Statistical Physics, Second Revised and Enlarged Edition

146 Chemical thermodynamics

since thejobisdone already in (13.13). But what aboutN,the average number of
gas molecules in our open system? From (13.6) we see that the answer is

N=

(

∂(PV)

∂μ

)

T,V

Sinceμappears only oncein (13.13), the resultisimmediately accessible, namely

N=exp(μ/kkkBT)Z (13.14)

There are two important features of (13.14):

1. 
   

   In combination with(13.13),itleadstotherelationPV=NkkkBTfor theidealMB
   gas. As expected,theidealgas equation ofstateisvalid for an open system (where
   Nis determined from the givenμ), just as it is for the closed system (wherea
   fixedNcanbe usedtodefine aderivedμwhen necessary).

   
   


2. 
   

   We have seen above that for the open MBgas, the distribution isni =γγγi =
   exp[(μ−εi)/kkkBT].Replacingμfrom(13.14), we see that this can be written as

   
   

ni=

N

Z

exp(−εi/kkkBT)

the familiar form for the MB distribution, again with the subtle difference that

Nis now aderivednotagiven quantity. As expectedallthisis consistent with

our earlierdiscussion ofthemultiplierαandits relation to chemicalpotential,as

in(13.1).

1 3.4 Mixed systems and chemical reactions

Havingdiscussedopen one-component assemblies, we are now readyto consider
what happens when more than one substance ispresent. This will then enable us to
determine the equilibrium conditionsfor simplechemicalreactions. We shall find
that thechemicalpotentialplays a crucialroleinunderstandingchemicalreactions
(not too surprisingly when you think of the name!).

1 3.4.1 Free energy of a many-component assembly

In section 13.1, (equation (13.2)), we saw that, in a one-component assembly, we
couldwrite thechemicalpotentialμas theGibbsfree energyper particle, so that
G=Nμ.
The generalization to a many-component assembly is straight-forward from this
standpoint. Itisthat eachspecies (component),labelledbysinwhatfollows,hasits