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5.60_PS_8_2008_solutions

5.60_PS_8_2008_solutions - MASSACHUSETTS INSTITUTE OF...

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry Fall 2008 Problem Set 8 8.1 a) Consider a system of N particles that have just two states whose energies differ by a value ε 0 . What are the values of the heat capacity C (in units of Nk) at temperatures at which kT = ε 0 /10 and kT/ ε 0 = 10? At what limiting value(s) of T does the heat capacity equal zero? Based on your answers, sketch C vs. T. The y axis should be in units of Nk, and although you don’t need to get the magnitude right quantitatively, it should be reasonable. Assuming indistinguishable particles, q = 1 + e ε D / kT Q = q N N ! = (1 + e D / kT ) N N ! U = kT 2 ln Q T ⎟ = kT 2 (ln q N ln N !) T = NkT 2 ln q T ⎟ = NkT 2 q q T U = NkT 2 1 + e D / kT o kT 2 e D / kT = N o e D / kT 1 + e D / kT C = U T = N o o kT 2 1 + e D / kT ) 2 e 2 D / kT + 1 + e D / kT ) o kT 2 e D / kT C = N D 2 kT 2 e D / kT 1 + e D / kT () 2 At kT = D 10 , C Nk = 100 e 10 1 + e 10 2 = 0.0045 At kT = 10 D , C Nk = e 1 10 100 1 + e 1 10 2 = 0.0025 C=0 at T=0 and T=

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2 Heat Capacity versus Temperature b) Consider a 20-state system with the following nondegenerate energy levels. ε n = n ε 0 0 n 9 ε n = (n + 1000) ε 0 10 n 19 There are too many levels to carry out quantitative calculations by hand. In fact, no credit for exact calculations will be given. i) What are the approximate values of the single-particle partition function for values of T such that kT = 0, kT = 100 ε 0 , and kT = 10,000 ε 0 ? The problem describes twenty non-degenerate states: 0, ε D ,2 D ...9 D and 1010 D ,1011 D ...1019 D . At kT = 0 , only the ground state is populated and q 1 At kT = 100 D , the lowest ten states are about evenly populated, but the upper ten states are mostly unpopulated. Therefore, q 10 At kT = 10,000 D , all states are evenly populated, so q 20 .
3 ii) Sketch the approximate T dependence of the heat capacity. You should deduce the temperature ranges at which C is zero or nearly zero and at which it is large. The y axis should be in units of Nk, and although you don’t need to get the magnitude right quantitatively, it should be reasonable. Heat Capacity versus Temperature The x-axis is logarithmic in temperature in order to emphasize qualitative features of the heat capacity curve. We see two maxima – each of the two parts corresponding to a set of 10 grouped energy levels.

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5.60_PS_8_2008_solutions - MASSACHUSETTS INSTITUTE OF...

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