notes06-07-page25

notes06-07-page25 - (32 = Nkβ ~ ω 2 coth β ~ ω 2 − k...

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Problem: A simple model for a crystal is a “gas” of harmonic oscillators. De- termine A , S ,and U from the partition function for this model. Solution: For this model the crystal is modelled as a collection of harmonic oscillators so we need the partition function for the harmonic oscillator. Q crystal = q N HO = Ã 1 2sinh β ~ ω 2 ! (30) From our formulas for statistical thermodynamics A = kT ln Q crystal =+ NKT ln μ 2sinh β ~ ω 2 , (31) where we used properties of logs to pull the N out front and move the sinh term from to the numerator, S = ∂Q crystal ∂β + k ln Q crystal
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Unformatted text preview: (32) = Nkβ ~ ω 2 coth β ~ ω 2 − k ln μ 2 sinh β ~ ω 2 ¶ and U = − ∂Q crystal ∂β = N ~ ω 2 coth β ~ ω 2 . (33) Problem: Express the equation of state for internal energy for a Berthelot gas. Solution: The equation representing a Berthelot gas is P = nRT V − nb − n 2 a TV 2 . (34) We are interesting in an equation of state for U ( T, V ) . Writing out the total derivative of U ( T, V ) we get dU = μ ∂U ∂T ¶ V dT + μ ∂U ∂V ¶ T dV. (35) 13...
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