ch3 - Advanced Topics in Equilibrium Statistical Mechanics...

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Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson 3. Theory of Gases A. The Ideal Gas As we shall see, gases are much easier to deal with than liquids, because the low density of gases makes collisions and pair interactions of molecules infrequent. The strategy will be to derive an expansion in powers of particle density known as the virial expansion . The leading term, accurate at inFnite dilution, is the ideal gas and follows by setting U ( r N )=0. We shall begin by pursuing the thermodynamic properties of a monatomic ideal gas in the canonical ensemble . Q c ( N,V,T )= Z d r N e βU = V N Q = V N N ! λ 3 N T βA = ln Q =ln N !+ N ln( λ 3 T /V ) We are interested in the ”thermodynamic limit” of this expression, which corre- sponds to considering a system of macroscopic extent, i.e. with Avogadro’s number of atoms. We take this limit by simultaneously taking N →∞ , V , while holding the average particle number density ρ = N V constant. Stirling’s asymptotic approximation for the logarithm of large factorials, ln N ! N ln N N + ... , gives N ln N N ln V + N (ln λ 3 T 1) N ln( N/V )+ N (ln λ 3 T 1) The ideal gas equation of state follows from: p = ∂A ∂V ) N,T βp = N · 1 V ρ 1
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Thus, the ideal gas equation of state can be written in any of the alternate forms: βp = ρ, pV = Nk B T, pV = nRT where n is the number of moles of atoms. Next, we repeat this calculation in the grand canonical ensemble : Q G ( µ, V, T )= X N =0 z N N ! Q c ( N,V,T )= X N =0 ( zV ) N N ! = e The thermodynamic connection is βpV =ln Q G = or = z = e βµ λ 3 T . We thus have the pressure expressed as p ( z, T ), but a conventional equation of state is of the form p ( ρ, T ). Thus, we need an expression relating the activity z to the average density ρ , i.e. we need the function z ( ρ ). Recall that h N i = ln Q G ln z ) β,V = V ∂z ln z ) = Vz So z = h N i V ρ is the functional relationship between z and ρ in the ideal gas limit. It follows immediately that the equation of state for an ideal gas in the grand canonical ensemble is given as before by = ρ . Notice that we needed Stirling’s approximation in the canonical ensemble, but not in the grand canonical ensemble. Thus, results from the two agree only in the thermodynamic limit! This, however, is as expected, since fluctuations in the two ensembles are diFerent for systems of ±nite size and these will contribute small corrections to thermodynamic quantities that go to zero in the limit of an in±nite system. Lecture 4 Reading: McQuarrie — Chp. 12, Andersen handout B. Non-ideal gases — Cluster & Virial Expansions Now we are ready to talk about the problem of a real fluid with intermolecular interactions .
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ch3 - Advanced Topics in Equilibrium Statistical Mechanics...

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