10.6 Partition Function for a Gas
453
Thus,
A = −Nk B T ln (ez mol /N )
(10.74)
for our set of independent and indistinguishable molecules.
10.6.17 Ideal Gas Chemical Potential
Later, in considering air in the lungs, we will employ the chemical potential for an
ideal gas,
μ =
∂A
∂N
T ,V
= −k B T ln (z gas /N ) ,
(10.75)
μ = k B T ln
(N/V ))
3
T
− k B T ln z internal .
(10.76)
If there are several types of molecules in the system, each given a distinct k index,
then
μ k = k B T ln
(N k /V ))
3
T ,k
− k B T ln z internal .
(10.77)
Alternative to N k /V , we can use the ideal gas relations to find the chemical potential
in terms of concentration or pressure:
N k /V = N A n k /V = N A [C k ] = p k /(k B T ),
where N A is Avogadro’s number, [C k ] ≡ n k /V is the molar concentration of
molecules of type k, and p k is the partial pressure for molecules of type with index
k. So the chemical potential for the k-th component can be expressed in terms of
that component’s partial pressure as
μ k = μ
o
k + k B T ln (p k /p
o ) ,
(10.78)
where μ o
k depends on the temperature of the gas (but not its pressure), or, in terms
of the molar concentration [C k ] = n k /V of the k-th component, as
μ k = μ
c
k + k B T ln ([C k ]/[C
o
k ]) .
(10.79)