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Statistical mechanics: multiplicity 
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#1
Mar212, 12:20 AM

P: 61

1. The problem statement, all variables and given/known data
We have a surface that can adsorb identical atoms. There are N possible adsorption positions on this surface and only 1 atom can adsorb on each of those. An adsorbed atom is bound to the surface with negative energy [itex]\epsilon[/itex] (so [itex]\epsilon > 0[/itex]). The adsorption positions are far enough away to not influence each other. a) Give the multiplicity of this system for [itex]n[/itex] adsorbed atoms, with [itex]0 \leq n \leq N[/itex]. b) Calculate the entropy of the macrostate of n adsorbed atoms. Simplify this expression by assuming N >> 1 and n >> 1. c) If the temperature of the system is T, calculate the average number of adsorbed atoms. 3. The attempt at a solution a) [itex]\Omega(n) = \frac{N!}{n! (N  n)!}[/itex] b) [itex]S = k_b \ln \Omega(n) = k_b \ln \left(\frac{N!}{n! (N  n)!}\right)[/itex] Using Stirling's approximation: [itex]S \approx k_B ( N \ln N  N  n \ln n  n  (N  n) \ln (N  n)  (N  n) = k_B ( N \ln N  n \ln n  (N  n) \ln (N  n) [/itex] A Taylor expansion around n = 0 then gives: [itex] S \approx k_B ( \frac{n^2}{2N} + ...)\approx \frac{k_b n^2}{2N}[/itex] c) I'm not even sure if the previous stuff is correct, but I have no idea how to do this one. Any hints? 


#2
Mar312, 06:42 AM

HW Helper
P: 1,391

[tex]S \approx k_B ( N \ln N  N  n \ln n + n  (N  n) \ln (N  n) + (N  n) = k_B ( N \ln N  n \ln n  (N  n) \ln (N  n) [/tex] This change will result in some cancellations that help simplify your expression. Rewriting [tex](Nn)\ln(Nn) = \left(1\frac{n}{N}\right)N\ln N + (Nn)\ln\left(1\frac{n}{N}\right)[/tex] will help you make some more cancellations. Another mistake you made in your original attempt was that you expanded around n = 0, but you are told n is much greater than 1, so you can't do that expansion. What you can do, however, is assume that while n is much greater than 1, it is still much less that N, such that n/N is small, and you can expand the above logarithms in n/N. This will give you a simple expression for the entropy. To get the temperature, you need to write the entropy as a function of the total energy. Right now your entropy is a function of number. However, you are told how much energy there is per site, so you can figure out what the total energy is for n adsorbed atoms. Use this to rewrite the entropy in terms of the total energy. 


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