218 e. a. guggenheim and n. k....

218 E. A. Guggenheim and N. K. Adam.

may possibly be due to self reversal. At the greater retardations the red line also exhibits a variation in visibility which may be due to the same effect. Furthermore, the principal component of the second blue line becomes less homogeneous. Therefore, the Osram lamp, when excited with a current of 2 amperes, is definitely unsuitable as a standard source of monochromatic red radiation.

Some slight evidence was observed, at smaller retardations, of differing structure in the red line as produced from the Michelson lamp and the Osram lamp in the 2-ampere condition. In the case of the comparisons of the Michelson lamp and the Osram lamp in the 1-ampere condition, this effect, if present, was largely masked by incidental temperature disturbances which particularly affected the steel slides and screw of the Fabry-Perot interferometer during these comparisons. The intercomparisons of the red radiation emitted by the two sources are to be repeated with better apparatus and under conditions more suitable for the attainment of the very high precision which is required to discern any small differences of structure which may be present.

The Thermodynamics of Adsorption at the Surface of Solutions.

By E. A. Guggenheim and N. K. Adam.

(From the Sir William Ramsay Laboratories of Physical and Inorganic Chemistry, University College, London, W.C.l, and Imperial Chemical Industries, Ltd.)

(Communicated by F. G. Donnan, F.R.S.—Received September 13, 1932.)

Introduction.In Gibbs5* thermodynamic theory of surfaces, the general equations govern

ing equilibrium at interfaces were given, including equations governing adsorption. He pointed out, that in order to assign definite numerical values to the surface excess V of each component, it is necessary to choose a definite position for a certain mathematical surface, placed parallel to, and within, or near to, the inhomogeneous region between the two bulk phases. Gibbs5 most general equations apply to any possible position of this surface, but the particular equation commonly known as “ Gibbs5 adsorption equation '5 was

* ‘ Collected Works,’ vol. 1, p. 219, et seq.

on July 12, 2018http://rspa.royalsocietypublishing.org/Downloaded from

http://rspa.royalsocietypublishing.org/

Adsorption at the Surface of Solutions. 219

deduced by the aid of a particular choice of the position of this mathematical surface, such that the surface excess of one of the components vanished.

The object of this paper is to examine the form taken by Gibbs’ general equations, when other conventions relating to the position of this dividing surface are chosen, and to establish the quantitative relations between the values of the surface excess of each component, obtained on the various conventions used. Our aim in doing this is to attem pt to remove some of the confusion which has arisen, when interpreting the results of calculations of the r in terms of the physical structure of the surface. The particular convention relating to the position of the dividing surface used by Gibbs suffers from the disadvantage that it is not easily visualised in terms of the physical structure of the surfaces, and is also unsymmetrical in respect of the components of the solution. Some of the alternative conventions examined here seem to us more convenient in both these respects, and the values of T obtained by their aid are no more difficult to calculate from surface tension and vapour pressure data on solutions.

Confusion is likely to arise from the not uncommon definition of “ surface excess per unit area ” a t the surface of a solution, as the difference between the quantity of solute contained in a given volume of the solution containing unit area of free surface, and tha t in an equal volume without any free surface. I t will be shown in this paper tha t (except at infinite dilution) the numerical value of the surface excess defined in this manner is not the same as tha t obtained on the particular conventions used by Gibbs in obtaining the most well-known adsorption equation : indeed equations (39) to (44) show that the ratio of the two numerical values of T on Gibbs’ convention and on the above definition [ r ('> of our notation] becomes infinite as the solution approaches the composition of the pure second component, whose adsorption is being considered. The papers of Schofield and Rideal* and of Wynne-Jonesf are of interest as illustrating the difficulty of forming a definite physical conception of the surface excess as defined on Gibbs’ convention [ r 2(1) in our notation].

Dependence of Surface Excess T on Position of Geometrical Surface.

The discussion will be confined to systems consisting of two bulk phases in complete thermodynamic equilibrium, separated by a plane surface. In this case, the temperature T, the pressure P, and the chemical potential ix{

* ‘ Proc. Roy. Soc.,’ A, vol. 109, p. 61 (1925); ‘ Phil. Mag.,’ vol. 13, p. 806 (1932).t ‘ Phil. Mag.,’ vol. 12, p. 907 (1931).




of every species i will each have a value constant throughout the system. The two bulk phases will be completely homogeneous except in the immediate neighbourhood of the surface. We may therefore imagine a cylindrical column of unit cross-section, with its axis perpendicular to the bounding surface, connecting the homogeneous part of one phase with the homogeneous part of the other phase. This is shown diagrammatically in fig. 1. The first phase denoted by a is completely homogeneous up to and somewhat beyond the surface A and the second phase denoted by (3 is completely homogeneous, down to and somewhat beyond the surface B. Between the surfaces A and B, somewhere about S and S', is a region which is still homogeneous in the directions perpendicular to the axis of the cylindrical column but is not so in the direction parallel to this axis. The essential feature of Gibbs’ treatment

is to choose a definite geometrical surface such as S to which the thermo-dynamic properties of the system are referred as follows. Imagine a hypothetical system in which the phase a remains homogeneous right up to the surface S and the phase S remains homogeneous right down to the surface S. In this hypothetical system the number of moles of the species i contained in the column between the surfaces A and B will be

a»<‘ + (# — *) Gf> (!)

A'

a-x

VA

iX

f>

-B

■ S------

S ' - -

a-xiiI

A

IX '

where a denotes the distance between A and B, x denotes that between A and S, {a — x) that between S and B, while cf, c f denote the number of moles per unit volume in the homogeneous interiors of the phases a and (3. In the actual

system the number of moles of the species i contained in the column between the surfaces A and B will in general not be given by the expression (1). We may define quantities I \ such that the number of moles of each species i between A and B in the actual system is

.V

OfFig. 1.

x c f + (a — x) cj* + IV (2)

The may then be positive or negative and are called the “ surface excess of the species i ”

For constant temperature and pressure of the whole system variations of



the quantities Tt and the surface tension y are related by the thermodynamic formula*

— dy = S r 4 dy(3.1)

where fx£ denotes the chemical potential of the species i at any part of the system. Provided the vapours of the various components i obey the laws of perfect gases, we may write as an alternative to (3.1)

— dy = RT S T log p £, (3.2)i

where R is the gas constant and p t the partial vapour pressure of the species i at any part of the system.

Suppose now that instead of the surface S we should choose a different surface S' to which to refer the thermodynamic behaviour of the system. Corresponding to this surface we should define quantities r ' f such that the number of moles of the chemical species i between A and B is

x 'c f + (a — x') c f + F £, (4)

where x' denotes the distance AS'. Now the actual number of moles of each species i between A and B is obviously independent of a purely conventional choice between the surfaces S and S'. Hence the expressions (2) and (4) must be identical, that is

x c f + (a — x)cf + r £ = x 'c f + (a — x') c f + T 't- (5)or


F i — l j — {x x) (c/ c£“). (6)

Comparing the relations of the form (6) for the various chemical species present, we obtain

r , - r f r \ - r k _c f — Gia Ck — Gka (7)

Corresponding to the relations (3.1) or (3.2) we have

— dy — 21 I1',-. d\Xii (8.1)

— dy — RT S r'£ d log p t.i

(8.2)For (3) and (8) to be simultaneously true it is necessary that

s (r '<- r i)d[r< = onr i(9.1)

S (T ' i— Yd d log pi = 0.i

(9.2)

* Gibbs, loc. cit.9 vol. 1, equation (508).



Substituting from (6) and (9) we find as necessary conditions

(x' — a?) £ G / — C 0 (10.1)or

(x' — x)£ (cf — c/) d log = 0. (10.2)

I t is easily seen that equation (10.1) is in accordance with the Gibbs-Duhem* relation when we have at constant temperature and pressure

S c,“ d\Xi 0 (11)

S c / d[L{ = 0. (12)i

Similarly (10.2) is in accordance with the Duhem-Margules relation

S cf° dlog Pi — 0(13)i

in the phase a, and£ c / d log Pi = 0 (14)i

in the phase (3fTo sum up : when the geometrical surface S is displaced a distance 8x, the

values of the various become altered by amounts §r\. given by


8T{ = 8x. (c / - c l) (15)but the expressions

£ d\XiX

(16.1)occurring in (3.1), or

£ Tf dlog Pii

(16.2)

occurring in (3.2), remain invariant so that the relations

— dy = £ Tf d[i{i

(17.1)

— dy = RT £ r f d log p { (17.2)i

remain unaffected.Liquid-vapour Interface.—Let us now suppose that the phase a is a slightly

volatile liquid and the phase [3 is vapour. I t is generally accepted that the density of matter in the non-homogeneous region is comparable with that in the homogeneous part of the liquid phase while the density in the vapour phase is in comparison negligibly small. In this case some of the above formulae

* Gibbs, loc. cit.9 vol. 1, equation (98). t Of. Gibbs, loc. cit.f vol. 1, pp. 233-237.



become simplified, and the physical meaning of the various becomes much clearer.

We may in this case neglect c f ,omit the index in and write

a5*cf + r f (18)

for the number of moles of the species iabove the surface A. I t is now possible to redefine the I\- in a manner which makes no explicit reference to the geometrical surface S and which makes clearer the physical significance of the T,. If we compare a portion of the liquidcontainingunit area of surface withanother very nearly equal portion in the interior of the liquid then the former will contain T, moles of each species imore than the latter.

The indefiniteness in the values of the I \ was according to the original definition due to indefiniteness relating to the exact position of the geometrical surface S. According to the new definition it is due to indefiniteness about the quantity of liquid in the interior with which one compares the portion of liquid containing unit area of surface. According to either definition there is one degree of freedom in the assignment of values to the various lb. The variations 8 I\ corresponding to changes in position of the dividing surface up to which the hypothetical system remains homogeneous, are subject to the relation


which is the form now taken by (15). In terms of the mole fractions Nf, Nk, ... in the interior of the liquid phases this may be written

(19.2)

It follows that all linear combinations of the form

Nfc I\ - r k (20)

are invariant with respect to displacement of the geometrical surface S.Alternative Conventions Relating to the TV.—As Gibbs pointed out one may

choose the position of the geometrical surface S such tha t for one component, say 1, the value of Fx shall be zero. According to this convention a portion of the liquid containing unit area of surface contains T,- moles of each species

more than a portion in the interior which contains exactly the same quantity of the species 1. The values of the F; corresponding to this convention,




referred to henceforth as convention 1, will be specified by writing r <(1> [Gibbs, used r <(1)]. We have by definition

IY11 = 0. (21).

Similarly, if we define IP as the excess number of moles of the species i in a portion of liquid with unit surface area over a portion in the interior containing exactly the same number of moles of the species 2, we shall use the notation r / 2> and shall refer to this convention as convention 2. We then have by definition

r 2(2) = 0. (22)

A convention more symmetrical with respect to the various species present is the following. A portion of the liquid with unit area of surface contains I \ moles of each species i more than a portion in the interior containing exactly the same total number of moles of all species. Using the notation T /N) for this convention, to be referred to as convention N, we have by definition

S I 7 N> = 0. (23)»

According to yet another convention we may define as the number of moles of the species iin a portion of the liquid containing unit area of surface more than in a portion in the interior of exactly the same mass. Corresponding to this convention, to be referred to as convention m, we shall use the notation r / M) and we have by definition

Z Mf IYM> = 0, (24)i

where m4 denotes the molar mass (“ molecular weight ”) of the species i.Finally we may define IP as the number of moles of the species in a portion

of the liquid containing unit area of surface more than in a portion in the interior of exactly the same volume. This convention will be called convention v, and the notation r f(v) will be used. If we may with sufficient accuracy assume the surface tension to be independent of the pressure, it follows thermodynamically that change of surface area takes place without expansion or contraction. In this case the relation between the IP(V> is

S v< r,.<v> = 0, (25)i

where v* denotes the partial molar volume of the species i.We have cited five alternative conventions for fixing the values of the r , to

which correspond respectively the notations r <<1), i y 2), r / N), r / M), IYV).




Each convention is defined by one of the equations (21) to (25) which are all linear relations of the general form

S a, T, = 0. (26)

One of the primary causes of our writing this paper is the common practice of defining lb according to convention v and then applying to the T /v) relations that are exact only for the I t is only with convention v that the geometrical surface S coincides exactly with the physical boundary of the liquid.

Relations between the Pi on Different Conventions.

We shall now investigate how the various conventional sets of I\. are related one to another. In this investigation we shall confine ourselves to a system of only two components 1 and 2. The relation (26) then becomes

ai rv + ota r 2 = 0, (27)

while the Gibbs formula (3.2) becomes

- r i ^ logPi + r 2 ^ log^2 (28)

and the Duhem-Margules relation (13) may be written

Ni d log + n 2 d log p 2 = 0.

If we eliminate Tx and px from (27), (28), (29) we obtain

If on the other hand we eliminate P2 and p2 we obtain

- jyT (h = r x d log { 1 +- t \ » v oc2 n 2J

Prom (30) it follows that( 0 2 ^ !I ax NXJ

(29)

(30)

(31)

(32.1)

is invariant with respect to displacement of the Gibbs surface S. Similarly from (31) we see that

+ (32.2)l a2 n2J

is invariant as regards displacement of the surface S.

VOL. CXXXIX.—A. Q




According to (20) the quantityNi r 2 — n2 I \ (33)

is also invariant as regards displacement of the surface S.The forms taken by (27) and the values of oq, «2 corresponding to the various

kinds of Tj, Fa are given by

OII

to II o (34)

i y 2) = o ax = 0 (35)

i y N> + r 2<N) = o ocj == a2 (36)

m , ryM) + Mo r 2(M> = o _ Ogm2 (37)

Vl i y v> + v2 r 2<v> = o ii (38)

Substituting the appropriate values of oq, a 2 from relations (34) to (38) into (30), we obtain for conventions 1, N, m , v, the following relations, of which the first is, of course, the ordinary Gibbs adsorption equation

1 3y RT 3 l ^ 7 2 r 2(i) = i y N) j i + — J

— r w Ij + N2?L2\ = I. Nx Mj

p (v)1 2 I N, v j (39)

For convention 2, of course, r 2(2> is zero in accordance with equation (22).Introducing the mean molar mass m and the mean molar volume v defined

bym = n x m x + n 2 m 2 (40)

V — N j Vj -I- n 2 v 2 (41)

the relations (39) may be written

h=RT 0 log p 2

_ nx r 2n) i y N> = 2s. r 2(M> = ^ i y v> (42.1)

P (2) 1 2 . 0. (42.2)

Similarly by substituting from the relations (34) to (38) into (31) we obtain

— i k dyRT 0 log Pl n2 I\<2> = W") = - I \

M.

i y 1' = o.

(M) I\<V> (43.1)

(43.2)



The relations (29), (36), (42) and (43) may all be combined in the form

__ i dy _ ^2 dy _ p (n> _ __ p (n>R T 0 1 o g f t ' RT a log f t 2 1

= nx r2(1> = - n2 rx(2) - - f2<m> = - — rx(5i>

Adsorption at the Surface Solutions. 227

- I r w ------ I r <— „ 1 2 — „ 1 1

1Y: r2(2) = o.

(44.1)

(44.2)

Equations (44) give the relations between all the F for both components.It is easy to check that (44) is in agreement with the invariance of the

expression (33), that is

r2<N> = - rx<N> = nx r2<N> - n2 rx<H>= nx r2(i> = - n2 rx(2)= nx r « - n2 r «= nx r2<v> - n2 rxw- nx r 2<x> - n2 Fx(x) (45)

where the index X refers to any convention for fixing the values of the F other than the five conventions 1, 2, N, M, v.

Relations for very Dilute Component.—It is of special interest to consider the behaviour of the various Fx and T2 for mixtures differing only slightly from the pure substances 1 or 2. If the amount of the species 2 is very small compared with that of the species 1, we have approximately

and consequently by (44.1)

J _ h . = p <K>RT a log f t 2

y 0) _1 2 — y (M) _ y (v)2 2 *

(46)

(47)

Thus in the immediate neighbourhood of n2 — 0 four of the conventions give identical values for F 2.

For small values of n 2 where the surface film contains relatively few molecules of component 2, it is reasonable to expect that the number of molecules of 2 in the surface which at these great dilutions is practically equal to r2(v), and therefore to the other F2) will be proportional to n 2. We may consider the surface film as a region in which the molecules have a different energy from



228

those in the interior, then Boltzmann’s distribution law will hold good, the concentrations of component 2 in the surface and the interior bearing a constant ratio to each other, determined by the work of adsorption. Therefore, for small values of n2

r 2 a n2. (48)

This is the analogue of Henry’s law and Nernst’s distribution law for surface films ; it is well known to be verified by experiment.

For small values of n2 we may therefore write

r y N) = r 2(1) - r « = i y v> = n2, (49)

where k2 is independent of n2. Similarly for small values of Ny we may write

_ imn) = _ p x2 = _ i Mm) = _ r/v ) _ /olNl, (50)

where kt is independent of NvRelations for almost Pure Component.—Formulae (49) and (50) give the

behaviour of the various F of the dilute component. By comparing these with (44) we can obtain directly the behaviour of the F of the almost pure component. We obtain for small values of n2 using (40)

- r 1<N> = - n 2 r v 2> = - ^ r 1(M) = - - 1 r 1(v) - k 2 n2 (5i)m2 v2

from wfiicli we see that the values of — — 1VM), — Fx(v) tend to zeroas n2 tends to zero and nx to unity, but — I \ (2) on the other hand tends towards the constant value Jc2. Similarly in the neighbourhood of = 0 we have the relations.

F2in' = Nir 2(1> = —2 Fo(M) - ^ r 2<v> = (52)Ml - Vj

The r - N Curves.—From the definitions of F(N), F(M) and F(v) or alternatively from formula (44.1) we see that as long as the two species are of comparable molar mass and molar volume the values of F(N), F(M) and T(v) for each species will at any given composition be of the same order of magnitude. The curves for F(N), T(M), for each species plotted against the mole fraction n2 will be qualitatively similar. In particular according to the relations (49), (50), (51), (52), they all pass through both the origins n2 = 0 and = 0with finite slopes. For F(v) the geometrical surface S coincides with the physical boundary of the liquid and if the two species are of comparable molar mass and molar volume this will be approximately true also for T(IS) andr(M).

E. A. Guggenheim unci N. K. Adam. on July 12, 2018http://rspa.royalsocietypublishing.org/Downloaded from


Adsorption. at the Surface ooq_ . 7

The behaviours of F 2(1) and of T1(2) are, however, quite different. For solutions very dilute with respect to the species 2 there is no appreciable difference between r 2(1) and the F 2 of the other conventions. But for solutions consisting mainly of the species 2, with only a little of the species 1, whereas F2(n), IY m), r 2(v) tend to zero with N1? T2(1) on the other hand tends to the finite value Jcv For such solutions the distance between the geometrical surface S and the physical boundary of the liquid is not inappreciable. From the data for the system water-alcohol, discussed in detail below, one finds using (15) that for almost pure alcohol (species 2) containing a little water (species 1) the geometrical surface corresponding to r 2(1) is distant about 1 *3 A. from the physical boundary of the liquid, a distance which cannot be regarded as negligible compared with the thickness of the lion-homogeneous film. In the case of r x(2) in the neighbourhood of pure water (nx = 1) the distance between the geometrical surface S ai\d the boundary is according to (15) about 40 A. which is possibly even greater than the total thickness of the lion-homogeneous film.

The quantities T2(1) and I \ (2) were introduced by Gibbs for mathematical convenience, but those wishing to obtain a physical picture of the 64 surface excess ” T will probably find T(v), T(M) or T(N) easier to visualize. Should one, however, prefer to use r 2(1) and r x(2) then in considering the curves obtained by plotting these against the mole-fraction it should be remembered that the conventions 1 and 2 are themselves sufficient to account for the comparatively complicated shapes of the curves, without there being any physical peculiarity in the system. It is to be emphasised that the curves for F2(1) and F1'2) are thermodynamically associated with curves for the F(N), 1 (M) and F(v) and the last three will have simpler forms than the first two.

Relationship to Osmotic Pressure.—For a solution of only two components, such as we have been considering, if we choose to regard one component 1 as solvent and the other 2 as solute, we can obtain differential relations between the F and the osmotic pressure 11 of the solution. The exact thermodynamic relation between the osmotic pressure II and the partial vapour pressure px of the solvent* is

vx dll = u± dpv (55)

where iq denotes the molar volume of the solvent vapour. Provided the vapour obeys the laws of perfect gases (55) becomes

vx dll = RT d log pv (56)

* Porter, ‘ Proc. Roy. Soc.,’ A, vol. 79, p. 519 (1907).




Substituting from (56) into (44.1) we obtain

N 2 __ p (N) --- --- T1 (N)v ^ a n 2 ' 1

- Nl r2(l) = - n2 iy»M P (M) M p (M),, 1 2 ------- — J 1

— r 2<T> = - L r^ ), (57)

The relation between F 2(v) and IT may be written in a particularly simple form, namely,

n (v) __ ^ 2 dy __ dy,2 v a n 2 an (58)

where c2 denotes the number of moles of solute in unit volume of solution. Formula (58) has been obtained by Milner,* by an elementary proof involving a reversible cycle, and is generally quoted in text-books. I t is, however, not made clear that the simple formula (58) applies only to r 2(v), whilst the corresponding formulae for T2(1), T2(N), T2(M) are according to (57) somewhat more complicated.

Application to the System Water-Ethyl Alcohol.

The necessary data have been taken from the surface tension measurements of Bircumshawf and the vapour pressure measurements of Dobson. J Before using Dobson’s data, they were examined to see whether the vapour pressure measurements for the two components were compatible with the Duhem- Margules relation (29). I t was found that the dotted curves of fig. 2 could be drawn through the experimental points, the tangents (drawn as full lines) being chosen so as to be in exact agreement with the Duhem-Margules relation, the maximum deviation of any experimental point from the curve chosen being 0*5 mm. of mercury pressure and nearly all the points lying within 0*2 mm. of the selected curve.

Table I gives the actual data used in calculating the various T. The suffix1 refers to water and 2 to alcohol. Column 1 gives the mole fractions ; columns2 and 3 the vapour pressures used in millimetres of mercury ; column 4 the

* ‘ Phil. Mag.,’ vol. 13, p. 90 (1907). t ‘ J. Chern. Soc.; p. 887 (1922). t ‘ J. Chem. Soc.,’ p. 2806 (1925).



A dsorption at the Surface of Solutions. 231

values of — ^ or of —2 ^ taken from either curve ; as already mentionedPi 3ni Pz 9n2

the two curves were selected to conform with the thermodynamic requirement

Mole fraction, alcohol Fra. 2.

Table I.

| P v Pi- Pl d \_ ^2 dp%

P'l 9 2y- - " 3n2 -

dp*jy»> x iow

in moles/cm.1.r,(*» x io<s>moles/sq. A.

0 23-75 0-0 100 72-2 0-0 0-0 0-0 0-00 1 21-7 17-8 0-76 36-4 11-8 15-6 6-3 3-80-2 20-4 26-8 0-41 29-7 6-5 16-0 6-45 3-90-3 19-4 31-2 0-37 27-6 5-4 14-6 5-9 3-60-4 18-35 34-2 0-355 26-35 4-4 12-6 5-1 3-10-5 17-3 36-9 0-41 25-4 4-25 10-5 4-25 2-60-6 15-8 40-1 0-53 24-6 4-5 8-45 3-4 2-06| 0-7 13-3 43-9 0-655 23-85 4-7 7-15 2-9 1-750-8 10-0 48-3 0-77 23*2 4-75 6-2 2-5 1-5

i 0-9 5-5 53-3 0-915 22-6 5-0 5-45 2-2 1-331-0 0-0 59-0 1-00 22-0 5-2 5-2 2-1 1-27




that these quantities should be equal. In the fifth column are given the values of the surface tension, obtained from a large scale plot of Bircumshaw’s data.

From this plot we read the values of —- n 2 0y/0N2 given in the sixth column. The values of — p2 dy/dp2 were obtained by two methods : either by dividing

the values of — n2 0y/0N2 by those of ^ or by plotting y against log p2p2 on2

and taking the slopes of this curve directly. It is to be observed that if the

solutions were ideal, the values of — p 2 dy/dp2 in column 6 and of — n2 0y/0N2

in column 7 would be identical, while the values of ^ in column 4 wouldP2 ^ 2

be unity throughout. The eighth column gives the values of r 2(1) in moles per square centimetre, obtained by dividing the values in the seventh column by RT which is 2*48 X 1010 ergs. We estimate the uncertainty in calculating the individual values of P2(1) to be of the order 0*4 X 10“10 moles per square centimetre, but the smoothed curve given in fig. 3a is probably rather more accurate than this. The last column gives F2(1) in molecules per square A.

Table II shows the values of all the F calculated from the values of r 2(1) given in Table I, by equations (44). The partial molar volumes used in the calculation of the T(v) were obtained from tables of the density of alcohol-

Table II.(All the P are given in moles/cm.2 X 1010).

N2 r 2( n) = - r y n) r 2(M) - / y m> r j y ) - / y v ) A d ) - j y 2)

0 0 0 0 0 0 0 0 0 0 0 0 0-0 2200 05 5-55 5 1 5 13-2 5 1 14-8 5-85 1110 1 5-7 4-95 12*6 4-75 14 05 6-3 570-2 5 1 5 3-95 101 3-6 11-35 6-45 25-70-3 4 1 5 2-8 7-2 2-45 8-05 5-9 13-750-4 3 05 1-9 4-8 1-55 5-3 5 1 7-60-5 2 1 1-2 3 05 0-95 3-25 4-25 4-250-6 1-35 0-7 1-8 0-55 1-9 3-4 2-250-7 0-9 0*4 1 1 0-3 J -1 2-9 1-250-8 0-5 0*2 0-55 0 1 5 0-6 2-5 0-60-9 0-2 0 1 0-25 0 05 0-25 2-2 0-251 0 0 0 0 0 0 0 0 0 0-0 2-1 0-0

water mixtures. Fig. 3a shows the values of all the different T2, the adsorption of alcohol; fig. 3b shows the negative adsorption of water — Fv The following features are to be particularly noted. The four T2 curves all coincide in the neighbourhood of n2 = 0, as also do the four Fx curves in the neighbourhood of nx = 0. The curves for IVN), r 2(M), r 2(v)> all approach the nx = 0 origin




with finite but different slopes, but the r2(1) curve approaches the nx = 0 axis horizontally, cutting it at the height 2-1 x 10-10 moles per squarecentimetre numerically equal to the gradient of the — T1 curves in the neighbourhood of n2 = 1. Similarly the curves for — rx(N), — I V M), — ri(y), all approach the n2 = 0 origin with finite but different slopes, while the — r / 21 curve cuts the n2 = 0 axis a t the height k2 — 220 X 10"10 equal to the slope

Mole fraction, alcoholFio. 3a.—Adsorption of Alcohol. 3b.—Adsorption of Water.

of the T2 curves at the n2 = 0 origin. Finally the curves for T2<N> and — r 2(N> are identical.

Seeing that the curves for the r(N), T(M), r(v), must pass through both the origins n2 = 0 and Nj = 0 with finite slopes, the general shape of these curves is of the simplest kind. As already pointed out the somewhat less simpleshapes of the r2(1) and IY2) curves are entirely due to the peculiarities of the conventions 1 and 2.

A Possible Structure for the Surface of the Solutions.

The simplest theory of the structure of the surface is that the non-homogeneous layer is a film only one molecule thick. To investigate this theory it is most convenient to adopt the convention of placing the Gibbs’ geometrical surface S so as to separate the homogeneous liquid from the unimolecular

vol. cxxx ix .—A. R



layer outside it. We shall refer to this as convention u and use the notation p w r 2(D). I t is obvious that r x(u) and r 2(u) are actually the number of moles of water and alcohol respectively in unit area of the unimolecular layer. The T(c) unlike the r (1), T(2), r (N), r (M), T(v) do not satisfy a relation of the form (27). To deduce the relations between the T(u) and T(N) we therefore have to use a procedure somewhat different from our former one. From the relation (45) for IYX) and T2<x) we have for I \ (u) and r (c)

_ jmn) = r 2<N) = Ni r 2w - n2 i y u>. (59)

If in a solution of given composition the areas occupied by a mole of water in the unimolecular layer are ax and by a mole of alcohol a2 then

AjiY*) + A2r 2<°> = 1. (60)

By solving (59) and (60) simultaneously we obtain

r (c) _ A1 r 2<N) + N2 (61)NXAX + N2A2

F (u) _ A2I i 'N> + (62)NXAX + NaA2

As one would expect in the immediate neighbourhood of n2 = 0i y u> = r 2(N) = Fa(M) = r 2(v) = rgd) (63)

and in the immediate neighbourhood of Nx = 0

i\<u> = r x(N) = r 1(M) = r x(v) = (64)

The values of the r (u), unlike those of the r (1), T(2), T(N), T(M), T(v) cannot be calculated from purely thermodynamic data. For to use (61) and (62) we must know the values of a± and a2 to be inserted at each composition. Actually we know these values only roughly. As an approximation we must assume that ax and a2 are independent of the composition. For the alcohol we may safely assume a2 to be about 0*12 X 1010 sq. cm. per mole, equivalent to 20 A.2 per molecule, this being approximately the area occupied by a long chain aliphatic alcohol molecule orientated perpendicular to the surface in the insoluble unimolecular surface films.* For water it is more difficult to decide on the most likely value for Av If we take as the area occupied by a water molecule that of the side of a cube, whose volume is equal to the mean volume occupied by a single molecule in the liquid state, the relevant value of ax is almost exactly one-half that assumed for a2.

Since, however, the molecules are presumably not cubic the true value of


* See Adam, “ Physics and Chemistry of Surfaces,” p. 50 (1930).



Adsorption at the Surface of Solutions. 2S5

may be either greater or smaller according to how the molecules are orientated. We have made calculations on the three alternative assumptions >—

(1) Aj equal to one-third of(2) A± equal to one-half of a2.(3) ax equal to two-thirds of a2.

The results of these calculations are given in Table III, which is in three sections corresponding respectively to the three alternative values of Ar. In each section the values of r2(u) and of ri(u) calculated by means of formulae (61) and (62) are given in the first and second columns. In the third column. . . r (c) .in each section are given the values of ------- 2 - , the molecule fraction of

1 1 i 1 2alcohol in the unimolecular layer. It will be seen at once that the value assumed for A1? the area of the water molecule is of small importance. According to each of the three assumptions concerning the value of av as the mole fraction of alcohol in the interior increases to about 0-1 or 0 • 2 the mole fraction of alcohol in the unimolecular layer increases rapidly to about 0-7. It then remains almost unchanged until the mole fraction in the interior is over 0-5, It then increases steadily to the value of 1 - 0 in pure alcohol

The very small apparent decrease in r2(u) as n2 increases from about 0*2 to about 0*4 is probably not real It is probably due to the use of inexact values for ax and a2. It is extremely likely that the true values of a1 and a$

Table IIL(All F in 10 10 moles/cm,2).

Aj = 0 04 x 10“ c m .2. A | = 0 *06 X 1010 c m .2. Ai = 0 08 X 10“ c m .2.a 2 = 0-12 x 1010 c m .! . a 2 — 0 12 X 1010 c m .2. Aa = 0-12 X 10“ c m .2.

r tm . r / i o Ador t m

r 2ooA w i y n )

r 2ooA (D) + A (D) r xw + r \ (v ) r ^ v ) + r zm

0 0-0 25-0 0 00 0 0 16-7 0 00 0 0 12-5 0-00•05 6-2 6-4 0-49 6-1 4-5 0-57 6 0 3-5 0-631 6-8 4-6 0-60 6*7 3-3 0-67 6*6 2-6 0-72l'2 7-25 3-25 0-69 7 1 2*5 0-74 7-0 2 1 0-773 7-25 3-25 0-69 7-0 2-7 0*72 6-9 2-2 0-764 7-25 3-25 0-69 6-9 2-9 0*71 6-7 2-4 0-745 7-3 3 1 0-70 6-9 2-9 0-71 6-7 2-5 0-736 7-45 2-65 0-74 7 1 2-5 0-74 6-8 2-3 0-75• 7 7*65 2 0 0-79 7*4 1-9 0-80 715 1-8 0-808 7-9 1-3 0-86 7-65 1-4 0-85 7-5 1-25 0-86

9 8 1 0-7 0-94 8-0 0-7 0-92 7-9 0-65 0-930—

8-35 0 0 1 0 0 8-35 0 0 100 8 *35 0 0 1-00



236 Adsorption at the Surface of Solutions.

depend somewhat on the relative numbers of alcohol and water molecules in the surfaces just as the partial molar volumes and v2 vary with the composition of the solution.

In spite of a certain degree of uncertainty as to the exact values of ax and a2, the results of our calculations do seem to show that our theory that the only part of the liquid that is not homogeneous is a unimolecular layer leads to reasonable results. As the mole fraction of alcohol increases in the interior so it does in the unimolecular layer. The increase is perhaps not as steady as one might have expected, but this departure from the highest conceivable simplicity is not altogether surprising when one compares it with the complicated dependence of the partial vapour pressures or the partial molar volumes on the composition in the water-alcohol system. The alternative assumption made by Schofield and Rideal* that at mole fractions of alcohol greater than about 0*3 the surface contains a complete unimolecular layer of alcohol leads to the conclusion that below this unimolecular film there must be a layer containing excess of water. This theory seems to us unnecessarily complicated.

When our article was ready for the press a paper by Butler and Wightman j* appeared with new measurements of the surface tension of water-alcohol mixtures at 25° C. confirming the values of Bircumshaw. Values for r 2(1> are computed, agreeing fairly well with our own computations. Butler and Wightman also discuss the significance of these values in connection with the theory that the non-homogeneous part of the system is a unimolecular layer. Their formula (5) is a special case of our (45) and is equivalent to our (59), while their formula (4) is the same as our (60).

Summary.

The convention commonly used in fixing Gibbs' geometrical dividing surface, in which the surface excess F1(1) of one of the components is arbitrarily taken as zero, is ill adapted to forming a physical picture of the structure of the surface. Several other conventions for fixing the interface have been examined. The relations between the values of T obtained on these various conventions are derived from Gibbs' general equations and formulae are given for calculating each one of them.

The structure of the surface of water-alcohol mixtures is calculated approximately from the assumption that the inhomogeneous layer does not extend below the first layer of molecules.

* ‘ Proc. Roy. Soc.,’ A, vol. 109, p. 61 (1925). t ‘ J . Chem. Soc.,’ p. 2089 (1932).



218 e. a. guggenheim and n. k....

Documents