Statistical Physics

150 10 Dense Gases – Ideal Gases at Low Temperaturef (E I ,N I ) is proportional to the number of microscopic states of the heatbath:f (E I ,N I ) ∝ Ω II (E t − E I ,N t − N I )Ω II (E t ,N t )[ ]1=exp {S II (E t − E I ,N t − N I ) − S II (E t ,N t )}k B[ { 1 ∂S II≃ exp −E Ik B=exp∂E − N I[− 1]k B T (E I − µN I )}]∂S II∂N. (10.3)That is, the chemical potential µ of the heat bath controls the probabilityand hence controls the number of molecules in system I. The correspondingdistribution is called the grand canonical distribution.The normalized probability isf (E,N)= 1 [Ξ exp − 1]k B T (E − µN) . (10.4)The denominator of the normalization coefficient,∞∑∫ ∞[(E − µN)Ξ(T,µ) ≡ exp −k B T=N=00∫ ∞∞∑N=00]Ω I (E,N)dE[exp − [E − µN − S ]I(E,N)T ]dE, (10.5)k B Tis called the grand partition function. If system I is macroscopic, the integrandis sharply peaked around the minimum of the argument of the exponentialfunction in the last line, and the system should almost always be found at thevalues E = E ∗ and N = N ∗ for which the minimum is realized. Furthermore,since system I and the heat bath (system II) are in thermal equilibrium, thetemperature T and the chemical potential µ are also those of system I.The free energy associated with this grand partition function can be writtenasJ = −k B T ln Ξ(T,µ) . (10.6)The value of J can be formally evaluated by expanding the integrand aroundthe maximum at (E ∗ ,N ∗ ) up to second order in the deviation, neglectingterms of order one with respect to terms of order N. The result isJ = E ∗ − S(E ∗ ,N ∗ )T − µN ∗ . (10.7)Since E ∗ = U is the internal energy and µN ∗ = G = U − ST + PV is theGibbs free energy, we arrive at the conclusion thatJ = −PV . (10.8)