Thermodynamics and Chemistry

CHAPTER 11 REACTIONS AND OTHER CHEMICAL PROCESSES

11.1 MIXINGPROCESSES 307

12

(a)

A B

1 2

(b)

A

A+B

B

1 2

(c)

A+B

Figure 11.3 Reversible mixing process for ideal gases A and B confined in a cylinder.

Piston 1 is permeable to A but not B; piston 2 is permeable to B but not A.

(a) Gases A and B are in separate phases at the same temperature and pressure.

(b) The pistons move apart at constant temperature with negative reversible work,

creating an ideal gas mixture of A and B in continuous transfer equilibrium with the

pure gases.

(c) The two gases are fully mixed at the initial temperature and pressure.

Butpiis equal toniRT=V (Eq.9.3.3). Therefore, if a fixed amount ofiis in a container
at a given temperature,Sidepends only on thevolumeof the container and is unaffected by
the presence of the other constituents of the ideal gas mixture.
When Eq.11.1.24is applied to apureideal gas, it gives an expression for the molar
entropy
SiDSiRln.p=p/ (11.1.25)

wherepis equal tonRT=V.
From Eqs.11.1.24and11.1.25, and the fact that the entropy of a mixture is given by the
additivity ruleSD

P

iniSi, we conclude thatthe entropy of an ideal gas mixture equals
the sum of the entropies of the unmixed pure ideal gases, each pure gas having the same
temperature and occupying the same volume as in the mixture.
We can now understand why the entropy change is positive when ideal gases mix at
constantTandp: Each substance occupies a greatervolumein the final state than initially.
Exactly the same entropy increase would result if the volume of each of the pure ideal gases
were increased isothermally without mixing.
The reversible mixing process depicted in Fig.11.3illustrates this principle. The initial
state shown in Fig.11.3(a) consists of volumeV 1 (A) of pure ideal gas A and volumeV 1 (B)
of pure ideal gas B, both at the sameTandp. The hypothetical semipermeable pistons are
moved apart reversibly and isothermally to create an ideal gas mixture, as shown in Fig.
11.3(b). According to an argument in Sec.9.3.3, transfer equilibrium across the semiper-
meable pistons requires partial pressurepAin the mixture to equal the pressure of the pure
A at the left, and partial pressurepBin the mixture to equal the pressure of the pure B at
the right. Thus in intermediate states of the process, gas A exerts no net force on piston 1,
and gas B exerts no net force on piston 2.
In the final state shown in Fig.11.3(c), the gases are fully mixed in a phase of volume
V 2 DV 1 (A)CV 1 (B). The movement of piston 1 has expanded gas B with the same reversible
work as if gas A were absent, equal tonBRTlnåV 2 =V 1 (B)ç. Likewise, the reversible
work to expand gas A with piston 2 is the same as if B were absent:nARTlnåV 2 =V 1 (A)ç.
Because the initial and final temperatures and pressures are the same, the mole fractions in