6.3: Ideal Gas

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Recall the canonical partition function expression for the ideal gas:

\[Q(N,V,T) = {1 \over N!} \left[{V \over h^3}\left({2\pi m \over \beta}\right)^{3/2}\right]^{N} \nonumber \]

Define the thermal wavelength \(\lambda (\beta)\) as \[\lambda(\beta) = \left({\beta h^2 \over 2 \pi m}\right)^{1/2} \nonumber \] which has a quantum mechanical meaning as the width of the free particle distribution function. Here it serves as a useful parameter, since the canonical partition can be expressed as

\[Q(N,V,T) = {1 \over N!}\left({V \over \lambda^3}\right)^N \nonumber \]

The grand canonical partition function follows directly from \(Q(N,V,T)\):

\[{\cal Z}(\zeta,V,T) = \sum_{N=0}^{\infty}{1 \over N!}\left({V\zeta \over \lambda^3}\right)^N = e^{V\zeta/\lambda^3} \nonumber \]

Thus, the free energy is

\[{PV \over kT} = \ln {\cal Z} = {V \zeta \over \lambda^3} \nonumber \]

In order to obtain the equation of state, we first compute the average particle number \(\langle N \rangle\)

\[\langle N \rangle = \zeta {\partial \over \partial \zeta}\ln {\cal Z}= {V \zeta \over \lambda^3} \nonumber \]

Thus, eliminating \(\zeta\) in favor of \(\langle N \rangle \) in the equation of state gives

\[PV = \langle N \rangle kT \nonumber \]

as expected. Similarly, the average energy is given by

\[ E = -\left({\partial \ln {\cal Z}\over \partial \beta}\right )_{\zeta V} = {3V\zeta \over \lambda^4}{\partial \lambda \over \partial \beta} ={3 \over 2}\langle N \rangle kT \nonumber \]

where the fugacity has been eliminated in favor of the average particle number. Finally, the entropy

\[S(\mu,V,T) = k\ln {\cal Z}(\mu,V,T) - k\beta\left({\partial\ln {\cal Z} (\mu, V, T) \over \partial \beta} \right)_{\mu, V} = {5 \over 2}\langle N \rangle k + \langle N \rangle k\ln \left[{V\lambda^3 \over \langle N \rangle} \right] \nonumber \]

which is the Sackur-Tetrode equation derived in the context of the canonical and microcanonical ensembles.