Statistical Physics, Second Revised and Enlarged Edition

Idealgases in thegrand ensemble 141

Method 2 (Canonicalensemble). The given variables are nowT,V andN.
As discussed in section 12.2, we define an assembly partition function ZZA
(equation (12.5)) as

ZZA=

∑

k

(EEk,V,N)exp(−EEk/kkkBT)

Usingthe maximum term approximation (i.e. removingthe sum andsettingEEk=U)

then gives

kkkBTlnZZA=kkkBTln−U=−F

as already derived (equation (12.7)).

Method 3 (Grand canonical ensemble). The starting parameters are nowT,Vand

μ.Followingthedefinition ofZZZGin (13.4), andagainusing the maximum term

approximation, we nowderive thefollowingresult:

kkkBTlnZZZG=kkkBTln+μN−U=TS+G−U=PV (13.5)

Here,we have used the identification ofμNwithGas discussed above in section 13.1.

The resultisthat the appropriate thermodynamic energyfunctionissimplyPV.

Sinceitis somewhat unfamiliar as a thermodynamicfunction,itis worthstressing

thatPV as afunctionofT,Vandμis a thoroughly useable and useful idea. As

discussedin section 13.1, sinceG=F+PV =U−TS+PVandalso, (13.2),

G=Nμ,we maywritePV=Nμ−U+TS.SincedU=TdS−PdV+μdNwe

have immediately

d(PV)=SdT+PdV+Ndμ (13.6)

Hence aknowledgeofZZZGenables (using (13.5) and (13.6)) an effective and direct

route to the determination of other thermodynamic quantities, such asS,Pandin

particularN, the (equilibrium) number of particles in our open system.

1 3.3 Idealgases in thegrand ensemble

It is instructive to rederive some of the results for ideal gases using this new method,

toindicate the power andgenerality ofthis somewhat more sophisticatedapproach

to statisticalphysics. The power comes about since, as we shallnow see, theabsence

ofrestrictionsonNandUmakes it easy to evaluateZZZGfor an ideal gas.

1 3.3.1 Determination ofthegrand partitionfunction

Weshall labelthe one-particle statesbyj,andtheir corresponding energiesεεj.For

anidealgas these energiesdependonVonly(fittingwavesintoboxes!). Note thatin