1549380323-Statistical Mechanics Theory and Molecular Simulation

Thermodynamics 267

so that
A(N,V,T) =−PV+μN, (6.2.9)

which agrees with Euler’s theorem. Similarly, the Gibbs free energyG(N,P,T) is a
homogeneous function of degree 1 inNonly, i.e.G(λN,P,T) =λG(N,P,T). Thus,
from Euler’s theorem,

G(N,P,T) =N

∂G

∂N

=μN, (6.2.10)

which agrees with the definitionG=E−TS+PV=μN. From these two examples, we
see that Euler’s theorem allows us to derive alternative expressionsfor extensive ther-
modynamic functions such as the Gibbs and Helmholtz free energies.As will be shown
in the next section, Euler’s theorem simplifies the derivation of the thermodynamic
relations of the grand canonical ensemble.

6.3 Thermodynamics of the grand canonical ensemble

In the grand canonical ensemble, the control variables are the chemical potentialμ, the
volumeV, and the temperatureT. The free energy of the ensemble can be obtained
by performing a Legendre transformation of the Helmholtz free energyA(N,V,T). Let
A ̃(μ,V,T) be the transformed free energy, which we obtain as

A ̃(μ,V,T) =A(N(μ),V,T)−N

(

∂A

∂N

)

V,T

A ̃(μ,V,T) =A(N(μ),V,T)−N(μ)μ. (6.3.1)

SinceA ̃is a function ofμ,V, andT, a small change in each of these variables leads
to a change inA ̃given by

dA ̃=

(

∂A ̃

∂μ

)

V,T

dμ+

(

∂A ̃

∂V

)

μ,T

dV+

(

∂A ̃

∂T

)

μ,V

dT. (6.3.2)

However, from the first law of thermodynamics,

dA ̃= dA−Ndμ−μdN

=−PdV−SdT+μdN−Ndμ−μdN

=−PdV−SdT−Ndμ, (6.3.3)

and we obtain the thermodynamic relations

〈N〉=−

(

∂A ̃

∂μ

)

V,T

, P=−

(

∂A ̃

∂V

)

μ,T

, S=−

(

∂A ̃

∂T

)

V,μ

. (6.3.4)

In the above relations,〈N〉denotes the average particle number. Euler’s theorem can
be used to determine a relation forA ̃in terms of other thermodynamic variables. Since