Statistical Physics, Second Revised and Enlarged Edition

13 Chemical thermodynamics

In this chapter we will build on ideas introduced earlier in the book, notably the idea
ofchemicalpotential,andits relevance to adiscussion ofphase transitions (following
section 11.1). We shallextendthe conceptofthe ensembles (section 12.2) todiscuss
the grand canonical ensemble and an approach to open systems. These ideas come
togethertodiscusssimplechemicalreactionsandconditionsforchemicalequilibrium.

1 3.1 Chemicalpotentialrevisited

We met thechemicalpotentialμat theendofChapter 2. Theidea ofchemicalpotential
also has appeared frequently, notably in the discussions of FD gases (Chapter 8, where
it was calledthe Fermienergy) andofrealBE gases (Chapter 9, whereit washidden
intheBparameter). (The name chemicalpotentialwas not always used, perhaps
because of the physicist’s reluctance to mention chemistry?)
In statisticalphysics, theflavour ofchemicalpotentialisthatitisapotentialfor
particle number.Itisintimatelyassociatedwithα,theLagrangemultiplierintroduced
in conjunction with the number condition

∑

ni=N. In fact as we have seen it is
relatedby

α=μ/kkkBT (13.1)

In thermal physics (see, for example,Thermal PhysicsbyFinn, chapter 10), chemical
potential is defined as being an appropriate number derivative of an energy function.
Specifically,in terms ofinternalenergyUitisdefinedas

μ=

(

∂U

∂N

)

S,V

Putting together dU=TdS−PdV+μdNwith the definition of the Helmholtz free
energyF=U−TSweobtain

μ=

(

∂F

∂N

)

T,V

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