Skip to main content
Physics LibreTexts

4.7: Ideal Gas Statistical Mechanics

  • Page ID
    18756
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand\bes{\begin{equation}\begin{split}}\)
    \( \newcommand\ltwid{\propto}\)
    \( \newcommand\ees{\end{split}\end{equation}}\)
    \( \newcommand\mib{\mathbf}\)
    \( \newcommand\Sa{\textsf a}\)
    \( \newcommand\Sb{\textsf b}\)
    \( \newcommand\Sc{\textsf c}\)
    \( \newcommand\Sd{\textsf d}\)
    \( \newcommand\Se{\textsf e}\)
    \( \newcommand\Sf{\textsf f}\)
    \( \newcommand\Sg{\textsf g}\)
    \( \newcommand\Sh{\textsf h}\)
    \( \newcommand\Si{\textsf i}\)
    \( \newcommand\Sj{\textsf j}\)
    \( \newcommand\Sk{\textsf k}\)
    \( \newcommand\Sl{\textsf l}\)
    \( \newcommand\Sm{\textsf m}\)
    \( \newcommand\Sn{\textsf n}\)
    \( \newcommand\So{\textsf o}\)
    \( \newcommand\Sp{\textsf p}\)
    \( \newcommand\Sq{\textsf q}\)
    \( \newcommand\Sr{\textsf r}\)
    \( \newcommand\Ss{\textsf s}\)
    \( \newcommand\St{\textsf t}\)
    \( \newcommand\Su{\textsf u}\)
    \( \newcommand\Sv{\textsf v}\)
    \( \newcommand\Sw{\textsf w}\)
    \( \newcommand\Sx{\textsf x}\)
    \( \newcommand\Sy{\textsf y}\)
    \( \newcommand\Sz{\textsf z}\)
    \( \newcommand\SA{\textsf A}\)
    \( \newcommand\SB{\textsf B}\)
    \( \newcommand\SC{\textsf C}\)
    \( \newcommand\SD{\textsf D}\)
    \( \newcommand\SE{\textsf E}\)
    \( \newcommand\SF{\textsf F}\)
    \( \newcommand\SG{\textsf G}\)
    \( \newcommand\SH{\textsf H}\)
    \( \newcommand\SI{\textsf I}\)
    \( \newcommand\SJ{\textsf J}\)
    \( \newcommand\SK{\textsf K}\)
    \( \newcommand\SL{\textsf L}\)
    \( \newcommand\SM{\textsf M}\)
    \( \newcommand\SN{\textsf N}\)
    \( \newcommand\SO{\textsf O}\)
    \( \newcommand\SP{\textsf P}\)
    \( \newcommand\SQ{\textsf Q}\)
    \( \newcommand\SR{\textsf R}\)
    \( \newcommand\SS{\textsf S}\)
    \( \newcommand\ST{\textsf T}\)
    \( \newcommand\SU{\textsf U}\)
    \( \newcommand\SV{\textsf V}\)
    \( \newcommand\SW{\textsf W}\)
    \( \newcommand\SX{\textsf X}\)
    \( \newcommand\SY{\textsf Y}\)
    \( \newcommand\SZ{\textsf Z}\)
    \( \newcommand\Ha{\hat a}\)
    \( \newcommand\Hb{\hat b}\)
    \( \newcommand\Hc{\hat c}\)
    \( \newcommand\Hd{\hat d}\)
    \( \newcommand\He{\hat e}\)
    \( \newcommand\Hf{\hat f}\)
    \( \newcommand\Hg{\hat g}\)
    \( \newcommand\Hh{\hat h}\)
    \( \newcommand\Hi{\hat \imath}\)
    \( \newcommand\Hj{\hat \jmath}\)
    \( \newcommand\Hk{\hat k}\)
    \( \newcommand\Hl{\hat l}\)
    \( \newcommand\Hm{\hat m}\)
    \( \newcommand\Hn{\hat n}\)
    \( \newcommand\Ho{\hat o}\)
    \( \newcommand\Hp{\hat p}\)
    \( \newcommand\Hq{\hat q}\)
    \( \newcommand\Hr{\hat r}\)
    \( \newcommand\Hs{\hat s}\)
    \( \newcommand\Ht{\hat t}\)
    \( \newcommand\Hu{\hat u}\)
    \( \newcommand\Hv{\hat v}\)
    \( \newcommand\Hw{\hat w}\)
    \( \newcommand\Hx{\hat x}\)
    \( \newcommand\Hy{\hat y}\)
    \( \newcommand\Hz{\hat z}\)
    \( \newcommand\HA{\hat A}\)
    \( \newcommand\HB{\hat B}\)
    \( \newcommand\HC{\hat C}\)
    \( \newcommand\HD{\hat D}\)
    \( \newcommand\HE{\hat E}\)
    \( \newcommand\HF{\hat F}\)
    \( \newcommand\HG{\hat G}\)
    \( \newcommand\HH{\hat H}\)
    \( \newcommand\HI{\hat I}\)
    \( \newcommand\HJ{\hat J}\)
    \( \newcommand\HK{\hat K}\)
    \( \newcommand\HL{\hat L}\)
    \( \newcommand\HM{\hat M}\)
    \( \newcommand\HN{\hat N}\)
    \( \newcommand\HO{\hat O}\)
    \( \newcommand\HP{\hat P}\)
    \( \newcommand\HQ{\hat Q}\)
    \( \newcommand\HR{\hat R}\)
    \( \newcommand\HS{\hat S}\)
    \( \newcommand\HT{\hat T}\)
    \( \newcommand\HU{\hat U}\)
    \( \newcommand\HV{\hat V}\)
    \( \newcommand\HW{\hat W}\)
    \( \newcommand\HX{\hat X}\)
    \( \newcommand\HY{\hat Y}\)
    \( \newcommand\HZ{\hat Z}\)
    \( \newcommand\Halpha{\hat\alpha}\)
    \( \newcommand\Hbeta{\hat\beta}\)
    \( \newcommand\Hgamma{\hat\gamma}\)
    \( \newcommand\Hdelta{\hat\delta}\)
    \( \newcommand\Hepsilon{\hat\epsilon}\)
    \( \newcommand\Hvarepsilon{\hat\varepsilon}\)
    \( \newcommand\Hzeta{\hat\zeta}\)
    \( \newcommand\Heta{\hat\eta}\)
    \( \newcommand\Htheta{\hat\theta}\)
    \( \newcommand\Hvartheta{\hat\vartheta}\)
    \( \newcommand\Hiota{\hat\iota}\)
    \( \newcommand\Hkappa{\hat\kappa}\)
    \( \newcommand\Hlambda{\hat\lambda}\)
    \( \newcommand\Hmu{\hat\mu}\)
    \( \newcommand\Hnu{\hat\nu}\)
    \( \newcommand\Hxi{\hat\xi}\)
    \( \newcommand\Hom{\hat\omicron}\)
    \( \newcommand\Hpi{\hat\pi}\)
    \( \newcommand\Hvarpi{\hat\varpi}\)
    \( \newcommand\Hrho{\hat\rho}\)
    \( \newcommand\Hvarrho{\hat\varrho}\)
    \( \newcommand\Hsigma{\hat\sigma}\)
    \( \newcommand\Hvarsigma{\hat\varsigma}\)
    \( \newcommand\Htau{\var\tau}\)
    \( \newcommand\Hupsilon{\hat\upsilon}\)
    \( \newcommand\Hphi{\hat\phi}\)
    \( \newcommand\Hvarphi{\hat\varphi}\)
    \( \newcommand\Hchi{\hat\chi}\)
    \( \newcommand\Hxhi{\hat\xhi}\)
    \( \newcommand\Hpsi{\hat\psi}\)
    \( \newcommand\Homega{\hat\omega}\)
    \( \newcommand\HGamma{\hat\Gamma}\)
    \( \newcommand\HDelta{\hat\Delta}\)
    \( \newcommand\HTheta{\hat\Theta}\)
    \( \newcommand\HLambda{\hat\Lambda}\)
    \( \newcommand\HXi{\hat\Xi}\)
    \( \newcommand\HPi{\hat\Pi}\)
    \( \newcommand\HSigma{\hat\Sigma}\)
    \( \newcommand\HUps{\hat\Upsilon}\)
    \( \newcommand\HPhi{\hat\Phi}\)
    \( \newcommand\HPsi{\hat\Psi}\)
    \( \newcommand\HOmega{\hat\Omega}\)
    \( \newcommand\xhat{\hat\Bx}\)
    \( \newcommand\yhat{\hat\By}\)
    \( \newcommand\zhat{\hat\Bz}\)
    \( \newcommand\ehat{\hat\Be}\)
    \( \newcommand\khat{\hat\Bk}\)
    \( \newcommand\nhat{\hat\Bn}\)
    \( \newcommand\rhat{\hat\Br}\)
    \( \newcommand\phihat{\hat\Bphi}\)
    \( \newcommand\thetahat{\hat\Btheta}\)
    \( \newcommand\MA{\mathbb A}\)
    \( \newcommand\MB{\mathbb B}\)
    \( \newcommand\MC{\mathbb C}\)
    \( \newcommand\MD{\mathbb D}\)
    \( \newcommand\ME{\mathbb E}\)
    \( \newcommand\MF{\mathbb F}\)
    \( \newcommand\MG{\mathbb G}\)
    \( \newcommand\MH{\mathbb H}\)
    \( \newcommand\MI{\mathbb I}\)
    \( \newcommand\MJ{\mathbb J}\)
    \( \newcommand\MK{\mathbb K}\)
    \( \newcommand\ML{\mathbb L}\)
    \( \newcommand\MM{\mathbb M}\)
    \( \newcommand\MN{\mathbb N}\)
    \( \newcommand\MO{\mathbb O}\)
    \( \newcommand\MP{\mathbb P}\)
    \( \newcommand\MQ{\mathbb Q}\)
    \( \newcommand\MR{\mathbb R}\)
    \( \newcommand\MS{\mathbb S}\)
    \( \newcommand\MT{\mathbb T}\)
    \( \newcommand\MU{\mathbb U}\)
    \( \newcommand\MV{\mathbb V}\)
    \( \newcommand\MW{\mathbb W}\)
    \( \newcommand\MX{\mathbb X}\)
    \( \newcommand\MY{\mathbb Y}\)
    \( \newcommand\MZ{\mathbb Z}\)
    \( \newcommand\CA{\mathcal A}\)
    \( \newcommand\CB{\mathcal B}\)
    \( \newcommand\CC{\mathcal C}\)
    \( \newcommand\CD{\mathcal D}\)
    \( \newcommand\CE{\mathcal E}\)
    \( \newcommand\CF{\mathcal F}\)
    \( \newcommand\CG{\mathcal G}\)
    \( \newcommand\CH{\mathcal H}\)
    \( \newcommand\CI{\mathcal I}\)
    \( \newcommand\CJ{\mathcal J}\)
    \( \newcommand\CK{\mathcal K}\)
    \( \newcommand\CL{\mathcal L}\)
    \( \newcommand\CM{\mathcal M}\)
    \( \newcommand\CN{\mathcal N}\)
    \( \newcommand\CO{\mathcal O}\)
    \( \newcommand\CP{\mathcal P}\)
    \( \newcommand\CQ{\mathcal Q}\)
    \( \newcommand\CR{\mathcal R}\)
    \( \newcommand\CS{\mathcal S}\)
    \( \newcommand\CT{\mathcal T}\)
    \( \newcommand\CU{\mathcal U}\)
    \( \newcommand\CV{\mathcal V}\)
    \( \newcommand\CW{\mathcal W}\)
    \( \newcommand\CX{\mathcal X}\)
    \( \newcommand\CY{\mathcal Y}\)
    \( \newcommand\CZ{\mathcal Z}\)
    \( \newcommand\Fa{\mathfrak a}\)
    \( \newcommand\Fb{\mathfrak b}\)
    \( \newcommand\Fc{\mathfrak c}\)
    \( \newcommand\Fd{\mathfrak d}\)
    \( \newcommand\Fe{\mathfrak e}\)
    \( \newcommand\Ff{\mathfrak f}\)
    \( \newcommand\Fg{\mathfrak g}\)
    \( \newcommand\Fh{\mathfrak h}\)
    \( \newcommand\Fi{\mathfrak i}\)
    \( \newcommand\Fj{\mathfrak j}\)
    \( \newcommand\Fk{\mathfrak k}\)
    \( \newcommand\Fl{\mathfrak l}\)
    \( \newcommand\Fm{\mathfrak m}\)
    \( \newcommand\Fn{\mathfrak n}\)
    \( \newcommand\Fo{\mathfrak o}\)
    \( \newcommand\Fp{\mathfrak p}\)
    \( \newcommand\Fq{\mathfrak q}\)
    \( \newcommand\Fr{\mathfrak r}\)
    \( \newcommand\Fs{\mathfrak s}\)
    \( \newcommand\Ft{\mathfrak t}\)
    \( \newcommand\Fu{\mathfrak u}\)
    \( \newcommand\Fv{\mathfrak v}\)
    \( \newcommand\Fw{\mathfrak w}\)
    \( \newcommand\Fx{\mathfrak x}\)
    \( \newcommand\Fy{\mathfrak y}\)
    \( \newcommand\Fz{\mathfrak z}\)
    \( \newcommand\FA{\mathfrak A}\)
    \( \newcommand\FB{\mathfrak B}\)
    \( \newcommand\FC{\mathfrak C}\)
    \( \newcommand\FD{\mathfrak D}\)
    \( \newcommand\FE{\mathfrak E}\)
    \( \newcommand\FF{\mathfrak F}\)
    \( \newcommand\FG{\mathfrak G}\)
    \( \newcommand\FH{\mathfrak H}\)
    \( \newcommand\FI{\mathfrak I}\)
    \( \newcommand\FJ{\mathfrak J}\)
    \( \newcommand\FK{\mathfrak K}\)
    \( \newcommand\FL{\mathfrak L}\)
    \( \newcommand\FM{\mathfrak M}\)
    \( \newcommand\FN{\mathfrak N}\)
    \( \newcommand\FO{\mathfrak O}\)
    \( \newcommand\FP{\mathfrak P}\)
    \( \newcommand\FQ{\mathfrak Q}\)
    \( \newcommand\FR{\mathfrak R}\)
    \( \newcommand\FS{\mathfrak S}\)
    \( \newcommand\FT{\mathfrak T}\)
    \( \newcommand\FU{\mathfrak U}\)
    \( \newcommand\FV{\mathfrak V}\)
    \( \newcommand\FW{\mathfrak W}\)
    \( \newcommand\FX{\mathfrak X}\)
    \( \newcommand\FY{\mathfrak Y}\)
    \( \newcommand\FZ{\mathfrak Z}\)
    \( \newcommand\Da{\dot a}\)
    \( \newcommand\Db{\dot b}\)
    \( \newcommand\Dc{\dot c}\)
    \( \newcommand\Dd{\dot d}\)
    \( \newcommand\De{\dot e}\)
    \( \newcommand\Df{\dot f}\)
    \( \newcommand\Dg{\dot g}\)
    \( \newcommand\Dh{\dot h}\)
    \( \newcommand\Di{\dot \imath}\)
    \( \newcommand\Dj{\dot \jmath}\)
    \( \newcommand\Dk{\dot k}\)
    \( \newcommand\Dl{\dot l}\)
    \( \newcommand\Dm{\dot m}\)
    \( \newcommand\Dn{\dot n}\)
    \( \newcommand\Do{\dot o}\)
    \( \newcommand\Dp{\dot p}\)
    \( \newcommand\Dq{\dot q}\)
    \( \newcommand\Dr{\dot r}\)
    \( \newcommand\Ds{\dot s}\)
    \( \newcommand\Dt{\dot t}\)
    \( \newcommand\Du{\dot u}\)
    \( \newcommand\Dv{\dot v}\)
    \( \newcommand\Dw{\dot w}\)
    \( \newcommand\Dx{\dot x}\)
    \( \newcommand\Dy{\dot y}\)
    \( \newcommand\Dz{\dot z}\)
    \( \newcommand\DA{\dot A}\)
    \( \newcommand\DB{\dot B}\)
    \( \newcommand\DC{\dot C}\)
    \( \newcommand\DD{\dot D}\)
    \( \newcommand\DE{\dot E}\)
    \( \newcommand\DF{\dot F}\)
    \( \newcommand\DG{\dot G}\)
    \( \newcommand\DH{\dot H}\)
    \( \newcommand\DI{\dot I}\)
    \( \newcommand\DJ{\dot J}\)
    \( \newcommand\DK{\dot K}\)
    \( \newcommand\DL{\dot L}\)
    \( \newcommand\DM{\dot M}\)
    \( \newcommand\DN{\dot N}\)
    \( \newcommand\DO{\dot O}\)
    \( \newcommand\DP{\dot P}\)
    \( \newcommand\DQ{\dot Q}\)
    \( \newcommand\DR{\dot R}\)
    \( \newcommand\DS{\dot S}\)
    \( \newcommand\DT{\dot T}\)
    \( \newcommand\DU{\dot U}\)
    \( \newcommand\DV{\dot V}\)
    \( \newcommand\DW{\dot W}\)
    \( \newcommand\DX{\dot X}\)
    \( \newcommand\DY{\dot Y}\)
    \( \newcommand\DZ{\dot Z}\)
    \( \newcommand\Dalpha

    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Dbeta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Dgamma
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Ddelta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Depsilon
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Dvarepsilon
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Dzeta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Deta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Dtheta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Dvartheta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Diota
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Dkappa
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Dlambda
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Dmu{\dot\mu}\)
    \( \newcommand\Dnu{\dot\nu}\)
    \( \newcommand\Dxi{\dot\xi}\)
    \( \newcommand\Dom{\dot\omicron}\)
    \( \newcommand\Dpi{\dot\pi}\)
    \( \newcommand\Dvarpi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Drho{\dot\rho}\)
    \( \newcommand\Dvarrho{\dot\varrho}\)
    \( \newcommand\Dsigma{\dot\sigma}\)
    \( \newcommand\Dvarsigma{\dot\varsigma}\)
    \( \newcommand\Dtau{\var\tau}\)
    \( \newcommand\Dupsilon{\dot\upsilon}\)
    \( \newcommand\Dphi{\dot\phi}\)
    \( \newcommand\Dvarphi{\dot\varphi}\)
    \( \newcommand\Dchi{\dot\chi}\)
    \( \newcommand\Dpsi{\dot\psi}\)
    \( \newcommand\Domega{\dot\omega}\)
    \( \newcommand\DGamma
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\DDelta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\DTheta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\DLambda{\dot\Lambda}\)
    \( \newcommand\DXi{\dot\Xi}\)
    \( \newcommand\DPi{\dot\Pi}\)
    \( \newcommand\DSigma{\dot\Sigma}\)
    \( \newcommand\DUps{\dot\Upsilon}\)
    \( \newcommand\DPhi{\dot\Phi}\)
    \( \newcommand\DPsi{\dot\Psi}\)
    \( \newcommand\DOmega{\dot\Omega}\)
    \( \newcommand\Va{\vec a}\)
    \( \newcommand\Vb{\vec b}\)
    \( \newcommand\Vc{\vec c}\)
    \( \newcommand\Vd{\vec d}\)
    \( \newcommand\Ve{\vec e}\)
    \( \newcommand\Vf{\vec f}\)
    \( \newcommand\Vg{\vec g}\)
    \( \newcommand\Vh{\vec h}\)
    \( \newcommand\Vi{\vec \imath}\)
    \( \newcommand\Vj{\vec \jmath}\)
    \( \newcommand\Vk{\vec k}\)
    \( \newcommand\Vl{\vec l}\)
    \( \newcommand\Vm{\vec m}\)
    \( \newcommand\Vn{\vec n}\)
    \( \newcommand\Vo{\vec o}\)
    \( \newcommand\Vp{\vec p}\)
    \( \newcommand\Vq{\vec q}\)
    \( \newcommand\Vr{\vec r}\)
    \( \newcommand\Vs{\vec s}\)
    \( \newcommand\Vt{\vec t}\)
    \( \newcommand\Vu{\vec u}\)
    \( \newcommand\Vv{\vec v}\)
    \( \newcommand\Vw{\vec w}\)
    \( \newcommand\Vx{\vec x}\)
    \( \newcommand\Vy{\vec y}\)
    \( \newcommand\Vz{\vec z}\)
    \( \newcommand\VA{\vec A}\)
    \( \newcommand\VB{\vec B}\)
    \( \newcommand\VC{\vec C}\)
    \( \newcommand\VD{\vec D}\)
    \( \newcommand\VE{\vec E}\)
    \( \newcommand\VF{\vec F}\)
    \( \newcommand\VG{\vec G}\)
    \( \newcommand\VH{\vec H}\)
    \( \newcommand\VI{\vec I}\)
    \( \newcommand\VJ{\vec J}\)
    \( \newcommand\VK{\vec K}\)
    \( \newcommand\VL{\vec L}\)
    \( \newcommand\VM{\vec M}\)
    \( \newcommand\VN{\vec N}\)
    \( \newcommand\VO{\vec O}\)
    \( \newcommand\VP{\vec P}\)
    \( \newcommand\VQ{\vec Q}\)
    \( \newcommand\VR{\vec R}\)
    \( \newcommand\VS{\vec S}\)
    \( \newcommand\VT{\vec T}\)
    \( \newcommand\VU{\vec U}\)
    \( \newcommand\VV{\vec V}\)
    \( \newcommand\VW{\vec W}\)
    \( \newcommand\VX{\vec X}\)
    \( \newcommand\VY{\vec Y}\)
    \( \newcommand\VZ{\vec Z}\)
    \( \newcommand\Valpha{\vec\alpha}\)
    \( \newcommand\Vbeta{\vec\beta}\)
    \( \newcommand\Vgamma{\vec\gamma}\)
    \( \newcommand\Vdelta{\vec\delta}\)
    \( \newcommand\Vepsilon{\vec\epsilon}\)
    \( \newcommand\Vvarepsilon{\vec\varepsilon}\)
    \( \newcommand\Vzeta{\vec\zeta}\)
    \( \newcommand\Veta{\vec\eta}\)
    \( \newcommand\Vtheta{\vec\theta}\)
    \( \newcommand\Vvartheta{\vec\vartheta}\)
    \( \newcommand\Viota{\vec\iota}\)
    \( \newcommand\Vkappa{\vec\kappa}\)
    \( \newcommand\Vlambda{\vec\lambda}\)
    \( \newcommand\Vmu
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vnu
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vxi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vom
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vpi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vvarpi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vrho
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vvarrho
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vsigma
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vvarsigma
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vtau
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vupsilon
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vphi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vvarphi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vchi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vpsi
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\Vomega
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\VGamma
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\VDelta
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\VTheta{\vec\Theta}\)
    \( \newcommand\VLambda{\vec\Lambda}\)
    \( \newcommand\VXi{\vec\Xi}\)
    \( \newcommand\VPi{\vec\Pi}\)
    \( \newcommand\VSigma{\vec\Sigma}\)
    \( \newcommand\VUps{\vec\Upsilon}\)
    \( \newcommand\VPhi{\vec\Phi}\)
    \( \newcommand\VPsi{\vec\Psi}\)
    \( \newcommand\VOmega{\vec\Omega}\)
    \( \newcommand\BA{\mib A}\)
    \( \newcommand\BB{\mib B}\)
    \( \newcommand\BC{\mib C}\)
    \( \newcommand\BD{\mib D}\)
    \( \newcommand\BE{\mib E}\)
    \( \newcommand\BF{\mib F}\)
    \( \newcommand\BG{\mib G}\)
    \( \newcommand\BH{\mib H}\)
    \( \newcommand\BI{\mib I}}\)
    \( \newcommand\BJ{\mib J}\)
    \( \newcommand\BK{\mib K}\)
    \( \newcommand\BL{\mib L}\)
    \( \newcommand\BM{\mib M}\)
    \( \newcommand\BN{\mib N}\)
    \( \newcommand\BO{\mib O}\)
    \( \newcommand\BP{\mib P}\)
    \( \newcommand\BQ{\mib Q}\)
    \( \newcommand\BR{\mib R}\)
    \( \newcommand\BS{\mib S}\)
    \( \newcommand\BT{\mib T}\)
    \( \newcommand\BU{\mib U}\)
    \( \newcommand\BV{\mib V}\)
    \( \newcommand\BW{\mib W}\)
    \( \newcommand\BX{\mib X}\)
    \( \newcommand\BY{\mib Y}\)
    \( \newcommand\BZ{\mib Z}\)
    \( \newcommand\Ba{\mib a}\)
    \( \newcommand\Bb{\mib b}\)
    \( \newcommand\Bc{\mib c}\)
    \( \newcommand\Bd{\mib d}\)
    \( \newcommand\Be{\mib e}\)
    \( \newcommand\Bf{\mib f}\)
    \( \newcommand\Bg{\mib g}\)
    \( \newcommand\Bh{\mib h}\)
    \( \newcommand\Bi{\mib i}\)
    \( \newcommand\Bj{\mib j}\)
    \( \newcommand\Bk{\mib k}\)
    \( \newcommand\Bl{\mib l}\)
    \( \newcommand\Bm{\mib m}\)
    \( \newcommand\Bn{\mib n}\)
    \( \newcommand\Bo{\mib o}\)
    \( \newcommand\Bp{\mib p}\)
    \( \newcommand\Bq{\mib q}\)
    \( \newcommand\Br{\mib r}\)
    \( \newcommand\Bs{\mib s}\)
    \( \newcommand\Bt{\mib t}\)
    \( \newcommand\Bu{\mib u}\)
    \( \newcommand\Bv{\mib v}\)
    \( \newcommand\Bw{\mib w}\)
    \( \newcommand\Bx{\mib x}\)
    \( \newcommand\By{\mib y}\)
    \( \newcommand\Bz{\mib z}\)\)
    \( \newcommand\vrh{\varrho}\)
    \( \newcommand\vsig{\varsigma}\)
    \( \newcommand\ups{\upsilon}\)
    \( \newcommand\eps{\epsilon}\)
    \( \newcommand\ve{\varepsilon}\)
    \( \newcommand\vth{\vartheta}\)
    \( \newcommand\vphi{\varphi}\)
    \( \newcommand\xhi{\chi}\)
    \( \newcommand\Ups{\Upsilon}\)
    \( \newcommand\Balpha{\mib\alpha}\)
    \( \newcommand\Bbeta{\mib\beta}\)
    \( \newcommand\Bgamma{\mib\gamma}\)
    \( \newcommand\Bdelta{\mib\delta}\)
    \( \newcommand\Beps{\mib\epsilon}\)
    \( \newcommand\Bve{\mib\varepsilon}\)
    \( \newcommand\Bzeta{\mib\zeta}\)
    \( \newcommand\Beta{\mib\eta}\)
    \( \newcommand\Btheta{\mib\theta}\)
    \( \newcommand\Bvth{\mib\vartheta}\)
    \( \newcommand\Biota{\mib\iota}\)
    \( \newcommand\Bkappa{\mib\kappa}\)
    \( \newcommand\Blambda{\mib\lambda}\)
    \( \newcommand\Bmu{\mib\mu}\)
    \( \newcommand\Bnu{\mib\nu}\)
    \( \newcommand\Bxi{\mib\xi}\)
    \( \newcommand\Bom{\mib\omicron}\)
    \( \newcommand\Bpi{\mib\pi}\)
    \( \newcommand\Bvarpi{\mib\varpi}\)
    \( \newcommand\Brho{\mib\rho}\)
    \( \newcommand\Bvrh{\mib\varrho}\)
    \( \newcommand\Bsigma{\mib\sigma}\)
    \( \newcommand\Bvsig{\mib\varsigma}\)
    \( \newcommand\Btau{\mib\tau}\)
    \( \newcommand\Bups{\mib\upsilon}\)
    \( \newcommand\Bphi{\mib\phi}\)
    \( \newcommand\Bvphi{\mib\vphi}\)
    \( \newcommand\Bchi{\mib\chi}\)
    \( \newcommand\Bpsi{\mib\psi}\)
    \( \newcommand\Bomega{\mib\omega}\)
    \( \newcommand\BGamma{\mib\Gamma}\)
    \( \newcommand\BDelta{\mib\Delta}\)
    \( \newcommand\BTheta{\mib\Theta}\)
    \( \newcommand\BLambda{\mib\Lambda}\)
    \( \newcommand\BXi{\mib\Xi}\)
    \( \newcommand\BPi{\mib\Pi}\)
    \( \newcommand\BSigma{\mib\Sigma}\)
    \( \newcommand\BUps{\mib\Upsilon}\)
    \( \newcommand\BPhi{\mib\Phi}\)
    \( \newcommand\BPsi{\mib\Psi}\)
    \( \newcommand\BOmega{\mib\Omega}\)
    \( \newcommand\Bxhi{\raise.35ex\hbox{$\Bchi$}}\)
    \( \newcommand\RGamma{ \Gamma}\)
    \( \newcommand\RDelta{ \Delta}\)
    \( \newcommand\RTheta{ \Theta}\)
    \( \newcommand\RLambda{ \Lambda}\)
    \( \newcommand\RXi{ \Xi}\)
    \( \newcommand\RPi{ \Pi}\)
    \( \newcommand\RSigma{ \Sigma}\)
    \( \newcommand\RUps{ \Upsilon}\)
    \( \newcommand\RPhi{ \Phi}\)
    \( \newcommand\RPsi{ \Psi}\)
    \( \newcommand\ROmega{ \Omega}\)
    \( \newcommand\RA{ A}\)
    \( \newcommand\RB{ B}\)
    \( \newcommand\RC{ C}\)
    \( \newcommand\RD{ D}\)
    \( \newcommand\RE{ E}\)
    \( \newcommand\RF{ F}\)
    \( \newcommand\RG{ G}\)
    \( \newcommand\RH{ H}\)
    \( \newcommand\RI{ I}\)
    \( \newcommand\RJ{ J}\)
    \( \newcommand\RK{ K}\)
    \( \newcommand\RL{ L}\)
    \( \newcommand { M}\)
    \( \newcommand\RN{ N}\)
    \( \newcommand\RO{ O}\)
    \( \newcommand\RP{ P}\)
    \( \newcommand\RQ{ Q}\)
    \( \newcommand\RR{ R}\)
    \( \newcommand\RS{ S}\)
    \( \newcommand\RT{ T}\)
    \( \newcommand\RU{ U}\)
    \( \newcommand\RV{ V}\)
    \( \newcommand\RW{ W}\)
    \( \newcommand\RX{ X}\)
    \( \newcommand\RY{ Y}\)
    \( \newcommand\RZ{ Z}\)
    \( \newcommand\Ra{ a}\)
    \( \newcommand\Rb{ b}\)
    \( \newcommand\Rc{ c}\)
    \( \newcommand\Rd{ d}\)
    \( \newcommand\Re{ e}\)
    \( \newcommand\Rf{ f}\)
    \( \newcommand\Rg{ g}\)
    \( \newcommand\Rh{ h}\)
    \( \newcommand\Ri{ i}\)
    \( \newcommand\Rj{ j}\)
    \( \newcommand\Rk{ k}\)
    \( \newcommand\Rl{ l}\)
    \( \newcommand { m}\)
    \( \newcommand\Rn{ n}\)
    \( \newcommand\Ro{ o}\)
    \( \newcommand\Rp{ p}\)
    \( \newcommand\Rq{ q}\)
    \( \newcommand\Rr{ r}\)
    \( \newcommand\Rs{ s}\)
    \( \newcommand\Rt{ t}\)
    \( \newcommand\Ru{ u}\)
    \( \newcommand\Rv{ v}\)
    \( \newcommand\Rw{ w}\)
    \( \newcommand\Rx{ x}\)
    \( \newcommand\Ry{ y}\)
    \( \newcommand\Rz{ z}\)
    \( \newcommand\BBA{\boldsymbol\RA}\)
    \( \newcommand\BBB{\boldsymbol\RB}\)
    \( \newcommand\BBC{\boldsymbol\RC}\)
    \( \newcommand\BBD{\boldsymbol\RD}\)
    \( \newcommand\BBE{\boldsymbol\RE}\)
    \( \newcommand\BBF{\boldsymbol\RF}\)
    \( \newcommand\BBG{\boldsymbol\RG}\)
    \( \newcommand\BBH{\boldsymbol\RH}\)
    \( \newcommand\BBI{\boldsymbol\RI}\)
    \( \newcommand\BBJ{\boldsymbol\RJ}\)
    \( \newcommand\BBK{\boldsymbol\RK}\)
    \( \newcommand\BBL{\boldsymbol\RL}\)
    \( \newcommand\BBM{\boldsymbol }\)
    \( \newcommand\BBN{\boldsymbol\RN}\)
    \( \newcommand\BBO{\boldsymbol\RO}\)
    \( \newcommand\BBP{\boldsymbol\RP}\)
    \( \newcommand\BBQ{\boldsymbol\RQ}\)
    \( \newcommand\BBR{\boldsymbol\RR}\)
    \( \newcommand\BBS{\boldsymbol\RS}\)
    \( \newcommand\BBT{\boldsymbol\RT}\)
    \( \newcommand\BBU{\boldsymbol\RU}\)
    \( \newcommand\BBV{\boldsymbol\RV}\)
    \( \newcommand\BBW{\boldsymbol\RW}\)
    \( \newcommand\BBX{\boldsymbol\RX}\)
    \( \newcommand\BBY{\boldsymbol\RY}\)
    \( \newcommand\BBZ{\boldsymbol\RZ}\)
    \( \newcommand\BBa{\boldsymbol\Ra}\)
    \( \newcommand\BBb{\boldsymbol\Rb}\)
    \( \newcommand\BBc{\boldsymbol\Rc}\)
    \( \newcommand\BBd{\boldsymbol\Rd}\)
    \( \newcommand\BBe{\boldsymbol\Re}\)
    \( \newcommand\BBf{\boldsymbol\Rf}\)
    \( \newcommand\BBg{\boldsymbol\Rg}\)
    \( \newcommand\BBh{\boldsymbol\Rh}\}\)
    \( \newcommand\BBi{\boldsymbol\Ri}\)
    \( \newcommand\BBj{\boldsymbol\Rj}\)
    \( \newcommand\BBk{\boldsymbol\Rk}\)
    \( \newcommand\BBl{boldsymbol\Rl}\)
    \( \newcommand\BBm{\boldsymbol }\)
    \( \newcommand\BBn{\boldsymbol\Rn}\)
    \( \newcommand\BBo{\boldsymbol\Ro}\)
    \( \newcommand\BBp{\boldsymbol\Rp}\)
    \( \newcommand\BBq{\boldsymbol\Rq}\)
    \( \newcommand\BBr{\boldsymbol\Rr}\)
    \( \newcommand\BBs{\boldsymbol\Rs}\)
    \( \newcommand\BBt{\boldsymbol\Rt}\)
    \( \newcommand\BBu{\boldsymbol\Ru}\)
    \( \newcommand\BBv{\boldsymbol\Rv}\)
    \( \newcommand\BBw{\boldsymbol\Rw}\)
    \( \newcommand\BBx{\boldsymbol\Rx}\)
    \( \newcommand\BBy{\boldsymbol\Ry}\)
    \( \newcommand\BBz{\boldsymbol\Rz}\)
    \( \newcommand\tcb{\textcolor{blue}\)
    \( \newcommand\tcr{\textcolor{red}\)
    \( \newcommand\bnabla{\boldsymbol{\nabla}}\)
    \( \newcommand\Bell{\boldsymbol\ell}\)
    \( \newcommand\dbar{\,{\mathchar'26\mkern-12mu d}} \)
    \( \newcommand\ns{^\vphantom{*}}\)
    \( \newcommand\uar{\uparrow}\)
    \( \newcommand\dar{\downarrow}\)
    \( \newcommand\impi{\int\limits_{-\infty}^{\infty}\!\!}\)
    \( \newcommand\izpi{\int\limits_{0}^{\infty}\!\!}\)
    \( \newcommand\etc{\it etc.\/}\)
    \( \newcommand\etal{\it et al.\/}\)
    \( \newcommand\opcit{\it op. cit.\/}\)
    \( \newcommand\ie{\it i.e.\/}\)
    \( \newcommand\Ie{\it I.e.\/}\)
    \( \newcommand\viz{\it viz.\/}\)
    \( \newcommand\eg{\it e.g.\/}\)
    \( \newcommand\Eg{\it E.g.\/}\)
    \( \newcommand\dbar{\,{\mathchar'26\mkern-12mu d}} \)
    \( \def\sss#1{\scriptscriptstyle #1}\)
    \( \def\ss#1{\scriptstyle #1}\)
    \( \def\ssr#1{\scriptstyle #1}\)
    \( \def\ssf#1{\scriptstyle #1}\)
    \( \newcommand\NA{N_{\ssr{\!A}}}\)
    \( \newcommand\lala{\langle\!\langle}\)
    \( \newcommand\rara{\rangle\!\rangle}\)
    \( \newcommand\blan{\big\langle}\)
    \( \newcommand\bran{\big\rangle}\)
    \( \newcommand\Blan{\Big\langle}\)
    \( \newcommand\Bran{\Big\rangle}\)
    \( \newcommand\intl{\int\limits}\)
    \( \newcommand\half{\frac{1}{2}}\)
    \( \newcommand\third{\frac{1}{3}}\)
    \( \newcommand\fourth{\frac{1}{4}}\)
    \( \newcommand\eighth{\frac{1}{8}}\)
    \( \newcommand\uar{\uparrow}\)
    \( \newcommand\dar{\downarrow}\)
    \( \newcommand\undertext#1{$\underline{\hbox{#1}}$}\)
    \( \newcommand\Tra{\mathop{\textsf{Tr}}\,}\)
    \( \newcommand\det{\mathop{\textsf{det}}\,}\)
    \( \def\tket#1{|  #1 \rangle}\)
    \( \def\tbra#1{\langle #1|}\)
    \( \def\tbraket#1#2{\langle #1  |   #2 \rangle}\)
    \( \def\texpect#1#2#3{\langle #1 |   #2  |  #3 \rangle}\)
    \( \def\sket#1{|  \, #1 \,  \rangle}\)
    \( \def\sbra#1{\langle \,  #1 \, |}\)
    \( \def\sbraket#1#2{\langle \, #1  \, |  \, #2 \,  \rangle}\)
    \( \def\sexpect#1#2#3{\langle \, #1 \, | \,  #2  \, | \, #3 \, \rangle}\)
    \(\def\ket#1{\big| \, #1\, \big\rangle}\)
    \( \def\bra#1{\big\langle \, #1 \, \big|}\)
    \( \def\braket#1#2{\big\langle \, #1\, \big| \,#2 \,\big\rangle}\)
    \( \def\expect#1#2#3{\big\langle\, #1\, \big|\, #2\, \big| \,#3\, \big\rangle}\)
    \( \newcommand\pz{\partial}\)
    \( \newcommand\pzb{\bar{\partial}}\)
    \( \newcommand\svph{\vphantom{\int}}\)
    \( \newcommand\vph{\vphantom{\sum_i}}\)
    \( \newcommand\bvph{\vphantom{\sum_N^N}}\)
    \( \newcommand\nd{^{\vphantom{\dagger}}}\)
    \( \newcommand\ns{^{\vphantom{*}}}\)
    \( \newcommand\yd{^\dagger}\)
    \( \newcommand\zb{\bar z}\)
    \( \newcommand\zdot{\dot z}\)
    \( \newcommand\zbdot{\dot{\bar z}}\)
    \( \newcommand\kB{k_{\sss{B}}}\)
    \( \newcommand\kT{k_{\sss{B}}T}\)
    \( \newcommand\gtau{g_\tau}\)
    \( \newcommand\Htil{\tilde H}\)
    \( \newcommand\pairo{(\phi\nd_0,J\nd_0)}\)
    \( \newcommand\pairm{(\phi\nd_0,J)}\)
    \( \newcommand\pairob{(\Bphi\nd_0,\BJ\nd_0)}\)
    \( \newcommand\pairmb{(\Bphi\nd_0,\BJ)}\)
    \( \newcommand\pair{(\phi,J)}\)
    \( \newcommand\Hz{H\nd_0}\)
    \( \newcommand\Ho{H\nd_1}\)
    \( \newcommand\Htz{\Htil\nd_0}\)
    \( \newcommand\Hto{\Htil\nd_1}\)
    \( \newcommand\oc{\omega_\Rc}\)

    \(\newcommand \gtwid{\approx}\)

    \( \newcommand\index{\textsf{ind}}\)
    \( \newcommand\csch{\,{ csch\,}}\)
    \( \newcommand\ctnh{\,{ ctnh\,}}\)
    \( \newcommand\ctn{\,{ ctn\,}}\)
    \( \newcommand\sgn{\,{ sgn\,}}\)
    \( \def\tmapright#1{\xrightarrow \limits^{#1}}\)
    \( \def\bmapright#1{\xrightarrow\limits_{#1}}\)
    \( \newcommand\hfb{\hfill\break}\)
    \( \newcommand\Rep{\textsf{Re}\,}\)
    \( \newcommand\Imp{\textsf{Im}\,}\)
    \( \newcommand\ncdot{\!\cdot\!}\)
    \( \def\tmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits^{#1}}\ }\)
    \( \def\bmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits_{#1}}\ }\)
    \( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

    \( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
    \( \def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}\)
    \( \def\qabc#1#2#3{\pz^2\! #1\over\pz #2\,\pz #3}\)
    \( \def\rabc#1#2#3#4{(\pz #1,\pz #2)\over (\pz #3,\pz #4)}\)
    \( \newcommand\subA{\ns_\ssr{A}}\)
    \( \newcommand\subB{\ns_\ssr{B}}\)
    \( \newcommand\subC{\ns_\ssr{C}}\)
    \( \newcommand\subD{\ns_\ssr{D}}\)
    \( \newcommand\subAB{\ns_\ssr{AB}}\)
    \( \newcommand\subBC{\ns_\ssr{BC}}\)
    \( \newcommand\subCD{\ns_\ssr{CD}}\)
    \( \newcommand\subDA{\ns_\ssr{DA}}\)
    \( \def\lmapright#1{\ \ \smash{\mathop{\hbox to 55pt{\rightarrowfill}}\limits^{#1}}\ \ }\)
    \( \def\enth#1{\RDelta {\textsf H}^0_\Rf[{ #1}]}\)
    \( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
    \( \newcommand\longrightharpoonup{\relbar\joinrel\rightharpoonup}\)
    \( \newcommand\longleftharpoondown{\leftharpoondown\joinrel\relbar}\)
    \( \newcommand\cds{\,\bullet\,}\)
    \( \newcommand\ccs{\,\circ\,}\)
    \( \newcommand\nsub{_{\vphantom{\dagger}}}\)
    \( \newcommand\rhohat{\hat\rho}\)
    \( \newcommand\vrhhat{\hat\vrh}\)
    \( \newcommand\impi{\int\limits_{-\infty}^\infty\!\!\!}\)
    \( \newcommand\brangle{\big\rangle}\)
    \( \newcommand\blangle{\big\langle}\)
    \( \newcommand\vet{\tilde\ve}\)
    \( \newcommand\zbar{\bar z}\)
    \( \newcommand\ftil{\tilde f}\)
    \( \newcommand\XBE{\RXi\ns_\ssr{BE}}\)
    \( \newcommand\XFD{\RXi\ns_\ssr{FD}}\)
    \( \newcommand\OBE{\Omega\ns_\ssr{BE}}\)
    \( \newcommand\OFD{\Omega\ns_\ssr{FD}}\)
    \( \newcommand\veF{\ve\ns_\RF}\)
    \( \newcommand\kF{k\ns_\RF}\)
    \( \newcommand\kFu{k\ns_{\RF\uar}}\)
    \( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
    \( \newcommand\muB{\mu\ns_\ssr{B}}\)
    \( \newcommand\mutB{\tilde\mu}\ns_\ssr{B}\)
    \( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
    \( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
    \( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
    \( \newcommand\xhihOZ
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\jhz{\HJ(0)}\)
    \( \newcommand\nda{\nd_\alpha}\)
    \( \newcommand\ndap{\nd_{\alpha'}}\)
    \( \newcommand\labar
    ParseError: invalid DekiScript (click for details)
    Callstack:
        at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1
        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.07:_Ideal_Gas_Statistical_Mechanics), /content/body/p[1]/span, line 1, column 23
    
    \)
    \( \newcommand\msa{m\ns_\ssr{A}}\)
    \( \newcommand\msb{m\ns_\ssr{B}}\)
    \( \newcommand\mss{m\ns_\Rs}\)
    \( \newcommand\HBx{\hat\Bx}\)
    \( \newcommand\HBy{\hat\By}\)
    \( \newcommand\HBz{\hat\Bz}\)
    \( \newcommand\thm{\theta\ns_m}\)
    \( \newcommand\thp{\theta\ns_\phi}\)
    \( \newcommand\mtil{\widetilde m}\)
    \( \newcommand\phitil{\widetilde\phi}\)
    \( \newcommand\delf{\delta\! f}\)
    \( \newcommand\coll{\bigg({\pz f\over\pz t}\bigg)\nd_{\! coll}}\)
    \( \newcommand\stre{\bigg({\pz f\over\pz t}\bigg)\nd_{\! str}}\)
    \( \newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}\)
    \( \newcommand\vbar{\bar v}\)
    \( \newcommand\BCE{\mbox{\boldmath{$\CE$}}\!}\)
    \( \newcommand\BCR{\mbox{\boldmath{$\CR$}}\!}\)
    \( \newcommand\gla{g\nd_{\RLambda\nd}}\)
    \( \newcommand\TA{T\ns_\ssr{A}}\)
    \( \newcommand\TB{T\ns_\ssr{B}}\)
    \( \newcommand\ncdot{\!\cdot\!}\)
    \( \newcommand\NS{N\ns_{\textsf S}}\)

    The ordinary canonical partition function for the ideal gas was computed in Equation [zideal]. We found

    \[\begin{split} Z(T,V,N)&={1\over N!}\prod_{i=1}^N\int\!{d^d\!x\ns_i\, d^d\!p\ns_i\over (2\pi\hbar)^d}\>e^{-\beta\Bp_i^2/2m}\\ &={V^N\over N!}\left(\int\limits_{-\infty}^\infty\!\!{dp\over 2\pi\hbar}\>e^{-\beta p^2/2m}\right)^{\!\!Nd}\\ &={1\over N!}\bigg({V\over\lambda_T^d}\bigg)^{\!N}\ , \end{split}\]

    where \(\lambda\ns_T\) is the thermal wavelength:

    \[\lambda\ns_T=\sqrt{2\pi\hbar^2/m\kT}\ .\]

    The physical interpretation of \(\lambda\ns_T\) is that it is the de Broglie wavelength for a particle of mass \(m\) which has a kinetic energy of \(\kT\).

    In the GCE, we have

    \[\begin{split} \Xi(T,V,\mu)&=\sum_{N=0}^\infty e^{\beta\mu N}\,Z(T,V,N)\\ &=\sum_{N=1}^\infty {1\over N!}\Bigg({V e^{\mu/\kT}\over \lambda_T^d}\Bigg)^{\!\!N} =\exp\Bigg({V e^{\mu/\kT}\over \lambda_T^d}\Bigg)\ . \end{split}\]

    From \(\Xi=e^{-\Omega/\kT}\), we have the grand potential is

    \[\Omega(T,V,\mu)=-V\kT\,e^{\mu/\kT}\big/\lambda_T^d\ .\]

    Since \(\Omega=-pV\) (see §6.2), we have

    \[p(T,\mu)=\kT\,\lambda_T^{-d}\,e^{\mu/\kT}\ .\]

    The number density can also be calculated:

    \[n={N\over V}=-{1\over V}\pabc{\Omega}{\mu}{T,V}=\lambda_T^{-d}\,e^{\mu/\kT}\ .\]

    Combined, the last two equations recapitulate the ideal gas law, \(pV=N\kT\).

    Maxwell velocity distribution

    The distribution function for momenta is given by

    \[g(\Bp)=\Big\langle {1\over N}\sum_{i=1}^N\delta(\Bp\ns_i-\Bp)\Big\rangle\ .\]

    Note that \(g(\Bp)=\big\langle\delta(\Bp\ns_i-\Bp)\big\rangle\) is the same for every particle, independent of its label \(i\). We compute the average \(\langle A\rangle=\Tra \big(A e^{-\beta\HH}\big)/\Tra e^{-\beta\HH}\). Setting \(i=1\), all the integrals other than that over \(\Bp\ns_1\) divide out between numerator and denominator. We then have

    \[\begin{split} g(\Bp)&={\int\!\!d^3\!p\ns_1 \>\delta(\Bp\ns_1-\Bp)\,e^{-\beta\Bp_1^2/2m}\over \int\!\!d^3\!p\ns_1 \>e^{-\beta\Bp_1^2/2m}}\\ &=(2\pi m\kT)^{-3/2}\,e^{-\beta\Bp^2/2m}\ . \end{split}\]

    Textbooks commonly refer to the velocity distribution \(f(\Bv)\), which is related to \(g(\Bp)\) by

    \[f(\Bv)\,d^3\!v = g(\Bp)\,d^3\!p\ .\]

    Hence,

    \[f(\Bv)=\bigg({m\over 2\pi\kT}\bigg)^{\!3/2}\,e^{-m\Bv^2/2\kT}\ .\]

    This is known as the Maxwell velocity distribution. Note that the distributions are normalized, viz.

    \[\int\!\!d^3\!p\>g(\Bp)=\int\!\!d^3\!v\,f(\Bv)=1\ .\]

    clipboard_e39663cecce98ac534e5842a6d41e0fef.png
    Figure \(\PageIndex{1}\): Maxwell distribution of speeds \(\varphi(v/v_0)\). The most probable speed is \(v_{\ssr{MAX}}=\sqrt{2}\>v_0\). The average speed is \(v_{\ssr{AVG}}=\sqrt{\frac{8}{\pi}} \>v_0\). The RMS speed is \(v_{\ssr{RMS}}=\sqrt{3}\>v_0\).

    If we are only interested in averaging functions of \(v=|\Bv|\) which are isotropic, then we can define the Maxwell speed distribution, \(\ftil(v)\), as

    \[\ftil(v)=4\pi\,v^2 f(\Bv)=4\pi\bigg({m\over 2\pi\kT}\bigg)^{\!3/2}\, v^2\,e^{-mv^2/2\kT}\ .\]

    Note that \(\ftil(v)\) is normalized according to

    \[\int\limits_0^\infty\!\!dv\,\ftil(v)=1\ .\]

    It is convenient to represent \(v\) in units of \(v\ns_0=\sqrt{\kT/m}\), in which case

    \[\ftil(v)={1\over v\ns_0}\>\varphi(v/v\ns_0)\qquad,\qquad \varphi(s)=\sqrt{\frac{2}{\pi}}\,s^2\,e^{-s^2/2}\ .\]

    The distribution \(\varphi(s)\) is shown in Figure \(\PageIndex{1}\). Computing averages, we have

    \[C\ns_k\equiv \langle s^k\rangle=\int\limits_0^\infty\!\! ds\>s^k\,\varphi(s) = 2^{k/2}\cdot {2\over\sqrt{\pi}}\,\RGamma\left(\frac{3}{2} + \frac{k}{2}\right)\ .\]

    Thus, \(C\ns_0=1\), \(C\ns_1=\sqrt{\frac{8}{\pi}}\), \(C\ns_2=3\), The speed averages are

    \[\blangle v^k\brangle =C\ns_k\,\bigg({\kT\over m}\bigg)^{\!k/2}\ .\]

    Note that the average velocity is \(\langle\Bv\rangle=0\), but the average speed is \(\langle v\rangle=\sqrt{8\kT/\pi m}\). The speed distribution is plotted in Figure \(\PageIndex{1}\).

    Equipartition

    The Hamiltonian for ballistic (massive nonrelativistic) particles is quadratic in the individual components of each momentum \(\Bp\ns_i\). There are other cases in which a classical degree of freedom appears quadratically in \(\HH\) as well. For example, an individual normal mode \(\xi\) of a system of coupled oscillators has the Lagrangian

    \[L=\half{\dot\xi}^2-\half\,\omega_0^2\,\xi^2\ ,\]

    where the dimensions of \(\xi\) are \([\xi]=M^{1/2} L\) by convention. The Hamiltonian for this normal mode is then

    \[\HH={p^2\over 2} + \half\, \omega_0^2\,\xi^2\ ,\]

    from which we see that both the kinetic as well as potential energy terms enter quadratically into the Hamiltonian. The classical rotational kinetic energy is also quadratic in the angular momentum components.

    Let us compute the contribution of a single quadratic degree of freedom in \(\HH\) to the partition function. We’ll call this degree of freedom \(\zeta\) – it may be a position or momentum or angular momentum – and we’ll write its contribution to \(\HH\) as

    \[\HH\ns_\zeta=\half K\zeta^2\ ,\]

    where \(K\) is some constant. Integrating over \(\zeta\) yields the following factor in the partition function:

    \[\int\limits_{-\infty}^\infty\!\!\!d\zeta\>e^{-\beta K\zeta^2/2}= \bigg({2\pi\over K\beta}\bigg)^{\!1/2}\ .\]

    The contribution to the Helmholtz free energy is then

    \[\RDelta F\ns_\zeta=\half\kT\ln\!\bigg({K\over 2\pi \kT}\bigg)\ ,\]

    and therefore the contribution to the internal energy \(E\) is

    \[\RDelta E\ns_\zeta={\pz\over\pz\beta}\big(\beta\,\RDelta F\ns_\zeta\big)= {1\over 2\beta}=\half \kT\ .\]

    We have thus derived what is commonly called the equipartition theorem of classical statistical mechanics:

    We now see why the internal energy of a classical ideal gas with \(f\) degrees of freedom per molecule is \(E=\half f N\kT\), and \(C\ns_V=\half N\kB\). This result also has applications in the theory of solids. The atoms in a solid possess kinetic energy due to their motion, and potential energy due to the spring-like interatomic potentials which tend to keep the atoms in their preferred crystalline positions. Thus, for a three-dimensional crystal, there are six quadratic degrees of freedom (three positions and three momenta) per atom, and the classical energy should be \(E=3N\kT\), and the heat capacity \(C\ns_V=3N\kB\). As we shall see, quantum mechanics modifies this result considerably at temperatures below the highest normal mode (phonon) frequency, but the high temperature limit is given by the classical value \(C\ns_V=3\nu R\) (where \(\nu=N/\NA\) is the number of moles) derived here, known as the Dulong-Petit limit.


    This page titled 4.7: Ideal Gas Statistical Mechanics is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.