Particle number fluctuations 275which contains the average particle number〈N〉instead ofNas would appear in the
canonical ensemble. Similarly, the average energy is given by
E=−
∂
∂βlnZ(ζ,V,T) =−∂
∂βV ζ
λ^3=
3 V ζ
λ^4∂λ
∂β=
3
2
〈N〉kT. (6.5.9)Finally, in order to compute the entropy,Zmust be expressed in terms ofμrather
thanζ, i.e.
lnZ(μ,V,T) =Veβμ
λ^3. (6.5.10)
Then,
S(μ,V,T) =klnZ(μ,V,T)−kβ(
∂lnZ(μ,V,T)
∂β)
μ,V=k
Veβ
λ^3−kβ[
V μeβμ
λ^3−
3 Veβμ
λ^4∂λ
∂β]
. (6.5.11)
Using the facts that
Veβμ
λ^3=
V ζ
λ^3=〈N〉,
∂λ
∂β=
λ
2 β, (6.5.12)
we obtain
S=k〈N〉−kβ〈N〉kTlnζ+kβ3
2
〈N〉
β=
5
2
〈N〉k−〈N〉kln(
〈N〉λ^3
V)
=
5
2
〈N〉k+〈N〉kln(
V
〈N〉λ^3)
, (6.5.13)
which is the Sackur–Tetrode equation derived in Section 3.5.1. Note that because the
1 /N! is includeda posterioriin the expression forQ(N,V,T), the correct quantum
mechanical entropy expression results.
6.6 Particle number fluctuations in the grand canonical ensemble
In the grand canonical ensemble, the total particle number fluctuates at constant
chemical potential. It is, therefore, instructive to analyze thesefluctuations, as was
done for the energy fluctuations in the canonical ensemble (Section 4.4) and volume
fluctuations in the isothermal-isobaric ensemble (see Problem 5.2 in Chapter 5). Parti-
cle number fluctuations in the grand canonical ensemble can be studied by considering
the variance
∆N=
√
〈N^2 〉−〈N〉^2. (6.6.1)
In order to compute this quantity, we start by examining the operation