PS3Sol_f10 - Physics 112 Fall 2010 Professor William...

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Unformatted text preview: Physics 112 Fall 2010 Professor William Holzapfel Homework 3 Solutions Problem 1: Energy Fluctuations a) Kittel 3.4 The easiest way to approach to this problem is probably to start with the right side of the answer. By defining β = 1 τ , ∂U ∂τ V = ∂U ∂β V ∂β ∂τ V =- 1 τ 2 ∂U ∂β . Using U = < > = ∑ n n e- β n /Z =- 1 Z ∂Z ∂β we can write: τ 2 ∂U ∂τ V =- ∂U ∂β V = ∂ ∂β 1 Z ∂Z ∂β = 1 Z ∂ 2 Z ∂β 2- 1 Z 2 ∂Z ∂β 2 . (1) Next, simply note that < > = X n n e- β n /Z =- 1 Z ∂Z ∂β < 2 > = X n 2 n e- β n /Z = 1 Z ∂ 2 Z ∂β 2 and so by (1) τ 2 ∂U ∂τ V = < 2 >- < > 2 = ( 4 ) 2 as desired. b) Using our result from a) and the fact that U = 3 2 Nτ for an ideal gas, we have < (Δ U ) 2 > = τ 2 ∂U ∂τ V = τ 2 · 3 2 N 1 and so p < (Δ U ) 2 > U = q 3 2 Nτ 2 3 2 Nτ = r 2 3 N . Using q 2 3 N = 0 . 1 we get N = 200 3 . So to have the energy fluctuations of an ideal gas to be on the order of 10% or more, you need to have fewer than 67 particles. Problem 2, Kittel 3.6: Diatomic Molecules a) As the problem reminds you, it is important to remember that the partition func- tion is defined as a sum over all microstates , not energy levels. To write the partition function as a sum over energy levels, you must include the degeneracy (or multi- plicity) of each level. We know for a molecule with total angular momentum j the possible values of j z are- j,- j + 1 , ··· , j- 1 , j . This gives a multiplicity of 2 j + 1 for each energy level at fixed j , so the partition function is: Z R = ∞ X j =0 (2 j + 1) e- j ( j +1) /τ . b) For the high temperature limit τ the thermal energy is much larger than the spacing between energy levels, so we can convert the sum into an integral over j : Z R ( τ ) = ∞ X j =0 (2 j + 1) e- j ( j +1) /τ ≈ Z ∞ (2 j + 1) e- j ( j +1) /τ dj =- τ Z ∞ ∂ ∂j e- j ( j +1) /τ dj =- τ e- j ( j +1) /τ ∞ . = τ 2 c) In the low temperature limit, τ , higher j terms in the partition function quickly decay exponentially, so we can approximate the sum by just keeping the first two terms: Z R ( τ ) = ∞ X j =0 (2 j + 1) e- j ( j +1) /τ = 1 + 3 e- 2 /τ + 5 e- 6 /τ + ··· ≈ 1 + 3 e- 2 /τ ....
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This note was uploaded on 11/18/2010 for the course PHYSICS 112 taught by Professor Steveng.louie during the Fall '06 term at Berkeley.

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PS3Sol_f10 - Physics 112 Fall 2010 Professor William...

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