6.3: Ideal Gas
- Page ID
- 5212
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.