GUEST AUTHOR A. G. Williamson University of
Otago
Dunedin, N e w Zeolond
II
Textbook Errors, 67
Raoult'r Law and the Thermodynamic Definition of Ideal Mixing
I t is frequently implied and sometimes stated explicitly in textbooks1 of physical chemistry that for liquid mixtures Raoult's Law and the "thermodynamic" definition of ideal mixing as given by equaSuggestions of material suitahle for this column and guest columns suitahle for publication directly should he sent with as many details as possible, and particularly with references t o modern texthooks, to WilliamH. Eberhardt, Schoolof Chemistry, Georgia Institute of Technology, Atlanta, Georgia 30332. Since the purpose of this column is to prevent the spread and continuation of errors and not the evalustion of individual texts, the source of errors discussed will not be cited. I n order to he presented, an error must occur in itt least two independent recent standard hooks.
tions (1) and (2) below are equivalent when the vapors behave as perfect gases. The following thermodynamic analysis shows that this is not so and that the apparent identity of these two definitions of ideal mixing arises from neglect of the effect of pressure on the chemical potential of the liquid phase. Raoult's law for a liquid mixture is usually stated in the form Pyi(T,P,zi) = Pio(T,Pio)zi
(1)
where Py,(T,P,x,) is the partial pressure of component i in the vapor in equilibrium with the liquid mixture containing mole fraction xi of component i and Pto X (T,P,O) is the vapor pressure of pure component i.
Volume 43, Number 4, April 1966
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21 1
Equation (1) is often taken as a definition of an ideal mixture. The thermodynamic definition of ideal mixing is a statement of the relation between the chemical potential of component i of the mixture, the chemical potential of the pure liquid component and its mole fraction in the mixture at given temperature and pressure pi(T,P,zi) = piS(T,P)
+ R T ln xi
(2)
For our comparison of equations (1) and (2) we shall need the following relations: The condition for equilibrium between two phases u and B
,:'
(3)
=
for each component which occurs in both phases. The expression for the variation of chemical potential of component i of a gas mixture with pressure and composition yi ai'(T,P,yi)
=
-+(TI
+ RTInfdT,P,yi)
(4)
where pi+ is the chemical potential at unit fugacity and J, is the fugacity of the ith oampouent in the mixture. For a pure gas equation (4) becomes @(T,P) = pi+(T)
+ R T ln J,(T,P)
(5)
which differs from Raoult's law [eqn. (I)] by a factor exp { VIo(T)(P-PCo)/(RT) 1. Thus even when the vapors are perfect gases, equations (1) and (2) are not equivalent definitions of ideal mixing. It should be noted, however, that the expression given in many texts f* = fc0zi
(14)
where f, andf," are the fugacities of component i in the liquid mixture and in the pure liquid respectively at the same pressure (and temperature) is identical with equation (2). The commonly made error is in the assumption that the fugacities referred to in eqn. (14) can be equated to the fugacities of component i in the mixture a t the vapor pressure P of the mixture and of the pure component at its own vapor pressure P,' respectively. I t is of interest to consider the sign and magnitude of the deviations from Raoult's law (eqn. 1) implied by the thermodynamic definition of ideal mixing (eqn. 2) for a binary mixture of liquids whose vapors are perfect gases. For such a mixture
and when the gases me perfect, equations (4) and (5) reduce to
+ R T In Pyi
(6)
+ RT In P
(7)
rig(T,P,yi) = pi+(T) and pio(T,P) = ai+(T)
respectively. The expression for the variation of chemical potential of a pure liquid with pressure a t constant temperature pi0(T,P) = pi"(T,P')
+
(8)
where Vdois the molar volume of the liquid a t temperature T and pressure P. For the relatively small pressure changes which we are considering V." may be regarded as independent of pressure and equation (8) may be written in the form rs-'(T,P)
= piq(T,P') + ViD(T)(P- P')
(9)
Combining eqns. (2), (3), (4) and (5) we can obtain the relations between the fugacity of component i in the equilibrium vapor and its chemical potential in the ideal liquid mixture piS(T,P)
+ R T In zi = p;+(T) + RT In Jt(T,P,yi)
(10)
and between the fugacity of the vapor in equilibrium with pure liquid component i and the chemical potential of the liquid where P and Pioare the vapor pressures of the mixture of composition x, and of the pure component respectively a t temperature T. I.'rom equations (9), (lo), and (11) it follows that
+ R T In zi +
RT ln fdT,P,yi) = R T ln J;(T,PL0)
Viq(T)(P
- Pi")
(12)
where f,(T, P, y,) is the fugacity of component i in the equilibrium vapor a t the vapor pressure P of the mixture of composition x, and f,(T,P,") is the fugacity of the vapor in equilibrium with pure liquid i a t the vapor pressure P," of the pure liquid. If the vapors are perfect gases equation (12) becomes R T In Pyi(T,P)
c
R T In Pio(T,Pi')
+ RT In zi + Vio(T)(P - Pi") (13)
21 2 / Journal of Chemical Education
where Py, and Py, are the partial pressures of i and j in the equilibrium vapor. Since the diierences between eqn. (I) and eqn. (2) are small we can, to a very good approximation, substitute in the third term of eqn. (13) to obtain RT In Pyi(T,P)
-- R T In Pi"(T,Pi0) + RT ln zi +
Vi0(T)lP~'(T)- P
