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hw04soln - 1 Homework 4(due November 1 Problem 4-1(4 pts...

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1 Homework 4 (due November 1) Problem 4-1 (4 pts.) Find the mean—square fl uctuation of the number of particles in a volume V , for a system with the given equation of state: P = P ( ρ, T ) Apply your result to ideal and van der Waals gases Solution = T log X N exp μ F µN T D ( N h N i ) 2 E = T μ 2 ∂µ 2 V,T = T μ ∂N ∂µ V,T V dP = Ndµ + SdT μ ∂µ ∂N V,T = V N μ ∂P ∂N V,T = 1 ρV μ ∂P ∂ρ V,T D ( N h N i ) 2 E = T μ ∂N ∂µ V,T = TρV μ ∂P ∂ρ 1 V,T Ideal gas P = : D ( N h N i ) 2 E = ρV = N VdW gas, P = Tρ/ (1 v o ρ ) ±ρ 2 / 2 : D ( N h N i ) 2 E = ρV " 1 (1 v o ρ ) 2 ±ρ # 1 Problem 4-2 (6 pts.) Consider a binary mixture of liquids with the same volume per molecule v 0 . Its free energy is given by F ( φ ) = Z d r v o h log φ + T (1 φ ) log (1 φ ) + ± 2 (1 φ ) φ i . Here φ is the local composition, i.e. the fraction of molecules of type 1. Find: a) the critical temperature and composition.
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