Statistical Physics, Second Revised and Enlarged Edition

148 Chemical thermodynamics

SubstitutingPVfrom (13.17)gives

NNs=kkkBT

∂

∂μs

(lnZZZG)=exp(μskkkBT)Z(s) (13.18)

Comparing(13.17) and(13.18) shows thatlnZZZG=

∑

sNNsandhence

PV

kkkBT

=

∑

s

NNs

is the appropriate generalization of the equation of state. The same result can be
written as the‘law ofpartialpressures’

P=

∑

s

Ps=

∑

s

(NNskkkBT/V) (13.19)

i.e. the total pressurePof the ideal gas mixture is the sum of the pressuresPswhich
eachcomponent wouldexertintheabsence oftheothers.

1 3.4.3 Equilibrium in chemical reactions

Insection13.4.4weshallconsiderareactioninthegaseousphase, sothatthetreatment
ofthelast section remains relevant. However the memorable resultofthis section
requires no such restriction. To be specific, let us consider a reaction of the type

A+BAB

in which an atom of component A combines with an atom of component B to give a
moleculeofcomponent AB. (An apology.Thisis not sucha common reaction type,
especiallyin thegaseous phase, although it is the simplest. More familiar are reactions
such as H+HH 2 ,orH 2 +Cl 2  2 HCl,or2H 2 +O 2  2 H 2 O.
These can eachbe treatedbymethodswhichare entirelysimilarinprinciple,but
which have some differences in detail from our chosen simple reaction. For instance
thefirsthas twoidenticalatoms on theleft, andtheother two reactions similarly have
2s appearing.)
For a gaseous phase reaction, the results of the previous section, such as the law
ofpartialpressures, are allentirelyvalid,where weidentifyswithA,BandAB. But
now there is another requirement, that of chemical equilibrium, to consider. We can
write the free energy (equation (13.1 5 )) as

G=NNAμA+NNNBμB+NNABμAB (13.20)

Equilibrium means, as ever, that we shouldminimize the free energy (see
section 13.1 above,for example). In thegrandcanonicalensemble approach,the