Thermodynamics and Chemistry

CHAPTER 11 REACTIONS AND OTHER CHEMICAL PROCESSES

11.1 MIXINGPROCESSES 309

interactions is proportional tonBxB, and the number of A–B interactions is proportional to
nAxBCnBxA. The internal energy of the liquid mixture is then given by

U(mixt)DnAUA(g)CnBUB(g)

CnAxAkAACnBxBkBBC.nAxBCnBxA/kAB (11.1.29)

wherekABis the interaction energy per amount of A when each molecule of A is surrounded
only by molecules of B, or the interaction energy per amount of B when each molecule of
B is surrounded only by molecules of A.
The internal energy change for mixing amountsnAof liquid A andnBof liquid B is
now

ÅU(mix)DU(mixt)nAUAnBUB
DnAxAkAACnBxBkBBC.nAxBCnBxA/kABnAkAAnBkBB
DnA.xA1/kAACnB.xB1/kBBC.nAxBCnBxA/kAB (11.1.30)
With the identitiesxA 1 D xB,xB 1 D xA, andnAxB DnBxADnAnB=n
(wherenis the sumnACnB), we obtain

ÅU(mix)D

nAnB

n

.2kABkAAkBB/ (11.1.31)

If the internal energy change to form a mixture of any composition is to be zero, as it is for
an ideal mixture, the quantity.2kABkAAkBB/must be zero, which meanskABmust
equal.kAACkBB/=2. Thus, one requirement for an ideal mixture is thatan A–B interaction
equals the average of an A–A interaction and a B–B interaction.
If we write Eq.11.1.29in the form

U(mixt)DnAUA(g)CnBUB(g)C

1

nACnB

.n^2 AkAAC2nAnBkABCn^2 BkBB/ (11.1.32)

we can differentiate with respect tonBat constantnAto evaluate the partial molar internal
energy of B. The result can be rearranged to the simple form

UBDUBC.2kABkAAkBB/.1xB/^2 (11.1.33)

whereUBis given by Eq.11.1.28. Equation11.1.33predicts that the value ofUBdecreases
with increasingxBifkABis less negative than the average ofkAAandkBB, increases for the
opposite situation, and is equal toUBin an ideal liquid mixture.
When the excess molar volume and entropy are set equal to zero, the model describes
what is called aregular solution.^5 The excess molar Gibbs energy of a mixture isGmED
UmECpVmETSmE. Using the expression of Eq.11.1.31with the further assumptions that
VmEandSmEare zero, this model predicts the excess molar Gibbs energy is given by

GmED

ÅU(mix)

n

DxAxB.2kABkAAkBB/ (11.1.34)

This is a symmetric function ofxAandxB. It predicts, for example, that coexisting liquid
layers in a binary system (Sec.11.1.6) have the same value ofxAin one phase as the value
ofxBin the other.

(^5) Ref. [ 79 ].