(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Statistical mechanics: multiplicity

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