization ofliquid hydrogen at the boiling temperature deter-mined by the pressure of the expanded hydrogen ;and (Hvap—Hlig)=Lv is the difference in enthalpiesof saturated vapor and liquid in equilibrium at thepressure of the expanded hydrogen. Only a rela-tively small error is made in x if in place of H'and Hvav for the real gas at atmospheric pressureone uses the enthalpies H° and H°vap of hydrogenin the ideal gas state at the same temperatures aswould be used for H' and H7On.

x=H°-H

H°-H°vap+Lv(5.21)

For a temperature of precooling equal to 65° K,the error introduced by the approximation isabout 0.5 percent.

The lines of figure 13 that are roughly parallelto the inversion curve and converge with it atthe inversion point, 204.6° K, are lines showingthe pressure at which H°—H has reached a givenfraction of its maximum value for the giventemperature. As the inversion curve is the lineof maximum values of (H°—H) it is also theICO-percent line in this family of constant per-centage lines.

In the free expansion of a continuous flow ofgas not doing work against an external forcesystem, the maximum refrigeration is obtained byexpanding from the inversion pressure for thegiven temperature of the compressed gas. Thecurves of constant percentage of maximum valuesof (H°—H) are also curves of constant percentageof the maximum available refrigeration in anexpansion to low pressure.

Figure 13 makes apparent how greatly therefrigeration and the fraction of hydrogen liquefied(eq 5.21) by a Hampson type liquefier are increased

by lowering the temperature of the compressedhydrogen before it enters the final interchangerfrom which expansion of the hydrogen takes place.It is also seen that the condition of highest inversionpressure (92° K and 165 atm) is by no means themost favorable condition for liquefaction; a furthercooling of the compressed hydrogen by 32 degreesnearly doubles the refrigeration produced and morethan doubles the fraction liquefied. It is also seenfrom figure 13 that for the usual range of tempera-tures (55° to 90° K) to which compressed hydrogenis precooled before expansion in a Hampson-typeliquefier, about 95 percent of the maximum refrig-eration is obtained when the pressure of the com-pressed gas is only 75 percent of the inversionpressure.

VI. Viscosity and Thermal Conductivity

1. Viscosity and Thermal Conductivity of the GasNear Atmospheric Pressure

(a) Hydrogen

Values for the viscosity of gaseous normalhydrogen at atmospheric pressure for tempera-tures above the boiling point and at saturationpressure for two temperatures below the boilingpoint are given in table 24. These were calcu-lated using the empirical equation

poises (6.1)7 = 85.558X10"

for the viscosity at very low pressure,13 togetherwith values for the small differences betweenviscosities at atmospheric or saturation pressureand at very low pressure (see eq 6.17 and 6.16).The four constants of eq 6.1 were chosen on thebasis of experimental data near 20°, 90°, 300°,and 685° K. The value used for the viscosityof hydrogen at 685° K was 0.55 percent largerthan the experimental values of Trautz and Zink[99], as their value was based on Millikan's valuefor the viscosity of air which is now known to below by about this amount.

In figure 14 are plotted deviations of recentexperimental viscosity data from eq 6.1. Nochanges were made in the experimental data for

13 This viscosity at very low pressure is a true or bulk viscosity. The pres-sure effect mentioned here is not the familiar low pressure effect on theapparent experimental viscosity involving the accommodation coefficientand the limited size of experimental apparatus.


TABLE 24. Viscosity of gaseous hydrogen (H2)

T

°K1020304050

60708090 . .100 ._ . .

110 ._120130140150

160170180190200

210220230240250

V

Poises51.0X10-7

109.3160.7206.8248.9

287.6323.8357.9390.3421.1

450.8479.3507.0533.8559.8

585.2610.0634.3658.1681.4

704.3726.9749.0770.9792.4

T

°K260270 . . .280290 __300

310320330340 . .350

360370380 ._390400

420440 _.-460480500

520540560580.—600

V

Poises813.6X10~7

834.6855.3875.8896.0

916.0935 8955.4974.8994.0

1,0131,0321,0511,0691,087

1 1241,1601,1951,2301,264

1,2981,3311,3641,3971, 429

T

°K620640 .660680700

720 -740760780800

820840860880900

9209409609801,000

1,0201,0401,0601,0801,100

V

Poises1,461XMH1,4931 5241, 5551,585

1,6161 6461,6751,7051,734

1,7631,7921,8201,8481,876

1,9041,9321,9591,9862,013

2,0402,0662,0922,1182,144

the differences in density. Deviations of table24 values from eq 6.1 are represented in figure 14by the peaked curve, which is appreciably abovethe zero line between 10° K and 100° K and invery close agreement with it at higher tempera-tures. This peaked curve represents the viscosityat atmospheric pressure above the boiling pointand at saturation vapor pressure below the boilingpoint. Different reported values of viscosity atlow temperatures are so poorly in agreement thattheir comparison does not indicate the magnitudeof the peak, which has accordingly been obtainedfrom theory, using data of state. To limit thecrowding of experimental points in the figure,those plotted represent only data published since1928, but a few data obtained after 1928 havebeen omitted. The data of Trautz and co-workers[94 to 102] would be in better agreement with thezero line if increased by about one half percentfor the revision in the value for the viscosity ofair.

It has been pointed out by others that theSutherland formula

does nob fit the data for hydrogen over an extendedrange of temperature. This may be seen in figure14 in which the deviations of the Sutherland for-mula from eq 6.1 are represented by the curve belowthe zero line. The constant C was evaluated at300° K to represent the trend of the best data.

Values of the thermal conductivity of gaseousnormal hydrogen are given in table 25.

TABLE 25. Thermal conductivity of gaseous hydrogen at1 atm

T

°K10.20304050

60-708090100.

110120130140150

160170.180190200

210220230240250

K

cal cm-1

sec-i °C-i14.3X10"6

34.653 570.786.5

101.4116.1130.8145.9161.3

177.0192.9208.8224.6240.4

256.0271.4286.5301.1315.4

329.6343.5357.2370.7384.0

T

°K260270._.280290300

320340360380400

420440. .460480500

520540.560580600_

K

cal cm-1

se<rl °C-!397.0X10"6

409.7422 1434.2446.3

469.8492.8515537559

580601622643664

684705725745766

They were calculated from the equation

£=[1.8341-0.004458 T+ (1.1308 +

0.0008973T)Cp°]~-

v-vT^Y72

T+C(6.2)

(6.3)

In principle, a correction from low pressure toone atmosphere would be applicable, but it hasbeen omitted because the uncertainty of theexperimental values is much greater. In eq 6.3,M is the molecular weight, rj the viscosity givenby eq 6.1, O£ the specific heat in calories per moleper degree at constant pressure, and T the tem-perature in degrees Kelvin. This equation is anempirical representation of the data and was


oo

0

e

o

cc

1o

1

Io

©

©

o

o

o

o

o °

x

o oo

o Adzumi• Johnston a McCloskey© Keesom a KeesomQ Sutherland a Maaso Trautz a coworkers9 van Itterbeek a Claesc van Itterbeek a Paemale Wobser a Muller

x

200 4 0 0 600T °K

800

FIGURE 14. Viscosity of hydrogen.

obtained in several steps, which will be explainedin the discussion that follows.

In figure 15, curve A represents eq 6.3, whereascurves B and C are theoretical and are given forcomparison. Curve C is for Eucken's relation

i=(97-5)tf^/(4M), (6.4)

or its equivalent

k=(C°P+1.25R)r1/M. (6.5)

Chapman and Cowling [137] proposed the formula

(6.6)

(6.7)

which is equivalent to

k=[UnCoP+(3.75-2.5Un)R]v/M.

The transport of internal molecular energy of agas is supposed to be represented better theoret-

ically as a result of including the quantity Un,which is the ratio of mean free path lengths fordiffusion and viscosity.

Un is a pure number whose value was deter-mined theoretically for (1) smooth elastic spheresand (2) for molecules repelling as the inverse fifthpower of the distance (Maxwellian molecules),the values being 1.204 and 1.55, respectively.

For Un equal to 1, curve C is obtained, as eq6.6 and 6.7 then reduce to eq 6.4 and 6.5. CurveB of figure 15 is a graph of eq 6.7 with C/n = 1.4,a value indicated by a group of measurements ofthe conductivity near 300° K. It is evident thatthe main body of the experimental data is notconsistent with a constant value of Un. On thebasis of a value of 1.4 for Un near 300° K and ahigher value at 700° K, as indicated by a curverepresenting the data, the relation

= 1.1308+0.0008973T (6.8)


8

7

7* 6

'oCO

T£ 5O

"5° A

2

1

0

/T

//

•J

•

yy• Archer

oDickins

a Eckerle

©Eucken

« Gregory

« Gregory

e Gunther

c Johnsto

a> Kannulu

« Nothdur

• Schnekj

e Schleier

»Spence

e Vargaft

e Weber

• Winkelrr

n

a Archer

n a Grilly

yy

k a Martin

ft

er

macher

-Gregorya Oock

k a Parfenov

lann

0 100 200 300 400 500 600 700T °K

FIGURE 15. Thermal conductivity of hydrogen.

was adopted. It was found that the curve wasnot critically dependent on the functional form ofUu as a change to Uu = a-\-b^T altered thefinal curve negligibly between 300° and 700° K.

At temperatures somewhat below 100° K, theideal gas specific heat of hydrogen at constantpressure approaches the value (5/2)R character-istic of a monatomic gas. For this value of Cp,the Uu terms in eq 6.7 cancel and eq 6.4 to 6.7reduce to

(6.9)

This equation has been derived exactly for a forcethat at all distances is repulsive and proportionalto 1/r5. Enskog [132] has shown that for attractingrigid spheres (Sutherland molecules),

where C is the Sutherland constant in eq 6.2.Thermal conductivities of hydrogen measured atliquid air temperatures are a few percent lowerthan equations 6.4 to 6.9 would indicate. No the-oretical explanation of this is at hand, but theagreement of the three independent investigationsin this region indicates that the lower value is tobe accepted. To take account of this, a correction

factor 1/(1 + 3.2/T) has been included, having aform suggested by Enskog's theoretical result forattracting rigid spheres but with the constantchosen to fit these experimental data. The in-clusion of this factor also brings the final curvecloser to Eucken's experimental value at 20.96° K,which is still almost 12 percent lower than thecurve.

The curve as chosen to fit the thermal conduc-tivity data is not regarded as completely satisfac-tory. In the temperature range 270° to 400° K,the experimental data appear to fall into twogroups, one quite close to the curve adopted andthe other lower by about 7 percent. The lowergroup includes the most recent data.

Equation 6.4 to 6.9 make it evident that at low-temperatures where the specific heats of ortho andpara hydrogen differ, their thermal conductivitiesdiffer also. This difference in thermal conduc-tivity was the basis of the method of ortho-paraanalysis used by Bonhoeffer and Harteck [121].The temperature or electrical resistance of anelectrically heated wire carrying a given currentdetermines, after calibration, the ortho-para com-position of the hydrogen that surrounds the wirein a tube externally thermostated at liquid airtemperature. A small difference is to be expectedin the viscosities of ortho and para hydrogen byreason of small differences in their intermolecularforces manifested by small differences in vaporpressure, and density of the condensed states.

This difference in viscosities is small and was notdetected in the experiment undertaken by Harteckand Schmidt [122], in which an accuracy of 1percent was attained. In later developments ofthe so-called thermal conductivity method ofortho-para analysis, the pressure of the gas wasreduced to make the mean free path large com-pared with the diameter of the heated wire. Forthis condition the ordinary thermal conductivityis not the controlling factor.

(b) Deuterium

Several investigations have been made of theviscosity of deuterium at atmospheric pressure,the most recent being that of Van Itterbeek andVan Paemel [106, 107], published in 1940. Table26, which gives values for the ratio betweenviscosities of deuterium and hydrogen for severaltemperatures, was taken from the paper by VanItterbeek and Van Paemel.


TABLE 26. Ratio of viscosities for gaseous D2 and H2

r?(D2)/77(H2)

° K2839080 -702015 -12 5

1.401.381.371.361.241.241.24

The ratio of the thermal conductivity of deu-terium at 0° C to the thermal conductivity ofhydrogen also at 0° was determined by C. T.Archer [127] and by W. G. Kannuluik [130], whoobtained respectively, the values 0.7365 and0.7324. By using the mean of these values withappropriate values of Cv and 77, one obtains forUn in eq 6.7 for the thermal conductivity of D2

at 0° C the value 1.55. Archer also measured thethermal conductivity of various equilibrium mix-tures of H2, HD, and D2.

For two isotopic gases with identically the sameintermolecular forces, the classical theory valuesfor the ratio of their viscosities, and the ratio of theirthermal conductivities at temperatures where theirheat capacities are equal are

171/772= and kxik2= X (6.10)

For H2 and D2 these ratios have the values:*7D A?H = 1-414 and kDJku = 0.707, and are inde-

22 22

pendent of the intermolecular force field so longas it is the same for the two isotopes. Thedifference between the rotational heat capacitiesof H2 and D2 at low temperatures by itself makesthe ratio &D2/&H2 larger and thus has an effectopposite to but less than that of the smaller meanvelocity of D2 molecules caused by the greatermass. Using Eucken's eq 6.4 for k and makingallowance for the difference in heat capacities ofH2 and D2, one obtains 0.718 for kDJkU2 at 0° C.The classical theory values for these ratios ofthermal conductivities and viscosities are ap-proached closely at room temperatures. Theeffect of quantum mechanical interaction intransport phenomena can be described in termsof increase in the apparent size of the molecules.In classical theory the size of the molecule playsan important role, the viscosity and thermal con-ductivity decreasing as the size increases. For

hydrogen and deuterium, the quantum mechanicalincrease in apparent size is small at room temper-ature but becomes large at low temperature. Theincrease depends also upon the masses of thecolliding molecules and is larger for H2 than forD2 at the same temperature. It was pointed outin the section on the PVT data for deuterium thatthe quantum theory of second virial coefficientsincludes an effect interpretable classically as anincrease in apparent size of molecules, becomingvery large at low temperatures. The quantummechanically obtained increase in apparent sizewith lowering of temperature is not the same forviscosity as that associated with the second virialcoefficient, however. This is not surprising whenone considers that the increase in the meande Broglie wave length with decreasing tempera-ture increases the diffraction behind a scatteringmolecule; an effect that does not enter in thedetermination of the second virial coefficient, butwhich taken by itself would decrease the apparentsize of a scattering molecule for viscosity.

2. Viscosity and Thermal Conductivity of the Gasat High Pressures

There are no experimental data on the thermalconductivity of gaseous H2 at high pressures.For viscosity, however, experimental data ob-tained by Boyd [134] and Gibson [135] are avail-able. Gibson's data, which are for 25° C, aremore precise than those of Boyd and are plottedin figure 16. It will be seen that there is fairly goodagreement between these better experimentaldata and the curve representing the theoreticalformula due to Enskog. Differing approachesto the problem of relating viscosity and variablesof state will be found elsewhere [133, 136].

In elementary theory, the viscosity and thermalconductivity for a given gas are proportional tothe product of V, p, and A, where V is the meanmolecular velocity, p is the density, and Ais a suitable mean path length for the transferof momentum or energy. Although A is often takenas identical with the ordinary free path of molec-ular motion, it is actually greater by a smalldistance of the order of magnitude of a moleculardiameter, as at each collision the momenta andenergies are transferred an additional distancerelated to the diameters of the molecules involved.Thus instead of A decreasing as 1/p when p isincreased, which would make pA independent of


p, A decreases a little less slowly so as to makepA increase slightly as p is increased. Accord-ingly, both the thermal conductivity and theviscosity of a gas would be expected to increasewith increasing density, particularly when mul-tiple encounters between molecules occur fre-quently as in the case of high densities.

Enskog's theory was developed for a gas whosemolecules were assumed to be mutually attractingrigid spheres, for which the equation of state hasthe form

(6.11)

used by Enskog takes account of simultaneousencounters of three and four molecules as treatedby Boltzmann and Clausius.

According to Enskog's theory, the viscosityand thermal conductivity of a compressed gas arerelated to the viscosity r]0 and conductivity k0

at low pressure by the equations

and

(6.14)

(6.15)

o o

o

/

V/

/

/<o

/

o

e =FIGURE 16. Effect of density on viscosity of hydrogen at 25° C.

where the constants a and b are assumed to beindependent of T and p, and x is a function of pexpressed in the form of a power series in bo.The equation of state that was used is thus almostthe same as the Van der Waals equation

2 +. . . ) ] (6.12)

except for the details of the dependence of xupon p. The Van der Waals equation is derivedon the basis that simultaneous encounters ofthree or more molecules are rare enough to beneglected. Only at low pressures is this validand under this condition terms of the seconddegree and higher in bp are neglected in the deriva-tion. The function

(6.13)

It follows from eq 6.11, the equation of stateassumed for Enskog's theory, that

<••«•>

Thus, the value of bp% may be calculated from thetables of Z and (dZ/dT)p and the value of bp maythen be found with the help of eq 6.13.

Over the range of Gibson's experimental vis-cosity data very little change is made in the valuespredicted if simple power series expansions inbpx, obtained from equations 6.14 and 6.15, areused:

(6.17)

px)3

(6.18)


The coefficient of the last term of each equationwould be changed if higher order terms were addedto eq 6.13, 6.14, and 6.15. Dropping the lastterm of eq 6.17 for 77/770 does not significantlychange the agreement with Gibson's experimentaldata.

In order to show the general magnitude of thetheoretical effect of pressure on the viscosity andthermal conductivity of hydrogen the precedingequations have been evaluated for several addi-tional combinations of temperature and pressure,using data from the PVT tables. Table 27 givesthe values thus obtained. It is seen that thecalculated relative change in 77 and k with pressureis much more pronounced at the lower tempera-tures, for which large deviations from the idealgas law occur even at moderate pressures.

TABLE 27. Effect of pressure on viscosity and thermalconductivity of hydrogen

T

°K182022303038 __

40 ___404050 __5050

6070 __707080 _..9090 ._90

100_110150 _ _..250400600-

P

atm0.455.889

1.5651

2.0430.4

1

2.8037 2

1

3 5550

11

5.0650

11

6.5650 0

111111

v/v0

1.00451 00771. 01261 00371.00861.53

1.00211.00671 531.00151 00601 31

1 00121.00091.00511.111.000751 000651.00471 06

1.000561 000491.000341 000181.000101.00006

klkQ

1.01381 02251 03471 01141 02481 76

1 00681.01991 761 00481 01781 49

1 00371 00301.01551 221 00241 00211 01411 13

1. 00181 00161 00111 00061 00031.0002

3. The Viscosity of Liquid Hydrogen

The first determination of the viscosity ofliquid hydrogen was made in 1917 by Verschaffeltand Nicaise [138] from measurements of thelogarithmic decrement of the oscillatory rotationof a sphere in liquid hydrogen at 20.36° K.

Later, determinations were made of the viscosityof liquid hydrogen from 15° to 20° K, in 1938 byKeesom and Mac Wood [139] from measurementsof the logarithmic decrement of an oscillatingdisc, and in 1939 by Johns [140], using the capil-lary flow method. The reported viscosities are

250

230

210

190

> 150

130

110,

\o

\ ,

9

\ " \

o \

\ \

\ \?\ o -.

Viscosity of Liquid HydrogenVerschaffelt a Nicaise oKeesom a MacWoodJohns o

•ex.

o

15 16 17 18 19Temperature . °K

2 0 21

FIGURE 17. Viscosity of liquid hydrogen.

shown in figure 17. The values obtained byJohns are roughly 10 percent greater than thoseof Keesom and Mac Wood except near the boilingpoint, 20.4° K. There seems to be no clearindication in the papers reporting the measure-ments that either of these two later sets is lessdependable than the other. Accordingly a curveto represent the present most probable values ofthe viscosity of liquid hydrogen was drawnprincipally between the two sets. Near the boil-ing point the curve was drawn approximatelyparallel to that of Johns because it was felt thatthe lower value of Verschaffelt and Nicaise sup-ported the more regular variation of viscosity withtemperature as reported by Johns.

VII. Pressure Temperature Relations forTwo-Phase Equilibria for H2/ HD, andD2 as Single Components

In this section are presented data on (1) vaporpressures of solid and liquid H2, HD, and D2 withsuch derived constants as normal boiling tem-peratures and triple-point temperatures and pres-


sures; differences between the vapor pressures ofdifferent mixtures of o- and ;p-H2; and changes invapor pressures of ortho-para H2 mixtures result-ing from self conversion; (2) the pressure-tem-perature relations for the solid-liquid equilibriumof H2, HD, and D2. The data are presented in theform of equations, tables, and graphs.

1. Vapor Pressures, Boiling, and Triple Points 14

The present vapor-pressure data on the hydro-gens can be fitted with equations of the form

log10P=A+B/T+CT (7.1)

to within the accuracy of the experimental data.The millimeter of Hg at 0° C and standard gravity

14 Boiling-point and triple-point data from this section have been used inadvance of publication in the "Tables of Selected Values of Chemical Ther-modynamic Properties" prepared by the National Bureau of Standards inconjunction with the Office of Naval Research of the U. S. Navy Department.

is used in this section as the unit of vapor pressure.Temperatures are on the Kelvin Scale.

In tables 28 and 29 the vapor pressures, boilingpoints, and triple points of the different isotopicand ortho-para modifications of hydrogen arecompared.

(a) H2

The differences between the hydrogen vapor-pressure data reported in the literature [143 to146, 148] are the result, principally, of differencesin the temperature scales used by different observ-ers and of unknown differences in the ortho-paracomposition of the hydrogen.

The vapor-pressure data recently obtained [146]at the National Bureau of Standards are on thelow-temperature scale established at the NationalBureau of Standards and are for known ortho-

TABLE 28. Vapor pressures of the several isotopic varieties of hydrogen at integral temperatures and at their triple points andboiling points.

[Values marked (*) were obtained by extrapolation of the vapor-pressure equation to temperatures at which no data were available. The 0-H2 table isbased on an extrapolation with respect to composition.]

T

0 K1011

1213

13.81s

13.9571414 051516

16.604171818.69i18 723

192020.27s2O.39o20.454

212222 1332323.52723.57s

20.4° K Equil ib-r ium hydrogen

0. 21 percent0-H2

P

mm Hg1.935.6213.930 252.8

57.458.860 5

100.4161 2

209.3246 2360.6459.8464 9

510.1700.3760786.8801.7

937.01226. 61269 41574. 91784. 41803. 5

State

Solid*___Solid..__

dodo

Triplepoint.

Liquid.,dodo

-__do.___do

- dodo

-__do..___ do

do

do-__do.__-___do_.doLiquid*.

___do_______do.___

do_._do-._____do__ do

Normal hydrogen 75percent 0-H2

P

mm Hg1.7s5.0o

12.727 949.1

54.055.457 095.0

153 3

199.7235 2345.9442.0446 9

490 8675.7733.9760774.4

906.41189.01230 81529. 61734. 51753. 3

State

Solid*Solid

dododo

Triple point _Liquid

dododo

dododododo

dodododo

Liquid*

dodododododo

Orthohydrogen 100percent 0-H2

P

mm Hg

55.192.2

149 1

194.4229.2337.8432.3437 1

480 7662.6720.0745.7760

890.61170. 41211 81508. 41712. 21730. 8

State

Solid . . .dodododo

dodo

Triple point*Liquid*

do

dododododo

dododo

. dodo

dododo

_ . dodo.__...

Normal deuterium66.67 percent

0-D2

P

mm Hg0.05

20.73

2 144.61

5.245.445.6812.325 4

37.948.687.2

126.3128.5

145.1219.9244. 9256.2262. 5

322.2458. 5479. 6636.2749. 3760

State

Solid*dodododo

doSolid

dododo

dodododo

Triple point-

Liquiddododo

Liquid*

- dodododododo

20.4° K Equilibriumdeuterium 97.8percent 0-D2

P

mm Hg0.05..21

.752.204.73

5.375.575.8212.626.0

38.749 688.7

128.5130.3

147 2223.1248.4259.9266.2

326.9465.1486 5645.3760770.6

State

Solid*dodododo

doSolid

dododo

. . . dododo-__ —

Triple point.Liquid

dododo

..._ doLiquid*

_... dodododododo .

Hydrogen deuteride

P

mm Hg0.28.99

2.947.4614.6

16.316.817.534.465.2

92.8112.5176.4234.5237.5

264.7382.8420.9438.1447.7

536.2730.5760972.0

1120.11133. 8

State

Solid.*Solid.

Do.Do.Do.

Do.Do.Do.Do.Do.

Triple point.Liquid.

Do.Do.Do.

Do.Do.Do.Do.

Liquid.*

Do.Do.Do.Do.Do.Do.


453

TABLE 29. Boiling points and triple points of the hydrogens

20.4° K equilibrium hydrogen (0.21% 0-H2)38 percent o-H2, 62 percent p-B~2Normal hydrogen (75% 0-H2)OrthohydrogenNormal deuterium (66.67% 0-D2)20.4° K equilibrium deuterium (97.8% o-D2) _ParadeuteriumHydrogen deuteride

Boilingpoint

0 K20. 273

20.3220.39o20.4523. 57323. 527

23.6622.133

Triple point

0 K13.8I313.8613. 957

14.0518. 723

18.69i18.7816. 6O4

mm Hg52.853.054.055.1

128.5128.5128.592.8

para compositions. Only the NBS results are givenhere.

Normal hydrogen (75 percent o-H2, 25 percent

Liquid: log10P(mm Hg) =4.66687 —

Solid: log10P(mmHg)=4.56488-

(7.2)

(7.3)

20.4°K-equilibrium hydrogen (99.79 percent p-H2,0.21 percent o-H2:

Liquid: log10P(mm Hg)=4.64392 —

Solid: log10P(mmHg)=4.62438-

( 7 4 )

(7.5)

The triple-point temperatures and pressureswere determined experimentally with a low-tem-perature calorimeter with a platinum resistancethermometer for the temperature measurements.Equations 7.2 to 7.5 were made to fit these triplepoints, and are based on vapor pressure dataextending from 10.5° to 20.4° K. Although theequation for liquid normal H2 is based only onNational Bureau of Standards data below 20.4° K,the equation represents, within the limits of exper-imental accuracy, the Leiden data that extendnearly to the critical point, 33.19° K. As men-tioned in section IV, the vapor-pressure equationfor normal hydrogen was used in constructing thePVT relations for hydrogen. The experimentallydetermined triple-point temperatures and pres-

sures for TI-H2 and e-H2 are given in tables 28and 29.

Figure 18 is a diagram of differences between thevapor pressures of a 20.4°K equilibrium mixtureof o- and p-H2 (0.21 percent o-H2) and five differentmixtures of o- and p-H2 in the liquid state. Thevapor pressure of the 20.4°K equilibrium mixtureis denoted by P{e-H2 )and that of any other mixtureby ^mixture)- Each curve of the graph is for asingle mixture whose composition is indicated onthe graph by its o-H2 composition. The 75 percentcurve is for normal hydrogen. The vapor pressuredifferences AP are plotted as a function of thevapor pressure of the 20.4°K equilibrium hydrogen.The circles represent the experimental data.

Figure 19 shows the vapor pressure differencesof figure 18 extended into the solid range, for mix-tures of 38 and 75 percent ortho composition. Atthe extreme right of the figure, these mixtures andthe 6-H2 with which they are compared are allliquid. Passing to the left, the first sharp breakencountered on either curve corresponds to thetriple point of the mixture. The second sharpbreak corresponds to the triple point of e-H2. Tothe left of the last break, both materials are solid.Between the two breaks on either curve, the mix-ture is solid but the e-H2 is liquid.

O 100 200 300 400 500 600 700 800P(E-H2) MM OF HG

FIGURE 18. Vapor pressure differences for liquid ortho-paraH2 mixtures.


o

•2.0

1.0

/

/

/

/

/

75%

25%

• y/

>

10 20 30 40 50 60 70MM OF HG

80

FIGURE 19. Vapor pressure differences for solid ortho-paraH2 mixtures.

A comparison of the AP's for different mixturesof 0- and p-H2 in figures 18 and 19 shows that theAP's are not proportional to their correspondingdifferences in composition.

For ideal solutions the ratio AP/Ax, where Axis the difference in composition, is independent ofthe composition at constant temperature. Infigure 20 this ratio is plotted for four temperatures,the circles representing the experimental vaporpressure data as given by points on the smoothcurves of figure 18. Figure 20 shows that thevapor pressures of ortho-para mixtures differgreatly from ideal solution predictions.

The vapor pressure differences (Pe-H2—Pm) formixtures of o- and p-H2 of any composition at14.00°, 16.00°, 18.00° and 20.39° K may be calcu-lated from the isotherms of figure 20. Otherisotherms may be determined with the help offigures 18 and 19. By extending the isothermsof figure 20 to 100 percent o-H2, the vapor pres-sure of pure liquid o-H2 was determined. Thefollowing equation represents the vapor pressures:>f pure liquid o-H2 obtained in this way:

liquid: log10P(mm Hg)=4.65009 T

0.0211687 (7.6)

The triple-point temperature and pressure ofo-H2 were determined by a quadratic extrapolationof the triple point temperatures and pressures of

e-H2(20.4° K), m-H2(38 percent o-H2) and n-R2.The values thus obtained for o-H2 were 14.05°K and 55.1 mm Hg. These are in agreementwith eq 7.6 for the vapor pressure of liquid o-H2.

If linear extrapolation is used, omitting thevalues for m-H2, one obtains 14.00° K and 54.4 mmHg as lower limiting values of the triple point con-stants for o-H2. The triple point constants ofm-H2 were obtained by reading the values P(e-H2)and AP corresponding to the upper break in the38 percent curve. The difference of these is thetriple point pressure of m-H2. By substitutingP(e-H2) into the vapor pressure equation (eq 7.4)for liquid e-H2, the triple point temperature ofm-H2 is obtained. The uncertainties in these de-rived triple point constants of m-H2 and o-H2 aregreater than for the experimentally determinedvalues for e-H2 and n-H2.

The vapor pressure of a nonequilibrium mixtureof o- and p-H.2 changes slowly with time because ofthe slow conversion of a nonequilibrium mixture,liquid or solid, to the equilibrium composition. Atits normal boiling point, the vapor pressure ofn-H2 changes at the rate of 0.23 mm Hg per hour[148]. Paramagnetic substances increase the rateof conversion. The rate of increase of the vaporpressure at 20.4° K of a sample of hydrogen con-taining 0.01 percent oxygen was about three timesthat for pure hydrogen.

The interconversion of ortho and parahydrogenin the absence of molecular dissociation is the re-sult of an intra-molecular rearrangement of pro-

0.25 0.50 0 75CONCENTRATION OF o-H2

1.00

FIGURE 20. Deviations of vapor pressure of ortho-para H2mixtures from law of ideal solutions.


tons in the presence of a strong magnetic field,inhomogeneous on a scale of molecular dimensions.

As p-H.2 has no net nuclear magnetic moment,the self conversion of nonequilibrium mixturesresults only from the interaction of o-H2 mole-cules, which do have a nuclear magnetic moment,with each other and with p-R2 molecules. Hence,the ortho-para conversion in liquid and solid H2

is a bimolecular change.

-d[o-H2]/dt=k1[o-H2\2-k2[o-R2] [p-Ha] (7.7)

The velocity constant k2 is much smaller than k\in accord with the small equilibrium proportionof o-H2. At equilibrium, where d[o-H2]/dt is zero,k2/ki = [o-H.2]/[p-H.2]. Values of equilibrium con-centrations are given in table 12. For liquid hydro-gen the velocity constant kx for conversion is0.0114 per hour when concentrations are expressedin mole fractions. The value of k\ for solid H2,0.019 hr"1 [147], is larger than for liquid H2 butdecreases with time due to the immobility ofmolecules in the solid. The initial value of k\ isrestored however by melting and freezing.

(b) D2

The vapor pressures of normal and equilibriumdeuterium were measured [149] relative to thevapor pressure of liquid n-H2 from 14° to 20.4° K.As these measurements are independent of atemperature scale their functional relations aregiven. Vapor pressures are expressed in terms ofmm of Hg at standard conditions.

Normal deuterium (66.67 percent o-D2, 33.33percent p-T>2):

Liquid: log10 P(ft-D2) = —1.3376 + 1.3004 log10

P (n-R2). (7.8)

Solid: log10P(n-D2) =-1.9044 + 1.5143log10P(n-H2). (7.9)

20.4K Equilibrium deuterium (97.8 percento-D2, 2.2 percentp-D2):

Liquid: log10P(e-D2) = -1.3302 + 1.3000log10P (n-H2). (7.10)

Normal deuterium (66.67 percent o-D2, 33.33percent p-D2):

Solid: logioP(e-D2) = —1.8873 + 1.5106log10P (w-Ha). (7.11)

Substituting for logi0 P(n-H2) values given by eq.7.2 for liquid n-H2 the following equations forlogi0P(D2) are obtained:

Liquid: log10P(mm Hg)=4.7312- 5 8 ' 4y 1 9

+0.02671 T. (7.12)

Solid: log10P(mm Hg) = 5 .1626- 6 8 -^ 8 2

+ 0.03110T. (7.13)

20.4°K equilibrium deuterium (97.8 percento-Do, 2.2 percent p-D2)

Liquid: log10P(mm Hg) = 4 . 7 3 6 7 - 5 ^

+ 0.02670 T. (7.14)

6 7 ' y 1 9

(7.15)Solid: log10P(mm Hg) = 5.1625-

+0.03102T.

Tbe triple-point temperatures and pressures forD2 given in tables 28 and 29 were obtained bysimultaneous solution of tbe vapor pressure equa-tions for solid and liquid.

The self conversion of nonequilibrium mixturesof 0- and p-D2 proceeds at a very much slower ratethan for H2. Thus no increase in the vapor pres-sure of liquid n-D2 resulting from self conversionwas observed at 20.4° K over a period of 100hours [149]. The estimated probable error of twoobservations extending over 100-hour periods was±0.27 mm Hg. The small rate of self conversionof D2, compared with H2, is a result of the smallermagnetic moment of the deuteron compared withthe proton. The ratio of nuclear magnetic mo-ments D/H is 0.26. The relative rate of self con-version for the same displacements of D2 andH2 from the equilibrium ortho-para compositionis proportional, as to order of magnitude only, tothe fourth power of their relative magnetic mo-ments, that is to 0.005. Allowing for the smallerdisplacement of n-D2 from equilibrium composi-tion and the smaller difference between the vaporpressures of the ortho and para varieties of D2,the expected ratio of tbe rates of vapor pressurechange, n-D2 to n-H2, is of the order of 10~3.For a more detailed discussion see reference [149].

(c).HD

As the two nuclei of the HD molecule are dis-similar, hydrogen deuteride does not have orthoand para varieties. Measurements of the vapor


pressure of HD extend from 10.4° to 20.4° K[150]. The following vapor-pressure equationswere made to fit the triple-point temperature16.604° K measured with a platinum resistancethermometer in a calorimeter in which the solidand liquid phases were in equilibrium.

HD:Liquid: log10 P (mm Hg) = 5.04964—

(7.16)

Solid: logio P (mm Hg)=4.70260-

(7.17)

The triple-point pressure of HD given in tables28 and 29 can be obtained from either of theseequations.

(d) HT and DT

Tritium, T, the hydrogen isotope of atomicweight 3 is radioactive and has a half-lifetime of31 ± 8 years [151]. Its disintegration products area negative /3-p article and He3. Because of itscomparatively short half-life, the natural abun-dance of T in hydrogen is extremely small. Libbyand Barter [152] determined the vapor pressures ofHT and DT using T made by the irradiation of ablock of metallic Li with neutrons (Li6+n-^He4+T3). The tritium held by the Li as LiT wasliberated by the reaction of H2O or D2O with theLi block. Gaseous H2 or D2 with a trace of HT orDT was obtained. The gas was liquefied andthen evaporated, and the radioactivity of theevaporated vapor was measured as a function ofthe volume of the remaining unevaporated liquid.From a comparison of the radioactivity of thevapor leaving the liquid during different periodsof the evaporation, Libby and Barter calculatedthe vapor pressures of HT and DT, making use ofideal solution laws for this purpose. They ob-tained for the vapor pressures of HT and DT,254 ±1.6 and 123 ±6 mm Hg, respectively, at thenormal boiling temperature of hydrogen (20.39 °K).By extrapolation, they estimated that the vaporpressure of T2 at 20.39° K is 45 ± 10 mm Hg.

2. Pressure-Temperature Relations for Solid-LiquidEquilibrium

The melting, or freezing pressures, of w-H2,HD, and n-D2 given in table 30 are based on

smooth curves drawn through the experimentaldata (H2, [153 to 157]; HD [150]; D2 [174]) andcover the same ranges of pressure and temperatureas the data. Figure 21 is a diagram of the devia-tions of the data for n-H2 from the table. Thedashed line shows a 1-percent deviation from thetable and the full-line curve represents the devia-tion from the table of the equation

log1o(237.1+P) = 1.85904log10T+0.24731, (7.18),

where P is in kg cm"2.

TABLE 30. Melting temperature-pressure relations for71-H2, HD, and n-D2

T

°K13.9614151616.60

1718 _18.721920

2122.232425.

2627282930

3234363840

4550556065

707580

r

[kg cmr*0.071.4

33.267.3

103. 5142.3

183.6227.1

272.3318.6366.0415.0465.6

518572628685744

867996

1,1311,2741,422

1,8212.2582,7353,2493,801

4,3895,0145,674

H D

kg cm-*

0.13

14.252.6

92.9

n-T>2

kg cmr2

0.1713.956.0

100.0

Figure 22 is intended to show the relationbetween the melting pressures of TI-H2, HD, andn-T>2- The curve for n-H2 is a graph of tablevalues. The curves through the experimental


lON.

Ol O

l O

CM

u

cC"1

CL

I O1^

1

-100

O

o o o

y

o

o o

o

o

/

/

K o

o

o "" ~~ ^

o

o

/

o

oo

o ^^—.

o

o

o

o

o

o

o

o

O ' v ^

o

IO°K 20 30 40 T_ok 50 60 70 80

FIGURE 21. Melting pressure of n-H2 as a function of temperature.

data for HD and D2 were obtained by a simplevertical displacement of the H2 curve and showthat the differences in melting pressures of thethree isotopic varieties are only slightly dependentupon the temperature. These differences inpressure are 89.6 kg cm"2 for H2 and HD and170.6 kg cm"2 for H2 and D2. As the change ofmelting pressure with temperature, dP/dT, hasnearly the same value for H2, HD, and D2, ifcompared at the same temperature, it follows fromthe Clapeyron equations that Lf/8V, the ratio ofthe heat of fusion to the change in volume onmelting, also has nearly the same value for thethree isotopes when compared at the same valueof T. A similar statement can be made for Sf/8V,the ratio of the entropy of fusion to the change involume on melting.

The table values of melting pressure for HD

and D2 were obtained from curves drawn throughthe experimental data and not from the curves offigure 22.

VIII. PVT Data for the Condensed States

The available date of state for the condensedphases of H2, HD, and D2 are meager [158 to 166]and in general not accurate enough for the calcula-tion of reliable values of thermodynamical proper-ties. The data on the liquid, however, wereused in the construction of the liquid regions of thea versus p diagrams, figure 6, and the T versus Sdiagram, figures 31, 32, and 33.

1. Liquid H2/ HD, and D2

In table 31 are given the molar volumes ofliquid n-H2, p-H.2, HD, and n-D2 in equilibrium


2. CM

Q- —

BRICKWEDDE S SCOTT — o

CLUSIUS a BARTHOLOME — e

/

/

/

H2 /

/

/

/

/

/

HD /

/

/

//

/

/

/

/

/

D2 /

/

/

/

13 14 16 17 18 19 2 0 21

T, °KFIGURE 22. Melting -pressures of n-H2, HD, and n-D2.


TABLE 31. Molar volumes of normal hydrogen, parahydro-gen, normal deuterium, and hydrogen deuteride, in theliquid state

T

°K13.813.. .13.96141516

16.604171818.72319

2020.39 .222426 ._ _

28303233.19...

Volume of liquid at saturation pressure

cm* mole-1

26.10826.11926.40726. 721

27.06127.426

27.816

28. 23228.40129.23330.45131.995

34 05937.13843. 21166.95

p-H2

cm* mole-1

26.176

26. 22726.51826.836

27.17927. 549

27.945

28.368

n-T>2

cm* mole~l

23.16223. 237

23. 525

H D

cm* mole-1

24.48724. 59424.885

25.211

25. 572

with vapor from the triple point to the highesttemperature of measurement. From the triplepoint to 20.4° K, these equilibrium molar volumeshave been represented by the following equations,in which temperatures are on the Kelvin scale:

Normal hydrogen [163]:F(cm3 mole-1)=24.747-0.08005T+0.012716T2.

(8.1)Parahydrogen [163]:

y(cm3mole-1)=24.902-0.0888T+0.013104T2.(8.2)

Hydrogen deuteride [150]:F(cm3 mole"1)—24.886 — 0.30911T+0.01717T2.

(8.3)Normal deuterium [174]:

F(cm3 mole"1)=22.965-0.2460T+0.0137T2.(8.4)

Table values at 20.39° K and lower were calculatedfrom these equations. Values of the molar volumeof liquid normal hydrogen above 20.4° K wereobtained from the experimental data of Mathias,Crommelin, and Onnes [161] with the help of asensitive interpolation method based upon the use

of an empirical equation and a deviation graph.A change was made in the experimental databecause the value used by Mathias, Crommelin,and Onnes for the density of gaseous hydrogen atstandard conditions differs from that recom-mended in this paper on page 396.

Bartbolome [177] measured the molar volumesof liquid n-13.2 and n-T>2 as a function of pres-sure at three temperatures between 16° and 21°K. The measurements extended from the vaporpressure to nearly the freezing pressure. Smoothedvalues of molar volumes are given in tables 32 and33. Bartholome showed that isothermal changesin volume to about 9 percent of the volume of"saturated" liquid can be represented to withinthe precision of his measurements, ±0.05 cm3

mole"1 by Eucken's equation

_ i n (8.5)

in which V, the molar volume of the liquid, isexpressed as a function of the pressure P. v0 isthe molar volume extrapolated to zero pressure,and a is an empirical constant dependent uponthe temperature. Tables 32 and 33 includevalues of the molar volumes of liquid n-H2 andn-D2 at freezing pressure for the three tempera-tures of Bartholomews measurements.

TABLE 32. Molar volumes of liquid n-H2 for varioustemperatures and pressures.

Pressure

kg cm~2

0°10

5075

82 6100 - -125150151 98

175200225241 83

T= 16.43° K

cm* mole-1

26.8726.5926.2025.6625.20

25.08

B ^ 8 0 X 1 0 C m l 6 k g

T-18.24° K

cm* mole-1

27.5427.1826.7226.1025.59

25.1424.7124.3024.27

i i m o l e f l

" 3 ' 9 3 ^ 1 0 C m l 6 k g

T= 20.33° K

cm* mole-1

28.4327.9727.4026.6225.98

25. 4224.9124.47

24.0923.7623.4823.31

B - U e ' N l ° Cml6kg

° The values at zero pressure were obtained by extrapolation consistentwith the molar volumes at saturation vapor pressure given by eq. 8.1.


TABLE 33. Molar volumes of liquid n-T>2 for varioustemperatures and pressures

Pressure

kg cm~2

0°10203040

43.18__5060_69.4670

8090.98.67

T=19.70° K

cw3 mole-1

23.4423. 2423.0622.8922.74

22.70

mole'd=6.75X10~^—TT"T—

T=20.31° K

cms mole-1

23.6323.3723.1622.9722.79

22.6322.4922.36

„ motelcmiekg

T=20.97° K

cm3 mole-1

23.8423.5923.3523.1422.95

22.7722.60

22.45

22.3022.1622.05

cmiekg

« The values at zero pressure were obtained by extrapolation consistentwith the molar volumes at saturation vapor pressure given by eq 8.4.

2. Solid H2/ HD, and D2

i sThe crystal structure of solid hydrogenthought to be hexagonal close-packed, on the basisof an X-ray investigation of solid parahydrogenby the Debye-Scherrer method at the tempera-ture of liquid helium, conducted by Keesom, deSmedt, and Mooy [162].

Tables 34 and 35 contain all the available experi-

mental data of state on solid H2, HD, and D2.Molar volumes at 0° K were obtained by calcula-tion.

Molar volumes of the solid at the triple pointgiven in table 34 were obtained by subtractingthe volume changes on fusion from the triple pointvolumes of the liquid calculated from eq 8.1, 8.3,and 8.4. The volume changes on fusion, givenin table 34, were calculated using the Clapeyronequation with the calorimetrically measured heatsof fusion (section IX, 3), and dP/dT for the solid-liquid equilibrium at the triple point (sectionVII, 2).

Molar volumes of the solid in table 34 abovethe triple-point temperature were obtained fromBartholomews measurements of the change involume on fusion at the temperatures given intable 34, and the volumes of the liquid at meltingpressure given in tables 32 and 33.

The molar volumes of solid H2 and D2 at 4.2°K in table 34 were measured by Megaw [165] witha picnometer in which the solid H2 or D2 was sur-rounded with liquid helium, the volume of whichhad previously been measured as a function ofpressure at this temperature. The compressibili-ties of solid H2 and D2 at 4.2° K, given in table35, were calculated by Miss Megaw from theresults of these measurements.

TABLE 34.

T

°K

20.9720.3118.72

16 60

18.2416.4313.96

4.2

4.24.24.24.24.24.2

0

P

kg/cm2

98 769.50 174

126

152.082.6O.O73

010255075

100

0

Molar volumes of solid tt-B^, HD and n-T>2 and volume changes upon fusion

n-H.2

Volumeof solid

cm*/mole

22.2422.7823.25

22.65

22.6522.4922.3022.0321.8021.60

22.57

Volumechange on

fusion

cms/mole

2.032.302.85

H D

Volumeof solid

cm*/mole

21.84

Volumechange on

fusion

cmz/mole

2.65

7Z-D2

Volumeof solid

cms/mole

20.0720.2020.48

19.56

19.5619.5019.4119.2819.1619.06

19.49

Volumechange on

fusion

cms/mole

1.982.162.66

Remarks

j T and P for solid-liquid equilibrium.

n-T>2 triple point.

HD triple point.

\ T and P for solid-liquid equilibrium.

tt-H2 triple point.

Solid-vapor equilibrium.

Smoothed values based on direct experimentaldetermination.

By calculation.


461

TABLE 35. Experimentally determined compressibilities,

°f solid H* and D 2 at ^'®° K

Compressibility

At pressure 0 kg cm-2

At pressure 100 kg cm-2

A verage for range 0 to 100 kg cm-2.

H2 compressi-bility

kg-i cm2

(6.8±1.5)X10-4

3.2X10"4

(5.0 ±0.5) X10-4

D2 compressi-bility

kg~l cm2

(4.5 ±2) X10-42. IX10-*

(3.3 ±0.7) XHH

Miss Megaw calculated the expansivities ofsolid H2 and D2 at 4.2° and 11° K, given in table36, using the formula

dV d v

with the calorimetrically determined specific heatsat constant pressure and volume, and the com-pressibility measured at 4.2° K (table 35).

TABLE 36. Expansivities,

calculated from Cv—Cv

T

° K4.2

11

H 2

0. 24X10-2

.51X10-2

D2

0.17X10-2

.37X10-2

The compressibilities and expansivities of solidH2 and D2 are large when compared with values ofthese properties for other substances. This isascribed to the zero-point vibrational energy of thelattice which for hydrogen is an unusually largefraction of the negative potential energy of thelattice. This accounts also for an unusually largevariation in the compressibilities of H2 and D2 withpressure (see table 35), and for the variation withT and V of d In 6/d In V, which derivative of theDebye 6 is usually regarded as a constant for othersolids [165].

IX. The Thermal Properties of theCondensed Phases

In this section are included the calorimetricallymeasured properties: specific heats and heats offusion and vaporization.

1. Specific Heats of the Solids and Liquids(a) Hydrogen

The specific heats at saturation pressure of solidand liquid hydrogen were measured (1923) by

Simon and Lange [171] between 10° and 20° K,before the discovery of parahydrogen. Clusiusand Hiller [172] measured (1929) the specific heatsof solid and liquid parahydrogen over the samerange of temperatures and obtained the samevalues, within experimental error, for the specificheats of parahydrogen as had been obtained bySimon and Lange for supposedly normal hydrogen.Mendelsohn, Ruhemann, and Simon [173] meas-ured (1931) the specific heats of several mixturesof ortho- and parahydrogen between 2.5° and11.5° K. Their results on pure parahydrogenwere in agreement with the earlier measurementsof Clusius and Hiller, the data from 2.5° to

> of solid H2 and D2 y

<{c

- c

• 0.2 PER CENT ORTHOHYDROGEN

20 - . .50

y 75

_—-*•

0

- & •

- 0 -

ITy

/

f/

///

cr 0,8UJa.

3 0.6

0.4

0.3

0.2

O.I

I 2 3 4 5 6 7 8 9 10 II 12 13 14T °K

FIGUKE 23. Specific heat, C8, of solid H2 for variousortho-para compositions.

14° K fitting rather closely a Debye functionwith 9 = 91° K.

The data of Mendelsohn, Ruhemann, andSimon are shown in figure 23. It is seen that, attemperatures below 11° K, the specific heats ofmixtures containing orthohydrogen are largerthan for pure parahydrogen. This difference inspecific heats is connected with the multiplicityof states belonging to the lowest o-H2 rotationallevel, «/= 1. The different states, three in number,correspond to three different orientations of theangular momentum vector of an o-H2 moleculerelative to the electric field in the hydrogencrystal. At 0° K, all o-H2 molecules are in theorientation state of lowest energy. At tempera-


tures of the order of AE/k, where AE is thedifference in the energy of the states, the dis-tribution of o-H2 molecules over the three stateschanges rapidly with change of temperature.Along with this there is an absorption of energyand an increase in specific heat. As tempera-tures are approached that are high comparedwith AE/Jc, the distribution of o-H2 moleculesbecomes uniform over the three orientation states,and the specific heat of orientation approacheszero. It may be seen from figure 23 that 12° Kis effectively a high temperature for this distri-bution, and that at temperatures above 12° K thedistribution over the three J=l states must bepractically uniform.

The specific heats, C8, of liquid and solidhydrogen along the saturated vapor lines aregiven in table 37. The Cs curves of figures 24,25, and 26 for n-H2 at temperatures above 11° Krepresent this table.

In figures 25 and 26 the heat capacity, Cv, of

]

]

_ _ — • —

. •

a

h>

i. — — •

"~ i—

1

1

10 II 12 13 14 15 16 17 18 19 20 21 22T,°K

FIGURE 24. Specific heat, Ca, of solid and liquid H2, HD,and D2.

i

>*

I1

* <

4

10 12 16

T.'K

18 20

FIGURE 25. Specific heats, C8 and Cv, of solid H2 and D2.

solid and of liquid n-H2 at constant specified valuesof the density are compared with the heat capacity,C8, of solid and liquid n-H2 in equilibrium withsaturated vapor. It is to be noted that the Cv

curves of these two figures are not for Cv of solidand liquid H2 along a line of equilibrium of vaporand condensed phase. The Cv measurements onthe solid were made by Bartholome and Eucken[176] at the density of solid H2 at a melting temp-erature of about 19° K. The Cv measurementsfor the liquid were made by Eucken [169] and byBartholome and Eucken at densities ranging from0.034 to 0.077 g cm"3 (380 Amagats to 860 Ama-gats). The density of liquid w-H2 at its normalboiling point is 0.07097 g cm"3 (789.7 Amagats).

The difference between Cv in figure 25 for thesolid at constant density and Cv at densities ofthe solid along the solid-vapor equilibrium lineis small. The corresponding difference for theliquid is larger and, at the critical temperature33.19° K, is of the order of 1 or 2 cal mole"1 ""K"1

6

5

2

/

--

. co

r-

0

H?

-0

BARTHOLOME &EUCKEN— °

EUCKEN o

erro —v» , <

14 16 18 20 22 24 26 28 30 32 34 36T,°K

FIGURE 26. Specific heats, C9 and Cv, of liquid H2 and D2.


TABLE 37. Specific heats at saturation pressure of normalhydrogen, normal deuterium, and hydrogen deuteride inthe solid and liquid states

T

0 K1011

121313.96 .

13.96.14151616.60 -.

16.6017.1818.72

18.7219202122_

Hydrogen

C.

cal mole-*deg-1

0.5876

.951.161.37

3.313.313.463.63

3.834.04

4.274.50

State

Soliddodododo

Liquiddododo

. dodo

dodo

Deuterium

c.

cal mole-1

deg-i

0.881.001.18

1.391.631.90

2.212.562.84

4.804 865.085.305.52

State

Solid—do

do

.. do .dodo

do .dodo

Liquiddodo

. - d o — .

. . do. __

Hydrogen deuteride

a

cal molerideg~l

0.691.03

1.391.762.172.42

4.404.534.88

5.205.495.796.09

State

SolidDo.

Do.Do.Do.Do.

LiquidDo.Do.

oo

oo

o o

o

p

Cv along the liquid-vapor line being greater [176].The difference between Cs and Cv for hydrogen is

large when compared with the differences forother substances having higher boiling tempera-tures. In general, (Cs—Cv) is large for low-boiling substances because of their larger expan-sivities.

The Debye 6 in the Debye specific heat func-tion that fits the Cv data on solid H2 is 105° K.This may be compared with 91° K for Cs.

The specific heats at constant pressure of com-pressed liquid hydrogen and gaseous hydrogenwere measured by Gutsche [178] for temperaturesfrom 1 6 ° K t o 3 8 ° K and for pressures of about10, 25, 40, 60, 80, and 100 kg cm"2, using a calori-meter so arranged that approximate constancy ofpressure was maintained by manual operation ofvalves permitting fluid to pass from the calori-meter. As a result of this experimental procedure,the mass of hydrogen in the calorimeter wassmaller at the higher temperatures, and conse-quently the accuracy of measurement is probablylower at the higher temperatures.

20^ 22° 24° 2 6° 28° 30^ 32° % 34° 36° 38^ 40° 42T

FIGURE 27. Specific heat, Cp, of compressed liquid and gaseous H2.


In figure 27 are plotted Gutsche's experimentaldata with dotted curves as drawn by Gutsche inhis paper to represent the experimental data.The full line curves apply only to the vapor andwere obtained by calculation from the PVT corre-lations of preceding sections of the paper andspecific heats in the ideal gas state, table 8. Theheavy curve shows Cp for saturated vapor. Thefull-line curves beginning on this heavy curve, orsaturated vapor line, sloping downward towardthe right represent the specific heats, Cp, for thevapor at pressures of 5 and 10 kg cm"2. Parts ofsimilar curves also based on the PVT data areshown for 11 and for 13.41 kg cm"2, the criticalpressure.

For temperatures above the critical, the dashedcurves of Gutsche for 10, 25, and 40 kg cm"2 arequite different from the full line curves based onPVT data. The dashed curve for the gas at 10kg cm"2 is certainly incorrect at the highesttemperatures, as the actual deviation from theideal gas law for hydrogen is such as to increaseCp above the approximately 5 cal deg"1 mole"1

of the ideal gas at these temperatures.It is seen in figure 27 that Gutsche's experi-

mental values for the liquid scatter considerably.It is believed that Gutsche's recommended valuesof Cp for liquid hydrogen, represented by thedashed lines in figure 27, are too high. In figure 30are shown two sets of isobars, E and E', on atemperature-entropy diagram for liquid hydrogen.The full-line curves, E, were calculated fromGutsche's Cp data; the dashed curves, Ef', are thebest fit for all the thermal and state data on liquidhydrogen and are the ones used in the construc-tion of the temperature-entropy diagram. As(dS/dT)P=Cp/T, the two sets of isobars, Esmd E',imply different Cp& and show that Gutsche'svalues of Cp are too high to be consistent with theother data on liquid hydrogen. The differencesare of the order of 15 percent in the Cp'& of liquidhydrogen. The ratio Cp/Cv for liquid hydrogenin equilibrium with vapor was calculated fromthe velocity of sound in liquid hydrogen, and Cp

was obtained by combining this calculated valueof the ratio (Cp/Cv), with Cv from figure 26. Pittand Jackson [175] obtained the value 1,127 msec"1 for the velocity of sound in liquid hydrogenat 20.46° K. Using this with a value of (dV/dP)extrapolated from Bartholomews data (VIII), oneobtains a value of 5.07 cal deg"1 mole"1 for Cp

for liquid hydrogen in equilibrium with vapor(~1 atm) at 20.46° K.

This is slightly lower than would probably beobtained by extrapolating Gutsche's curves to1 atm.

(b) D2 and HD

In figure 24 the specific heats Cs at saturationpressure of liquid and solid w-D2 and HD arecompared with Cs for H2. The D2 measurementswere made by Clusius and Bartholome [174] andthe HD measurements by Brickwedde and Scott[150]. The solid D2 data are fitted, within experi-mental accuracy, by a Debye function with 0 =89°. The data on solid HD, however, can not befitted over the range of measurement with a singlevalue of 9. Thus 0 for Cs of HD at 16.3° K is 79°,whereas for 12.5° K, 0 is 98°. As the Debyefunction is intended to represent CVJ this failureto fit the Cs data is not surprising.

In figures 25 and 26 the specific heat Cv at con-stant volume of solid and liquid D2 is comparedwith Cs for D2 and Cv for H2. A Debye functionwith 0=97° fits within experimental accuracythe Cv data for solid D2. This value of 0 for solidD2 may be compared with 105° for solid H2. Ac-cording to the simple theory of lattice vibrations,which assumes simple harmonic restoring forcesin the lattice, 0 would be proportional to l/-y/Mand the 0's for H2 and D2 would be in the ratio^4/2= 1.41. The ratio of the experimental valueshowever, is 1.08. This is evidence that the latticerestoring forces in solid H2 and D2 are stronglyanharmonic.

2. Latent Heats of Vaporization

(a) Normal Hydrogen

Simon and Lange [171] measured the heat ofvaporization of normal hydrogen at severaltemperatures between the triple point and theboiling point. They found that heat of vaporiza-tion, in calories per mole, was given by

ZP=219.7 —0.27 (T—16.6)2, (9.1)

where T is the Kelvin temperature,

(b) Mixtures of o-H2 and p-H2

As orthohydrogen and parahydrogen are veryclosely related, it might be expected that theirmixtures would have properties related verysimply to those of the pure components. Never-


theless, the H2 vapor-pressure data of Brickweddeand Scott [146] given by the equations and graphsof Section 7 show that the ortho-para H2 mixturesdo not follow Raoult's law for ideal solutions. Asimple application of the Clapeyron equation inthe form applying to a pure substance indicatesthat the latent heat of vaporization and theinternal energy of the liquid and solid do not followa linear, but rather an approximately quadraticdependence upon the composition. This samequalitative result is obtained when account istaken of change of composition by fractionationduring vaporization. Functions approximatelylinear in x, the ortho mole fraction, are obtainedwhen Lmix—LeQ, the difference in latent heats, andEe(l—Emlx, the difference in the internal energy, aredivided by xmix—xeQ, the corresponding differencein the ortho mole fraction. The subscript "eq"indicates the ortho-para mixture that is atequilibrium at 20.4° K, containing 0.21 percentof ortho- and 99.79 percent of parahydrogen. Thesubscript "mix" refers to any other mixture forwhich data were obtained. When the line forAE/Ax is horizontal, it indicates that ideal solutionlaws apply. The line has a clear indication ofslope, as shown by the continuous lines in figure28, indicating that ideal solution laws do notapply. In the graph for AE/Ax} the points for theliquid include a contribution of about 7 percentrelated to change of composition due to fractiona-tion. The lower dashed line shows the resultwhen this correction is omitted. For the solid itwas thought proper to omit the correction forthis effect because departure from equilibriumdue to slowness of diffusion in the solid wouldmake it too uncertain. The upper dashed lineshows the result for the solid when such a correc-tion for fractionation is included.

The use of straight lines for AE/Ax, the divideddifference of the internal energy, has a theoreticaljustification apart from the fact that the scatteringof individual values is so great as to obscure theexact shape of the curve for the liquid. If theinternal energy of the liquid is a simple sum ofindependent energies of different molecular pairs,all of essentially equal probability of formation,then the energy has the form

E=x2E'00+2x(l-x) Eop+(l-x)2Epp. (9.2)

In this case, the differences EeQ—Emlx dividedby the corresponding differences in x for the mix-

tures of different compositions will be linearlydependent on x. The slope of this line is 2Eop~Eoo—Epp and the value of the ordinate atx=—xeq is 2 (Epp—Eop). From the curves infigure 28, it will thus be found that Epp—Eop is 0.7cal mole"1 and Epp—E00 is 4.2 cal mole"1 for theliquid. For the solid the corresponding values are0.6 cal mole"1 and 5.4 cal mole"1, respectively.The relative size of Epp—E00 as compared toEpp—Eop suggests that most of the deviation fromideal solution laws is due to special effects betweeno-H2 molecules.

From the scattering of the points plotted, itappears that ordinates are uncertain to 0.2 or 0.3cal mole"1 for the liquid and possibly to 1 calmole"1 for the solid. The use of the straight linefor AL/Ax in figure 29 is very nearly consistent withits use for AE/Ax and is allowed within the scatter-ing of the data. Combining the results for thedependence upon composition with the results ofSimon and Lange for normal hydrogen, the latentheat of vaporization of liquid hydrogen in caloriesper mole is approximately

217.0-0.27 (7T-16.6)2+1.4x+2.9^2 (9.3)

for any mixture of o-H2 andp-H2, where x is theorthohydrogen mole fraction.

LJ

//

/

/

/yy.

/// V y

/ "

o y

// y

/

12 °K *S O M o, i | 5 : ^

16 °K°Liquid at 18 °KQv

20 °KC20.39°K/>

0 0.2 0.4. 0.6 0,8 1.0

X, ortho-hydrogen mole fraction

FIGURE 28. Dependence of internal energy of solid andliquid H2 upon the ortho-para composition.


3 ftA&y

y #<^

/ p

/ #

Solid at \ 13I 13

Liquid at

- — co cd

|

/

/

V K <1° K ^5°K *

>° K ^i° K <\)° K cf).39°K p

0 0.2 0,4 0.6 0.8 1.0X, ortho-hydrogen mole fraction

FIGURE 29. Dependence of latent heats of vaporizationand sublimation of hydrogen upon the ortho-paracomposition.

The heats of fusion of para- and normal hydrogenare reported in table 38 as being equal within 0.03cal mole'1. On the basis of the two distinctstraight lines for liquid and solid hydrogen infigure 29, it would be expected that the differencewould be about 0.7 cal mole"1. The reason forthis discrepancy is not known, though it may sug-gest that the lines for the liquid and solid shouldbe more nearly identical.

TABLE 38. Latent heats of fusion

Substance

Normal hydrogen.__ParahydrogenNormal deuterium._Hydrogen deuteride

Heat offusion

cal mole-1

28.028.0347.038.1

°K13. 957

13. 8I318. 723

16. 6O4

mm Hg54.052.8

128.592.8

The manner in which the vapor pressuresdepend on composition and temperature hasformed the basis for the treatment of latent heatsof vaporization given in this section. Cohen andUrey [166] and Schafer [164] have given theoreticaldiscussions of the vapor pressures of ortho and

para H2 and D2. Cohen and Urey did not expectdeviations from the law of perfect solutions.Schafer suggested that forces connected with ro-tation within the crystal lattice might account forvapor-pressure differences.

(c) Normal Deuterium

Clusius and Bartholome [174] measured theheat of vaporization of normal deuterium, ob-taining the value 302.3 cal mole"1 at 19.70° K.

(d) Mixtures of o-D2 and p -D 2

The difference in latent heats of vaporizationand the approximate difference in internal ener-gies have been calculated from the vapor pressuresof the normal and the 20.4° K equilibrium mix-tures of ortho- and paradeuterium measured byBrickwedde, Scott, and Taylor [149]. PVT datafor deuterium as determined by Schafer were alsoused in the calculation. As there are data foronly two compositions, giving only one differenceof composition, it is not possible either to correctfor fractionation or to test for deviation fromRaoult's Law. It seems improbable that the lawholds for deuterium, as it does not hold for hy-drogen. The indicated differences in latent heatsof vaporization are smaller than for hydrogen.Thus, inorm—ieq=0.3 cal mole"1 for the liquidand 1.0 cal mole"1 for the solid. The same valuesare obtained for the differences in internal ener-gies, Eeq—Eaorm. Cohen and Urey [166] on thebasis of their theoretical calculations, concludedthat differences in binding energy between cor-responding forms should be half as great for D2

as for H2. Considering that the uncertainties inthe data for D2 are comparable with the magni-tudes themselves, the data can not be said toconflict with the theoretical preduction.

(e) Hydrogen Deuteride

Brickwedde and Scott [146] measured the heatof vaporization of hydrogen deuteride, obtainingthe value 257 cal mole"1 at 22.54° K.

3. Latent Heats of Fusion

The latent heats of fusion of hydrogen, para-hydrogen, normal deuterium, and hydrogen deute-ride were measured by Simon and Lange [171],Clusius and Hiller [172], Clusius and Bartholome[174], and by Brickwedde and Scott [150], respec-


tively, and are listed in table 38 with correspond-ing vapor pressures and temperatures.

X. The Temperature-Entropy Diagram

1. Data

Data of several different types were used indetermining the temperature-entropy diagram.For the vapor, and for the gas below a density of500 Amagats, values of the various quantitieswere obtained by interpolation from tables 14, 22,and 23. The particular difficulties encountered intreating the liquid region will be evident from the

45°

° 3d

15°

ISOCHOREV

w

Yy/ /

ISOBARS)

0/

//

URATION < URVE

uf

— BASED ON DATA NOT CONSISWITH T-S DIAGRAM

~ FINAL POSITION ON T-S D

TENT

AQRAM

S , CAL GMJoKJ

FIGURE 30. Discrepancies in the thermal data for H2 inthe region of the liquid.

following discussion. Discrepancies between thevarious data for the liquid are shown in figure 30.

Between the triple point and the boiling point,the entropy of liquid normal hydrogen at satura-tion pressure was obtained using calorimetricdata for the solid and liquid and adding a theoret-ical value for the entropy of mixing. The resultis shown as line B in figure 30. The entropy of theliquid was also calculated using the theoreticalentropy of the ideal gas, correcting to the state ofsaturated vapor and subtracting the latent heatof vaporization. The latent heat of vaporizationwas determined in two ways;—by direct calori-metric measurement and by using vapor pressuresand other data with the Clapeyron equation.Line A is based on calorimetric latent heats andline C on latent heats from vapor pressures. At20° K, line B indicates values 0.03 cal deg"1 g"1

greater than line A and 0.08 cal deg"1 g"1 greaterthan line C.

Lines of constant density could be obtained forthe compressed liquid by integrating Cv/T, begin-ning at line B. Values of Cv from figure 26 wereused. The results indicate that these constantdensity lines are approximately parallel at a giventemperature for densities less than 500 Amagats.Data of table 14 indicate that there is a similarparallelism for higher densities near the criticaltemperature.

Values of entropy of the liquid for various pres-sures along the 17.34° K and 19.28° K isothermswere obtained by integration of the equation

(dS/dP)T=-(dV/dT)P. (10.1)

The values used for (dV/dT)P were based onsmoothed values of volume for the liquid as givenin table 32 for the temperatures 16.43° K, 18.24° K,and 20.33° K. The constant of integration waschosen to fit line B. From the results, a set ofconstant pressure lines, of which the segment F istypical, was obtained for various pressures. Inaddition, a point that should have been on the 860Amagat density line was obtained by interpolationand a line D was drawn through it and throughthe 860 Amagat density point on line B as deter-mined by eq 8.1. The line marked Df representsthe final correlation.

An unsatisfactory set of values of entropy forthe liquid along constant pressure lines wasobtained by integrating the Cv data of Gutsche,figure 27. Curves E are the results for 25 and 60atm, while the final correlation gave curves E'.

2. Final Correlation

In the final correlation, the saturation curve Bwas accepted and the isochores were consideredparallel. The isochores at high density weregiven by integration of CJT, beginning on line B.The isochores at intermediate density wereobtained by interpolation between values at highdensity and values below 500 Amagats. Theinterpolation was made along the 35° K isothermfrom an entropy-density plot extending fromp=860 Amagats to p=340 Amagats.

The extension of curve B to temperatures higherthan were given by calorimetric data for the liquidwas made from the lower parts of the interpolatedisochores and the temperature-density relationsfor the liquid at saturation pressure given by eq8.1.


\

The constant pressure lines were determinedmainly from the vapor-pressure equation and theequation

(dSfdV)T= (dP/dT)v. (10. 2)

At lower temperatures the lines were in fairagreement with Bartholomews PVT data, whichserved to locate them more closely.

The lines of constant enthalpy were determinedfrom integrals of TdS under the constant pressurelines and were checked by integration along theisochores based on the equation

(dH/dT)v=T(dS/dT)v+V(dS/dV)T. (10.3)

The location of the curves within the dome isquite straightforward, as the fractionation of the

ortho-para mixture is too small to affect thesecurves significantly.

The resulting temperature-entropy diagram fornormal hydrogen is presented in composite formin figures 31, 32, and 33. The thermal units usedare based on the calorie, the Kelvin degree, andthe gram, with pressures in atmospheres and den-sities in Amagat units.

The diagram shows lines of constant enthalpy,pressure and density and, in the region of coex-istance of liquid and vapor, lines of constant"quality." The painstaking construction of thecurves pertaining to the liquid region, amountingto a correlation of the data for the liquid, has beenmade by Kobert N. Schwartz, who has also drawnthe remainder of the diagram on the basis of thetables of this paper.


12 136 7 8 9 10 IIENTROPY CAL GM^DEG"'

FIGURE 31. Temperature-entropy diagram for H2 in the region 0° to 130° K,

14 15


300

290

280

270

260

13 14 15 169 JO II 12ENTROPY CAL GM'V

FIGURE 32. Temperature-entropy diagram for H2 in the region 130° to 300° K.Properties of Hydrogen 471

600 _

580 _

56<J_

5 4 6 .

52<J_

50d_

48<5_

46<5_

440

28010 II

FIGURE 33.

17 18_L

19

472

12 13 14 15

ENTROPY CALGM^DECT1

Temperature-entropy diagram for H2 in the region 280° to 600° K.Journal of Research

XL References

1. Thermodynamic Properties of the Hydrogens inthe Ideal Gas State

[1] D. M. Dennison, Proc. Roy. Soc. (London) 115, 483(1927).

[2] F. Rasetti, Phys. Rev. 34, 367 (1929).[3] W. F. Giauque, J. Am. Chem. Soc. 52, 4808 (1930).[4] W. F. Giauque, J. Am. Chem. Soc. 52, 4816 (1930).[5] H. H. Hyman, Phys. Rev. 36, 187 (1930).[6] H. H. Hyman and C. R. Jeppesen, Nature 125, 462

(1930).[7] R. T. Birge and C. R. Jeppesen, Nature 125, 463

(1930).[8] R. Rydberg, Z. Physik 73, 376 (1931).[9] O. Klein, Z. Physik 76, 226 (1932).

[10] J. L. Dunham, Phys. Rev. 41, 721 (1932).[11] R. W. Harkness and W. E. Deming, J. Am. Chem. Soc.

54,2850(1932).[12] C. R. Jeppesen, Phys. Rev. 44, 165 (1933).[13] H. C. Urey and D. Rittenberg, J. Chem. Phys. 1, 137

(1933).[14] G. N. Lewis and M. F. Ashley, Phys. Rev. 43, 837

(1933).[15] C. R. Jeppesen, Phys. Rev. 45, 480 (1934).[16] K. Mie, Z. Physik 91, 475 (1934).[17] C. O. Davis and H. L. Johnston, J. Am. Chem. Soc.

56, 1045 (1934).[18] H. L. Johnston and E. A. Long, J. Chem. Phys. 2,

389 (1934).[19] R. Wildt, Z. Astrophysik 9, 176 (1934).[20] H. Beutler, Z. physik. Chem. [B] 27, 287 (1934).[21] H. Beutler, Z. physik. Chem. [B] 29, 315 (1935).[22] G. K. Teal and G. E. Mac Wood, J. Chem. Phys. 3,

760 (1935).[23] Y. Fujioka and T. Wada, Sci. Papers Inst. Phys.

Chem. Research (Komagome, Hongo, Tokyo) 27,210 (1935).

[24] C. R. Jeppesen, Phys. Rev. 49, 797 (1936).[25] I. Sandeman, Proc. Roy. Soc. Edinburgh 59, 130

(1938-39).[26] R. T. Birge, Rev. Modern Phys. 13, 233 (1941).[27] H. W. Woolley, J. Chem. Phys. 9, 470 (1941).[28] D. D. Wagman, J. E. Kilpatrick, W. J. Taylor, K. S.

Pitzer, and F. D. Rossini, J. Research NBS 34, 143(1945) RP1634.

[29] G. N. Lewis and M. Randall, Thermodynamics andthe free energy of chemical substances (McGraw-Hill Book Co., Inc., New York, N. Y., 1923).

[30] P. S. Epstein, Textbook on thermodynamics (JohnWiley and Sons, Inc., New York, N. Y., 1937).

[31] F. H. Mac Dougall, Thermodynamics and chemistry(John Wiley and Sons, Inc., New York, N. Y.,1939).

[32] S. Glasstone, Textbook of physical chemistry (D.Van Nostrand Co., Inc., New York, N. Y., 1940).

2. Thermal Measurements on Gaseous Hydrogen

[33] O. Lummer and E. Pringsheim, Wied. Ann. 64, 555(1898).

[34] M. Pier, Z. Elektrochem. 16, 897 (1910).[35] N. Bjerrum, Z. Elektrochem. 18, 101 (1912).[36] A. Eucken, Sitzber. preuss. Akad. Wiss. 1912, 141

(1912).[37] K. Scheel and W. Heuse, Ann. Physik 40, 473 (1913).[38] I. Langmuir and G. M. J. Mackay, J. Am. Chem.

Soc. 36, 1708 (1914).[39] I. Langmuir, J. Am. Chem. Soc. 37, 417 (1915).[40] J. M. Crofts, J. Chem. Soc. 107, 290 (1915).[41] M. C. Shields, Phys. Rev. 10, 525 (1917).[42] M. Trautz and K. Hebbel, Ann. Physik 74, 285 (1924).[43] J. H. Brinkworth, Proc. Roy. Soc. (London) 107, 510

(1925).[44] J. H. Partington and A. B. Howe, Proc. Roy. Soc.

(London) 109, 286 (1925).[45] F. A. Giacomini, Phil. Mag. 50, 146 (1925).[46] R. E. Cornish and E. D. Eastman, J. Am. Chem. Soc.

50,627(1928).[47] F. R. Bichowsky and L. C. Copeland, J. Am. Chem.

Soc. 50, 1315 (1928).[48] A. Eucken and K. Hiller, Z. physik. Chem. [B] 4,

142 (1929).[49] E. J. Workman, Phys. Rev. 37, 1345 (1931).[50] A. Eucken and 0. Mticke, Z. physik. Chem. 18, 167

(1932).[51] K. Wohl and M. Magat, Z. physik. Chem. 19, 117

(1932).[52] A. Farkas, L. Farkas, and P. Harteck, Proc. Roy. Soc.

[A] 144,481(1934).[53] W. T. David and A. S. Leah, Phil. Mag. 18, 307

(1934).[54] D. Rittenberg, W. Bleakney, H. C. Urey, J. Chem.

Phys. 3,48 (1934).[55] A. J. Gould, W. Bleakney, and H. S. Taylor, J. Chem.

Phys. 2,362 (1934).[56] R. W. Fenning and A. C. Whiffen, Phil. Trans. Roy.

Soc. (London), 238, 149 (1939).[57] H. L. Johnston, I. I. Bezman, and C. B. Hood, J. Am.

Chem. Soc. 68, 2367 (1946).[58] H. L. Johnston, C. A. Swanson, and H. E. Wirth,

J. Am. Chem. Soc. 68, 2373 (1946).

3. PVT Relations for Gaseous Hydrogens

[59] H. Kamerlingh Onnes and C. Braak, Commun. Phys.Lab. Univ. Leiden 97a (1906), 99a, 100a (1907).

[60] H. Kamerlingh Onnes and W. J. de Haas, Commun.Phys. Lab. Univ. Leiden 127c (1912).

[61] P. Kohnstamm and K. W. Walstra, Koninkl. Akad.Wetenschappen Amsterdam, Proc. 17, 203 (1914).

[62] H. Kamerlingh Onnes, C. A. Crommelin, and P. G.Cath, Commun. Phys. Lab. Univ. Leiden 151c(1917).

[63] L. Holborn, Ann. Physik 63, 674 (1920).[64] J. Palacios Martinez and H. Kamerlingh Onnes,

Commun. Phys. Lab. Univ. Leiden 164 (1923).[65] H. Kamerlingh Onnes and F. M. Penning, Commun.

Phys. Lab. Univ. Leiden 165b (1923).[66] C. A. Crommelin and J. C. Swallow, Commun. Phys.

Lab. Univ. Leiden 172a (1924).[67] L. Holborn and J. Otto, Z. Physik 23, 77 (1924).


[68] F. P. G. A. J. van Agt and H. Kamerlingh OnnesCommun. Phys. Lab. Univ. Leiden 176b (1925).

[69] F. P. G. A. J. van Agt, Commun. Phys. Lab. Univ.Leiden 176c (1925).

[70] L. Holburn and J. Otto, Z. Physik 33, 1 (1925).[71] L. Holburn and J. Otto, Z. Physik 38, 359 (1926).[72] G. P. Nijhoff and W. H. Keesom, Commun. Phys.

Lab. Univ. Leiden 188d (1927).[73] E. P. Bartlett, J. Am. Chem. Soc. 49, 687 (1927).[74] E. P. Bartlett, H. L. Cupples, and T. H. Tremearne,

J. Am. Chem. Soc. 50, 1275 (1928).[75] G. P. Nijhoff and W. H. Keesom, Commun. Phys.

Lab. Univ. Leiden 188e (1928).[76] E. P. Bartlett, H. C. Hetherington, H. M. Kvalnes,

and T. H. Tremearne, J. Am. Chem. Soc. 52, 1363(1930).

[77] A. van Itterbeek and W. H. Keesom, Commun. Phys.Lab. Univ. Leiden, 216c (1931).

[78] A. van Itterbeek, Commun. Phys. Lab. Univ. Leiden,Supp. 70b (1931).

[79] A. Michels, G. P. Nijhoff, and A. J. J. Gerver, Ann.Physik 13, 562 (1932).

[80] W. E. Deming and L. E. Shupe, Phys. Rev. 40, 848(1932).

[81] A. Michels and A. J. J. Gerver, Ann. Physik 16, 745(1933).

[82] W. E. Deming and L. S. Deming, Phys. Rev. 45, 109(1934).

[83] J. B. M. Coppock, Trans. Faraday Soc. 31, 913 (1935).[84] G. E. Uhlenbeck and E. Beth, Physica 3, 729 (1936).[85] K. Schafer, Z. physik. Chem. [B] 36, 85 (1937).[86] K. Schafer, Z. physik. Chem. [B] 38, 187 (1937).[87] J. de Boer and A. Michels, Physica 5, 945 (1938).[88] R. Wiebe and V. L. Gaddy, J. Am. Chem. Soc. 60,

2300 (1938).[89] F. G. Keyes, Gas thermometer scale corrections based

on an objective correlation of available data forhydrogen, helium, and nitrogen; from: Tempera-ture, its measurement and control in science andindustry, American Institute of Physics, (ReinholdPublishing Corporation, 1941).

[90] C. S. Cragoe, Slopes of the PV isotherms of somethermodynamic gases at pressures below twoatmospheres; from: Temperature, its measure-ment and control in science and industry, AmericanInstitute of Physics (Reinhold Publishing Corpora-tion, 1941); J. Research NBS 36, 495 (1941)RP1393.

[91] A. Michels and M. Goudeket, Physica 8, 347 (1941).[92] A. Michels and M. Goudeket, Physica 8, 353 (1941).[93] A. Michels and M. Goudeket, Physica 8, 387 (1941).

4. Viscosity of Gaseous Hydrogen

[94] M. Trautz and P. B. Baumann, Ann. Physik 2, 733(1929).

[95] M. Trautz and F. W. Stauf, Ann. Physik 2, 737 (1929).[96] M. Trautz and W. Ludewigs, Ann. Physik 3, 409

(1929).[97] M. Trautz and H. E. Binkele, Ann. Physik 5, 561

(1930).

[98] M. Trautz and A. Melster, Ann. Physik 7, 409 (1930).[99] M. Trautz and R. Zink, Ann. Physik 7, 427 (1930).[100] M. Trautz and F. Kurz, Ann. Physik 9, 981 (1931).[101] M. Trautz and K. G. Sorg, Ann. Physik 10, 81 (1931).[102] M. Trautz and R. Heberling, Ann. Physik 20, 118

(1934).[103] B. P. Sutherland and O. Maass, Canadian J. Research

6, 428 (1932).[104] H. Adzumi, Bui. Chem. Soc. (Japan) 13, 199 (1937).[105] A. van Itterbeek and Miss A. Claes, Nature 142, 793

(1938) and Physica 5, 938 (1938).[106] A. van Itterbeek and O. van Paemal, Physica 7,

265 (1940).[107] A. van Itterbeek and O. van Paemal, Physica 7, 273

(1940).[108] W. H. Keesom and P. H. Keesom, Physica 7, 29

(1940).[109] H. L. Johnston and K. E. McCloskey, J. Phys.

Chem. 44, 1038 (1940).[110] R. Wobser and F. Muller, Kolloid-Beihefte 52, 165

(1941).

5. Thermal Conductivity of Gaseous Hydrogen

[111] A. Schleiermacher, Wied. Ann. 36, 346 (1889).[112] A. Winkelmann, Wied. Ann. 44, 177 and 429 (1891).[113] P. A. Eckerlein, Ann. Physik 3, 120 (1900).[114] P. Gunther, Dissertation, Halle (1906).[115] A. Eucken, Physik. Z. 12, 1101 (1911).[116] A. Eucken, Physik Z. 14, 324 (1913).[117] S. Weber, Ann. Physik 54, 437 (1917).[118] E. Schneider, Ann. Physik 79, 177 (1926).[119] E. Schneider, Ann. Physik 80, 215 (1926).[120] H. Gregory and C. T. Archer, Proc. Roy. Soc. [A]

110, 91 (1926).[121] K. F. Bonhoeffer and P. Harteck, Z. physik. Chem.

|B] 4, 113 (1929).[122] P. Harteck and H. W. Schmidt, Z. physik. Chem.

[B] 21, 447 (1933).[123] B. G. Dickens, Proc. Roy. Soc. (London) [A] 143,

517 (1934)[124] W.. G. Kannuluik and L. H. Martin, Proc. Roy.

Soc. (London) [A] 144, 496 (1934).[125] H. S. Gregory, Proc. Roy. Soc. (London) [A] 149,

35 (1935).[126] W. Northdurft, Ann. Physik 28, 137 (1937)'.[127] C. T. Archer, Proc. Roy. Soc. 165, 474 (1938).[128] H. Spencer-Gregory and E. H. Dock, Phil. Mag. 25,

129 (1938).[129] N. B. Vargaftik and I. D. Parfenov, J. Exptl.

Theoret. Phys. (U. S. S. R.) 8, 189 (1938).[130] W. G. Kannuluik, Proc. Roy. Soc. (London) [A]

175, 36 (1940).[131] H. L. Johnston and E. R. Grilly, J. Chem. Phys. 14,

233(1946).

6. Viscosity and Thermal Conductivity of GaseousHydrogen at High Pressure

[132] D. Enskog, Kungl. Svenska Vetenskaps AkademiensHandl. 63, No. 4 (1921).

[133] H. B. Phillips, J. Math. Phys. 1, 42 (1922).


[134] J. H. Boyd, Jr., Phys. Rev. 35, 1284 (1930).[135] R. O. Gibson, Dissertation, Amsterdam (1933).[136] E. W. Comings and R. S. Egly, Ind. Eng. Chem. 32,

714 (1940).137] S. Chapman and T. G. Cowling, The mathematical

theory of nonuniform gases, (Cambridge at theUniversity Press, 1939).

7. Viscosity of Liquid Hydrogen

[138] J. E. Verschaffelt and C. Nicaise, Commun. Phys.Lab. Leiden 151g (1917).

[139] W. H. Keesom and G. E. MacWood, Physica 5, 745(1938).

[140] H. E. Johns, Can. J. Research 17 [A] 221 (1939).

8. Vapor Pressures

[141] H. Kamerlingh Onnes and W. H. Keesom, Commun.Phys. Lab. Univ. Leiden, 137d (1913).

[142] P. G. Cath and H. Kamerlingh Onnes, Commun.Phys. Lab. Univ. Leiden, 152a (1917).

[143] J. Palacios Martinez and H. Kamerlingh Onnes,Commun. Phys. Lab. Univ. Leiden, 156b (1922).

[144] F. Henning, Z. Physik 40, 775 (1926).[145] W. H. Keesom, A. Bijl, and Miss H. Van der Horst,

Commun. Phys. Lab. Univ. Leiden, 217a (1931).[146] F. G. Brickwedde and R. B. Scott, The vapor pres-

sures of mixtures of ortho and para hydrogen(Unpublished).

[147] E. Cremer and M. Polanyi, Z. physik. Chem. [B]21,459 (1933).

[148] R. B. Scott, F. G. Brickwedde, H. C. Urey, andM. H. Wahl, J. Chem. Phys. 2, 454 (1934).

[149] F. G. Brickwedde, R. B. Scott, and H. S. Taylor,J. Research NBS 15, 463 (1935) RP841; J. Chem.Phys. 3, 653 (1935).

[150] F. G. Brickwedde and R. B. Scott, Vapor pressures,specific heats, heats of transition and molecularvolumes of liquid and solid hydrogen deuteride(Unpublished).

[151] R. D. O'Neal and M. Goldhaber, Phys. Rev. 58,574 (1940).

[152] W. F. Libby and C. A. Barter, J. Chem. Phys. 10,184 (1942).

9. Melting Curves

[153] H. Kamerlingh Onnes and W. van Gulik, Commun.Phys. Lab. Univ. Leiden 184a (1926).

[154] W. van Gulik and W. H. Keesom, Commun. Phys.Lab. Univ. Leiden 193b (1928).

[155] F. Simon, M. Ruhemann, and W. A. M. Edwards,Z. physik. Chem. [B] 6, 331 (1929).

[156] W. H. Keesom and J. H. C. Lisman, Commun. Phys.Lab. Univ. Leiden 213e (1931).

[157] W. H. Keesom and J. H. C. Lisman, Commun.Phys. Lab. Univ. Leiden 321a (1932).

10. PVT Relations for Condensed Phases

[158] J. Dewar, Proc. Roy. Soc. [A] 73, 251 (1904).[159] H. Kamerlingh Onnes and C. A. Crommelin, Com-

mun. Phys. Lab. Univ. Leiden 137a (1913).[160] H. Augustin, Ann. Physik 46, 419 (1915).[161] E. Mathias, C. A. Crommelin, and H. Kamerlingh

Onnes, Commun. Phys. Lab. Univ. Leiden 154b(1921).

[162] W. H. Keesom, J. de Smedt, and H. H. Mooy, K.Akad. Wetensch. Amsterdam, Proc. 33, 8, 814(1930); Commun. Phys. Lab. Univ. Leiden 209d(1930).

[163] R. B. Scott and F. G. Brickwedde, J. Research NBS19, 237 (1937) RP1023.

[164] K. Schafer, Naturwissenschaften 26, 563 (1938).[165] Helen D. Megaw, Phil. Mag. 28, 129 (1939).[166] K. Cohen and H. C. Urey, J. Chem. Phys. 7, 157

(1939).

11. Thermal Properties of Condensed Hydrogen

[167] W. H. Keesom, Commun. Phys. Lab. Univ. Leiden137e (1911).

[168] A. Eucken, Verh. deut. Phys. Ges. 18, 18 (1916).[169] A. Eucken, Verh. deut. Phys. Ges. 18, 4 (1916).[170] W. H. Keesom and H. Kamerlingh Onnes, Commun.

Phys. Lab. Univ. Leiden 153a (1917).[171] F. Simon and F. Lange, Z. Physik 15, 312 (1923).[172] K. Clusius and K. Hiller, Z. physik. Chem. [B] 4,

158 (1929).[173] K. Mendelssohn, M. Ruhemann, and F. Simon,

Z. physik. Chem. [B] 15, 121 (1931).[174] K. Clusius and E. Bartholome", Z. physik. Chem.

[B]30, 237 (1935).[175] A. Pitt and W. J. Jackson, Can. J. Research 12,

686 (1935).[176] E. Bartholome* and A. Eucken, Z. Elektrochem. 42,

547 (1936).[177] E. Bartholomew Z. physik. Chem. [B] 33, 387 (1936).[178] H. Gutsche, Z. physik. Chem. [A] 184, 45 (1939).

12. Unclassified

1179] G. Rutledge, Phys. Rev. 40, 262 (1932).[180] H. J. Hoge and F. G. Brickwedde, J. Research NBS

22, 351 (1939) RP1188.[181] Mathematical Tables Project, Tables of Lagrangian

Interpolation Coefficients, (Columbia UniversityPress, 1944).

WASHINGTON, August 7, 1947.


compilation of thermal properties of hydrogen in its various

Documents