A Tractable Model for Studying Solution Thermodynamics Carl W. David University of Connecticut, Storrs, CT 06268 I t would be useful if examples existed which covered topics in solution thermodvnamics. Further. since the teaching of statistical thermodynamics is now becoming standard in nhvsical chemistm courses. it is of value to have examnles of the use of statistiia~thermodynamics that illustrate aspects of classical thermodynamics in an illuminating way. Where the manipulation of formulas in classical thermodynamics might appear sterile, since the identity of substances (what makes I p different from 02)is obscured, the use of more "realistic" examples from statistical thermodynamics can lead to a better appreciation of both the structure of the classical thermodynamics and the relationship of fundamental, theoretically derived models and thermodynamics. A nontrivial example exists that allows one to "see" analytical expressions for the fugacity, the chemical potential, and various partial molar quantities whose form is usually not snecified. In these formulas the atomic narameters bew m r manipulati\,e parameters inside the classiral tht-rmud\wnmirs. thercbv iendinv some sense of indiridualitv to the results a s they apply to aifferent kinds of molecul& Our example is a nonideal gaseous mixture, technically a solution, but of course not the kind of solution that we would really like to treat. Unfortunately, no model for liquid solutions exists that is tractable and simple enough for use in the classroom. A Nonldeal Gaseous Mixture
Consider a gaseous mixture whose equation of state is
where NA is the number of molecules of substance A, NB is the number of molecules of B, o~ is the hard sphere volume of A molecules, and o s is the hard sphere volume of B molecules. The partition function from which this equation of state came was Z =
(V - NAoA- N ~ ~ ~ ) ~2=mAhT * + ~ B3 N ~ / 2 2rmgkT N A N
where mAis the mass of A particles, and me is the mass of B particles. The Helmholtz Free Energy must he
where
+
N ~ / 2
( 7 ( ) 7 (') )
Journal of Chemical Education
+
where we have been careful to write G' a i a function of l ' a n d Nutice thnt in eoinr from .+l(T.\? to G1T.o) we are exnlir. .kly illustrating trhe Gfect of the Legendrctransformakon. The volume (which is an independent variable for the Helmholtz free energy) must be excised, and the pressure (which is an independent variable for the Gihbs free energy) must be present, that is, G = G(T,p). To find the chemical potential of substance A, we are required to take the partial derivative of G with respect to NAat constant T, p and NB, that is, D.
which equals
484
+
in the standard manner, since A = -KT In Z. As G = A pV and pV = (NA Ne)kT + (NAUA NB~B)P, i t follows that
We would get a similar expression for GB. Then it followsthat
that is,
-
-
where x~ is the mole fraction of A. As NB 0, the mixture approaches pure A ( x ~ I), and we obtain
which, a s p becomes
-
po (pa is the pressure a t the standard state)
Equation 8 can be recast into a more traditional form by noting that
which recovers V. Osmotic Pressure The osmotic pressure experiment can be treated using the results we have obtained. I t is true that we rarely think of gaseous mixtures and osmotic pressure at the same time, but the underlying thermodynamics remains the same even if the phases are nonordinary, therefore justifying this unothodox approach. Assume that there is a way to have a solution of A and B in equilibrium with pure A (solvent). The excess pressure on the solution, a, required to maintain this equilibrium is called the osmotic pressure. The fundamental idea is that the chemical potential of the A molecules in the solution must equal that of the A molecules in the pure solvent, that is, 'ye
E TAP + poo,lkT + lnp,
-' A S - In kT
= -
(21)
'A
(11)
so that one has from eq 7
which, in the limit x~
-
which simplifies to 1becomes
I t is interesting t o notice how nicely the fugacity emerges in this example, that is, if we define fA
=pem~RT
(14)
which properly hecomes fA
lim-=1 P-0
then in the standard state (p fA0
- -
(15)
P
PO^
PO,f~
wdkT
f ~ " )f, ~ is ' (16)
Equation 12 then becomes the familiar
= Z A ,and R = We have gone to a per mole basis, where NOUA Noh, where No is Avogadro's number. Notice that this last equation itself is the classical thermodynamics starting poi$ for the introduction of the fugacity. From eq 11we have (for the solution situation)
Partial Molar Volumes Taking the partial of ' A with respect t o p holding T,NA, and NB constant, we obtain an expression of the partial molar volume of A:
which, unfortunately, is a transcendental equation for the osmotic Dressure a. This form shows whv the standard treatments of osmotic pressure always lead to equations involving integrals that require assumptions about u being constant (in liquids). In our gaseous mixture, such an approximation is clearly invalid. Discussion Solution thermodynamics is not a trivial subject. An example such as the one discussed here is helpful in understanding where the difficulties lie and why they exist. In proceeding from the abstract to the concrete, there is a chance that the ideas concerning standard states ( p po), ideal gas limits p 0, etc., will seem less formal, and more understandable.
-
-
Caveat The original partition function (eq 2) from which this entire paper is drawn is itself modestly fraudulent. In one dimension a gaseous mixture of hard spheres (lines of different lengths) has an exact partition function similar to the one discussed here' hut that is as far as it goes. For threedimensional hard sphere mixtures, the partition function cited is incorrect! That does not diminish the value of the exercise, since a discussion of the difficulties of evaluating partition functions for three-dimensional systems of interacting particles is required under any and all circumstances. This example merely shows how one could proceed if in fact one were able to obtain the exact partition function for real systems.
'
David, C . W. J. Chem. Phys. 1964,40,2418. Tonks, E. Phys. Rev. 1936, 50, 955.
Volume 64
Number 6
June 1987
485
