Mixed systems and chemical reactions 149μsarefixed,sothat the requirementisthat (13.20) mustbe stationaryifwe make
small changes in the numbers of reacting molecules. We must satisfy the equation
dG=μAdNNA+μBdNNNB+μABdNNAB=0(13.21)However, such changes of number are not independent. If we increase the AB popu-
lationbyone, we are ofnecessityreducingboththe A populationbyone andtheB
population byone, as can be seen from the reaction equation. (Now does it become
clear how to generalize the approach to other reaction types?) In other words, the
allowablevariations arelimitedtodNNAB=−dNNA=−dNNNB.Substitution ofthis
restriction into (13.21) immediatelyleads to the conclusion that in equilibrium we
must have
μA+μB=μAB (13.22)Thisisafamous (andsimple) result. It tellsusthat when chemicalequilibriumis
reached,thechemicalpotentialsofthe reactants must satisfyarelationshipofthis
type. They are no longer independent.
13 .4.4 Thelaw ofmass action
When equilibrium in our reaction is reached, we know (from (13.22)) thatμA+μB=
μAB.Hence
exp(μA/kkkBT)·exp(μB/kkkBT)=exp(μAB)/kkkBT (13.23)Now we return to gases. We now assume that A, B andAB are all idealMB gases, so
that we can use the results ofsection 13.4.2. Equation (13.18) tellsushow the number
of each componentsis related to its chemical potential and its partition functionZ(s).
This canbe usedtoderivearelationbetween the numbers ofreacting componentsin
chemicalequilibrium. Infact, using(13.18), (13.23) canbewritten simplyas
NNA
Z(A)·
NNNB
Z(B)
=
NNAB
Z(AB)
whichcanbe rearrangedto read
NNAB
NNANNNB=
Z(AB)
Z(A)Z(B)
=K(V,T),say (13.24)This is the ‘law of mass action’ for the reactionA+BAB.It relates the number
ofeachcomponent to a so-calledequilibrium constantK(V,T)whichdependsonly
on the reaction, thevolume andthe temperature. Similarlaws ofmass action canbe
developed for other reactions using the same methods. OnceK(V,T)is known,the
law enables us to predict what will bethe equilibrium composition ofthe reacting
mixturefrom anygiven startingconditions.