172 Gases

This approximation is motivated by tiie form of the ideal gas mixture chemical potential

i n which P,- = y,P. The Lewis-Randall rule simplifies the mixture equations because it

eliminates the need to calculate the fugacities for different compositions, requiring only

the pure-state quantities. Many nonideal gas mixtures can be described well by this

approach at low to moderate pressures.

Probleiifis

Conceptual and thought problems

9.1. Consider the results for ideal gases derived from quantum mechanics. Write an

expression for the function ̂ (r, v) that includes the parameter A ( r ) .

(a) Show that the behavior of the entropy as T ^ 0 is unrealistic i n a quantum

sense. What approximation i n tiie derivation is responsible for this behavior?

(b) The Sackur-Tetrode equation is valid when the thermal de Broglie wave­

length is much less than the average molecular separation distance, A ( r ) <

(V/N)^'^, which is generally the case at high temperatures. Calculate the

temperature at which this inequality is first violated for an ideal gas at

atmospheric pressure and wi th a mass of 6.6 x 10"^'' kg/molecule (which is

typical of argon).

9.2. At constant T, V, and AT, does the entropy of an ideal gas increase or decrease with

molecular weight? Explain why in physical terms.

9.3. An ideal gas mixmre is pressurized at constant temperature. Indicate whether

each of the foUovnng increases, decreases, or stays the same, or there is not enough

information to tell.

(a) the total mixture heat capacity Cp = (Siï/9r)p_j;v)

(b) the chemical potential /;,• of each component

(c) the entropy of mixing, ASmix

9.4. A n ideal gas mixture is heated at constant pressure. Indicate whether each of the

following increases, decreases, or stays the same, or there is not enough infor­

mation to tell.

(a) the partial pressure P,- of the dominant component

(b) tiie total mixmre heat capacity Cp = (3H/3r)p,(;vi

(c) the chemical potential /<,• of each component

(d) the entropy of mixing, AS^ix

9.5. For a single-component system, indicate whether or not each of the following

functions provides enough information to completely specify all of tiie intensive

thermodynamic properties of a monatomic gas whose atomic mass is Icnown:

(a) s{T, P)

(b) f{T, P], where ƒ is the fugacify

(c) P{T, v) containing the l imit v ^ co

236 Solutions: fundamentals

Problems

Conceptual and thought problems

12.1. Expliciüy derive Eqn. (12.3) for the ideal mixing entropy of a C-component system.

12.2. A tiny amount of salt is added to l iquid water at standard conditions. Indicate

whether the given quantity increases, decreases, or stays the same, or tiiere is not

enough information to specify the answer witi iout Icnowing more about tiie

system. Consider tiiis to be an ideal solution.

(a) The pressure at which the water wiU freeze, at 0 °C.

(b) The chemical potential of the water.

(c) The partial molar volume of the water.

12.3. For an arbitrary binary mixmre of species 1 and 2, indicate whether or not each of

tiie following relations is true in general. Explain why or why not.

(a) f—^

(b) dNi\ ^ f d S \

(c) AV^^ < 0

(d)

(1^1 > o T,P

12.4. Consider an ideal binary solution of components A and B. Is each of tiie following

quantities positive, negative, or zero, or is tiiere not enough information to tell?

A / S , V , , , B

pure (b) AVmix = Vsoln - ^ Vi.

(c)

(d)

T,P,NB

T.P.I'A

B / r , p

Problems 239

dependent and given b y ^ = u,/{k,T) where w is shorthand for the expression

Z{WAB - [WAK + t f B B ) / 2 ) . Express your answer in terms of w.

12.12. A container is prepared wi th equal numbers of moles of two immiscible liquids

A and B (e.g., oU and water). A solute C wi l l dissolve in both liquids, and a tiny

amount of it is added to the overall system, totaling 0.01 moles. At equilibrium, i t

I S found that the mole fraction of C in A is 0.001 and tiiat feat i n B is 0.002 An

addtttonal 0.01 nroles of C is feen added. What are the new mole fractions^

Provide a formal proof of your reasoning.

12.13. Consider a l iquid solvent (species 1) with an amoum of solute (species 2)

dissolved in it. Explain how the activity coefficient ean be determined in tiie

following cases and wi t i i tiie indicated experimental data.

(a) The pure solute is a liquid at the same conditions. The entire solution is

brought to liquid-vapor equilibrium and the compositions of tiie phases as

well as tiie partial pressures are measured as a ftmction of temperature or

pressure. Assume tiie vapor behaves ideally.

(b) The pure solute is a solid at the same conditions. The freezing-point tempera­

ture is measured as a function of tiie solution composition

(c) In turn, prove tiiat y ,(r , P, x ,) can be determined given loiowledge of

12.14. Find an expression for tiie temperature dependence of tiie activity coefficiem of a component i i n solution,

(dlny.

dT P, allx,-

In light o fyour result and Example 12.2, which exerts a greater influence on tiie

activity coefficient, tiie temperature or tiie pressure?

Fundamentals problems

12.15. Explici t iyderiveEqn.(12.7)fromEqn.(12.4).Youmaywamtorecall t i iechainrule

12.16. Consider a solution of two components A and B at vapor-liquid equilibrium

Henry s law is an approximation tiiat states tiiat, i f component A is very dilute

then P A = IOCA. Here, P A is tiie partial pressure of A in tiie gas phase, x^ is tiie

solution-phase mole fraction of A, and Kis Henry's constant, specific to the two

components involved but independem of concentration. Assuming ideal gases

and solutions, express 7Cin terms of standard chemical potentials. What makes

Henry s law diff'erent from Raoult's law ( P A = P ^ A ^ X A ) !

244 Solutions: fundamentals

(d) Wi l l your answer in part (c) be different if mixing is performed at constant

total volume instead of constant pressure, i.e., if a partition between species 1

and 2 i n a rigid container is slowly removed?

12.32. Consider a container of l iquid that is held at its boiling temperature T at pressure

P. A small amount of solute is added, but the temperature and pressure are

maintained at their original values. Since the solute has elevated the boiling

temperature, the system wi l l condense entirely into a l iquid.

One way to return the mixmre to boiling is to reduce the pressure while

maintaining the temperature at the original T. Find an equation that can be

solved to f ind the new vaporization pressure P' as a function of solute mole

fraction x^oiute and Hquid number density p^ = v^\ Assume ideal solution behav­

ior for the solvent species, and that a negUgible amount of solute is i n tiie vapor

phase. To a fair approximation, the solvent vapor at these conditions can be

described by the equation of state

v a = ^ + b + cP

where b and c are constants.

12.33. The solubility of nonpolar gases in l iquid water is typically very low. Consider

nitrogen in particular. Its solubility is characterized by tiie equilibrium mole

fraction X N of dissolved nitrogen in the l iquid phase (with water mole fraction

= 1 - X N ) when tiie system is in equilibrium wi th a vapor phase (with

corresponding mole fractions yN and yw)- The mole fraction X N is typicaUy of

tiie order 10"^ The vapor pressure of water at 300 K is P^^ = 3.5 IcPa, and you

can assume that tiie impact of its temperature dependence on solubility is

relatively weak. Use ideal models i n what follows.

(a) At 1 bar and 300 K, estimate the mole fraction of water that is present i n the

vapor phase, y w (b) Show that tiie solubility of nitrogen is given approximately by

X N = C ( r , P ) ( l - y w ) P

where C(r, P) is a constant that is independent of concentrations. Find an

expression for C(r, P) i n terms of standard and pure chemical potentials.

(c) Do you expect the solubility to increase or decrease w i t i i temperamre?

Explain by finding the temperature dependence of C(r, P).

12.34. At standard conditions, water and butanol phase-separate so tiiat a mixmre of

tiiem forms two l iquid phases. To such a mixmre, acetic acid is slowly added;

wi th each increase i n the amount added its mole fraction in each phase is

determined experimentally.

(a) I t is found for small amounts of added acetic acid tiiat tiie mole fractions in

the two phases nearly always have the same ratio and can be described by a

distribution or partition coefficient K = x ^ / x ^ tiiat is approximately 3.4. Here

the mole fractions pertain to acetic acid and tiie superscripts B and W to tiie

Problems 245

butanol and water phases. Explain why this might be the case and find an

expression for K i n terms of thermodynamic quantities,

(b) A total of 0.5 moles of acetic acid is added to a mixture of 1 L each of water

and butanol. Find the mole fractions of acetic acid in each phase. The

densities of butanol and water are 0.81 g/ml and 1.0 g/ml, respectively.

Assume that you can neglect any water that enters the butanol phase and

vice versa.

12.35. The solubility of n-butanol in water is 7.7% by mass at 20 °C and 7.1% at 30 °C.

The solubility of water in n-butanol, on the other hand, is 20.1% at 20 °C and

20.6% at 30 °C. The heat of solution is tiie enthalpy change of preparing a mbrture

f rom pure components, AH^i^. Estimate tiie heat of solution per mole of n-

butanol in water when tiie former is dilute. Similarly, estimate the heat of

solution per mole of water in n-butanol when water is dilute. To begin, first

prove tiiat tiie enthalpy of mixing for a solvent 1 wi t i i a dilute amount of species

2 is given by

A^^mix TT , * , -k* , .k ^H2-h'2= hi* - H;

where Hz is the partial molar enthalpy of species 2 in tiie mixmre, is the molar

entiialpy of pure l iquid 2, and /z" is the molar entiialpy of l iquid 2 in solvent

when l iquid 2 is in an infinitely dilute state. Then, consider equilibrium between

water- and n-butanol-rich phases. Note tiiat tiie partial molar enthalpies can be

related to the T dependence of the chemical potentials.

F U R T H E R R E A D I N G

IC. Denbigh, The Principles of Chemical Equilibrium, 4tii edn. New York: Cambridge University Press (1981).

K. Dill and S. Bromberg, Molecular Driving Forces: Statistical Thermodynamics in Biology,

Chemistry, Physics, and Nanoscience, 2nd edn. New York: Garland Science (2010).

T. L. Hill, An Introduction to Statistical Thermodynamics. Reading, MA: Addison-Wesley (1960);

New York: Dover (1986).

D. A. McQuarrie, Statistical Mechanics. Sausalito, CA: University Science Books (2000).

A. Z. Panagiotopoulos, Essential Thermodynamics. Princeton, Nf: Drios Press (2011).

J. M. Smith, H. V. Ness, and M. Abbott, Introduction to Chemical Engineering Thermodynamics,

7Ü1 edn. New York: McGraw-Hill (2005).

J. W. Tester and M. Modell, Thermodynamics and Its Applications, 3rd edn. Upper Saddle River, Nf: Prentice Hall (1997).