Page 109 - Modern physical chemistry

5.12 The Chemical Potential 99

Example 5.7

Obtain Cp - Cv for an ideal gas. Rearrange the ideal gas equation to

V= nRT
P
and differentiate with respect to T at constant P:

nR
P
Substitute this result and that from example 5.6 into formula (4.81)

C Cv = (0 + p) n:; = nR.
p
-
Note that this agrees with the result obtained in example 4.4,

5. 12 The Chemical Potential

So far, we have considered systems in which the number of moles of each constituent
is constant. But in general, this condition does not apply. Reactions go on and material
may move into or out of the given system. Then one must introduce additional terms into
formula (5.76). and the equations derived from it.
Material always carries energy. The reversible rate of change in internal energy caused
by altering the number n i of moles of the ith constituent, while keeping all other inde-
pendent mole numbers, entropy 8, and volume V constant, is called the chemical poten-
tial Pi; thus
OE)
Pi= (- [5.93]
ani S,V,othern's
and
dE=TdS-PdV+ LPidni' [5.94]
i
Combining formula (5.94) with the differentials of equations (5.77). (5.78), (5.79) leads to

dH = TdS + VdP+ LPi dni' [5.95]
i

dA=-PdV-8dT+ LPidni, [5.96]
i

dG = V dP -8 dT + LPi dni' [5.97]
i
One may consider building up a system from nothing at constant pressure and tem-
perature. Throughout such a process, equation (5.97) reduces to

[5.98]