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Unformatted text preview: I One way is to calculate averages for a path between two states (1 and 2) and to integrate over the reversible path. I Integrating the internal energy along a line of constant density: ± A Nk B T ² 2± A Nk B T ² 1 = Z β 2 β 1 ± E Nk B T ² d β β = Z T 2 T 1 ± E Nk B T ² d T T . (37) I Integrating the pressure along an isotherm: ± A Nk B T ² 2± A Nk B T ² 1 = Z ρ 2 ρ 1 ± PV Nk B T ² d ρ ρ = Z V 2 V 1 ± PV Nk B T ² d V V . (38) I This has to be done accurately for many closely spaced points, and is hence rather expensive. Notes I In FrenkelLadd method [FrenkelLadd, J. Chem. Phys. 81, 3188 (1984)] absolute internal energy is calculated by constructing a potential function which depends on parameter λ : U = U ( r , λ ) . I Then ∂ A ∂λ = k B T ∂ ∂λ ± ln R d r exp (U ( r , λ ) / k B T ) ² = R d r ∂ U ∂λ exp (U / k B T ) R d r exp (U / k B T ) = ³ ∂ U ∂λ ´ . (39) I U needs to be constructed so that for λ = λ the answer is accessible through analytic methods (e.g., ideal gas or harmonic lattice). Notes...
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 Winter '12
 Kotakoski
 Chemistry, Thermodynamics, Energy, Fundamental physics concepts, J. Chem

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