Statistical Physics, Second Revised and Enlarged Edition

Mixed systems and chemical reactions 147

own chemicalpotentialμswhichisagaintheGibbsfree energyper particleofspecies
s.Hence we may always write for the total Gibbs free energy of the assembly

G=

∑

s

NNsμs (13.15)

The sum goes over allcomponentssofthe assembly (e.g.s=nitrogen, oxygen, carbon
dioxide, water, ...ifwe were consideringair), andthere areNNsparticles ofcomponent
s.It is worth stressing that (13.1 5 ) is of very general applicability, although we shall
onlyillustrateits usein gaseous assembliesinwhatfollows.

13.4.2 Mixedideal gases

Thedescription ofmixedidealMBgasesfollows naturally from theidea ofchemical
potentials and from the discussions of section 13.3 (especially section 13.3.3). The
grandcanonicalensemble now relates to an assemblyofgiven volumeV,given
temperatureTandgivenchemicalpotentialsμsforeachandeverycomponents.Since
we consider only ideal gases, there is no interaction between the various components.
Therefore the energylevelsfor eachcomponent are the same as they would beby
themselvesinthe ensemble. Hence thegrandpartitionfunctionZZZGfactorizes as
follows:

ZZZG=

∏

s

ZZZG(S)=

∏

s

⎡

⎣

⎡⎡

∏

j

exp(γγj(s))

⎤

⎦

⎤⎤

(13.1 6 )

In the first part of (13.16),ZZZG(S)represents the contributory factor toZZZGfrom com-
ponents,as mentionedabove. Thelast part comes about when weintroduce the
requirement that each component is agas in the MB limit. The factorsγγj(s)foreach

component are defined exactlyas in (13.7) with energyvaluesεε(js)appropriate to the
gas in question.
Asin section 13.3.3, we nowderive thermodynamic quantities suchasPandthe
numbersNNsof each component. Equation (13. 5 )forPVtellsusthat

PV

kkkBT

=lnZZZG=

∑

s

lnZZZG(s)=

∑

s

exp(μs/kkkBT)Z(s) (13. 17 )

whereZ(s)is defined as the one-particle (ordinary) partition function for component
s(compare (13.13) for the one-component equation).
The numbers ofeachgas component arederivedfrom the many-component
generalization of (13. 5 ) and (13.6), following (13.15)

NNs=

(

∂(PV)

∂μs

)

V,T,μ(other components)