Thermodynamics and Chemistry

CHAPTER 11 REACTIONS AND OTHER CHEMICAL PROCESSES

11.1 MIXINGPROCESSES 308

the final mixture areyADV 1 (A)=V 2 andyBDV 1 (B)=V 2. The total work of the reversible
mixing process is thereforewDnARTlnyACnBRTlnyB, the heat needed to keep the
internal energy constant isqDw, and the entropy change is

ÅSDq=TDnARlnyAnBRlnyB (11.1.26)

It should be clear that isothermal expansion of both pure gases from their initial volumes
to volumeV 2 without mixing would result in the same total work and the same entropy
change.
When we divide Eq.11.1.26bynDnACnB, we obtain the expression for the molar
entropy of mixing given by Eq.11.1.9withxireplaced byyifor a gas.

11.1.5 Molecular model of a liquid mixture

We have seen that when two pure liquids mix to form an ideal liquid mixture at the sameT
andp, the total volume and internal energy do not change. A simple molecular model of
a binary liquid mixture will elucidate the energetic molecular properties that are consistent
with this macroscopic behavior. The model assumes the excess molar entropy, but not
necessarily the excess molar internal energy, is zero. The model is of the type sometimes
called thequasicrystalline lattice model, and the mixture it describes is sometimes called
asimplemixture. Of course, a molecular model like this is outside the realm of classical
thermodynamics.
The model is for substances A and B in gas and liquid phases at a fixed temperature.
Let the standard molar internal energy of pure gaseous A beUA(g). This is the molar
energy in the absence of intermolecular interactions, and its value depends only on the
molecular constitution and the temperature. The molar internal energy of pure liquid A is
lower because of the attractive intermolecular forces in the liquid phase. We assume the
energy difference is equal to a sum of pairwise nearest-neighbor interactions in the liquid.
Thus, the molar internal energy of pure liquid A is given by

UADUA(g)CkAA (11.1.27)

wherekAA(approximately the negative of the molar internal energy of vaporization) is the
interaction energy per amount of A due to A–A interactions when each molecule of A is
surrounded only by other molecules of A.
Similarly, the molar internal energy of pure liquid B is given by

UBDUB(g)CkBB (11.1.28)

wherekBBis for B–B interactions.
We assume that in a liquid mixture of A and B, the numbers of nearest-neighbor mole-
cules of A and B surrounding any given molecule are in proportion to the mole fractionsxA
andxB.^4 Then the number of A–A interactions is proportional tonAxA, the number of B–B

(^4) This assumption requires the molecules of A and B to have similar sizes and shapes and to be randomly mixed
in the mixture. Statistical mechanics theory shows that the molecular sizes must be approximately equal if the
excess molar entropy is to be zero.