4 ln 04 032 ln 032 024 ln 024 004 ln 004 kb p0

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Unformatted text preview: et 7 UC Berkeley Page 7 (ii) What is the ratio of the multiplicity for the two distributions? (iii) Calculate the difference in the entropy of the two distributions in units of J mol-1 K-1. Answer: Because the number of molecules is large enough, this question is best answered using the probabilistic definition of entropy, rather than trying to calculate factorials. For the first distribution: N total = 10000 + 8000 + 6000 + 1000 = 25000 10000 8000 = 0.4 p1 = = 0.32 25000 25000 6000 1000 p2 = = 0.24 p3 = = 0.04 25000 25000 S = − N total × [ 0.4 ln 0.4 + 0.32 ln 0.32 + 0.24 ln 0.24 + 0.04 ln 0.04 ] kB p0 = = − N total × [ −0.367 − 0.365 − 0.343 − 0.129 ] = N total × 1.204 For the second distribution: N total = 10000 + 7000 + 5000 + 3000 = 25000 10000 7000 = 0.4 p1 = = 0.28 25000 25000 5000 3000 p2 = = 0.20 p3 = = 0.12 25000 25000 S = − N total × [ 0.4 ln 0.4 + 0.28 ln 0.28 + 0.20 ln 0.20 + 0.12 ln 0.12 ] kB p0 = = − N total × [ −0.367 − 0.356 − 0.322 − 0.254 ]...
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