140 Chemical thermodynamics
texp(βU)is a maximum, subject now to one restrictive condition
∑
nnj=N.Using
the Lagrange method with the one condition (undetermined multiplierα) gives pre-
cisely (13.3). Thelinkwiththe canonicalensemble approachisevident when we
recognize that, withinthe usualapproximations oflarge numbers, thisviewisadja-
cent to that based on the assembly partition functionZZAof (12. 5 ). We simply need
to approximate(U,V,N)byitslargest termtandtoidentifythephysicalmeaning
oftheβmultiplier, i.e. to setβ=− 1 /kkkBT.Equation (13.3) can thenbe seen tobe
effectively picking out the maximum term inZZA.This is almost equivalent to eval-
uatingZZAitself,since thereis a verysharp peakindeedasdiscussedearlier (note 3,
following(12. 5 )).
Method3. The interestingwayof openingout (13.3) is now taken to one final
stage. The equation certainly also gives the recipe to identify the distribution which
makes thefunctiontexp(αN)exp(βU)an unconditionalmaximum. Anduncondi-
tional maxima aregood news mathematically. Hence, the imaginative approach to
which this argument leads is to:
1.Define agrand partition function ZZZGby the expression
ZZZG=∑
k,l(EEk,V,NNNl)exp(μNNNl/kkkBT)exp(−EEk/kkkBT) (13.4)where the sum goes over all energies (labelledEEkin this chapter) and all particle
numbers (labelledNNNl)from zero toinfinity.- Note that the maximum term in (13.4) will be recovered from (13.4) if we again
equatetand,andgive thephysicalidentification tobothmultipliersαandβ. - Suggest that a new methodtodescribe equilibrium should bebasedon thefull
sum in (13.4), when this is convenient, rather than just on the largest term in the
peakeddistribution. We note that this methodhas allthe attributes todescribe
open systems, since the construction ofZZZGstarts off byassumingthatTandμ
are known and thatUandNare to be determined from them, by identifying the
maximum terminZZZGor (asis equivalentin practice)byusing the termsinZZZGas
statisticalweightsinanythermodynamic averagingprocess.
13.2.2 The connection to thermodynamics
This is straightforward, but important.
Method 1 (Microcanonical ensemble). The connection is via(1.5),S=kkkBln,
appropriate to the set variablesU,VandN.Hence
TS=kkkBTlnkkkBTlnt∗Thisisthebasisonwhichmost ofthisbooksofarisbased.