Physics Forums

## Statistical mechanics: multiplicity

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 $-\epsilon$ (so $\epsilon > 0$). The adsorption positions are far enough away to not influence each other.

a) Give the multiplicity of this system for $n$ adsorbed atoms, with $0 \leq n \leq N$.

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) $\Omega(n) = \frac{N!}{n! (N - n)!}$

b) $S = k_b \ln \Omega(n) = k_b \ln \left(\frac{N!}{n! (N - n)!}\right)$

Using Stirling's approximation: $S \approx k_B ( N \ln N - N - n \ln n - n - (N - n) \ln (N - n) - (N - n) = k_B ( N \ln N - n \ln n - (N - n) \ln (N - n)$

A Taylor expansion around n = 0 then gives: $S \approx k_B (- \frac{n^2}{2N} + ...)\approx -\frac{k_b n^2}{2N}$

c) I'm not even sure if the previous stuff is correct, but I have no idea how to do this one. Any hints?

PhysOrg.com  science news on PhysOrg.com  >> Front-row seats to climate change>> Attacking MRSA with metals from antibacterial clays>> New formula invented for microscope viewing, substitutes for federally controlled drug

Recognitions:
Homework Help

Quote by SoggyBottoms 					 				 				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 $-\epsilon$ (so $\epsilon > 0$). The adsorption positions are far enough away to not influence each other.  a) Give the multiplicity of this system for $n$ adsorbed atoms, with $0 \leq n \leq N$.  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) $\Omega(n) = \frac{N!}{n! (N - n)!}$  b) $S = k_b \ln \Omega(n) = k_b \ln \left(\frac{N!}{n! (N - n)!}\right)$  Using Stirling's approximation: $S \approx k_B ( N \ln N - N - n \ln n - n - (N - n) \ln (N - n) - (N - n) = k_B ( N \ln N - n \ln n - (N - n) \ln (N - n)$  A Taylor expansion around n = 0 then gives: $S \approx k_B (- \frac{n^2}{2N} + ...)\approx -\frac{k_b n^2}{2N}$  c) I'm not even sure if the previous stuff is correct, but I have no idea how to do this one. Any hints?
Part (a) looks good, but you made a couple of sign mistakes in part (b). In the terms in the denominator you forgot to distribute the negative sign to the second term in n ln n - n and (N-n) ln(N-n). I've corrected the signs here:

$$S \approx k_B ( N \ln N - N - n \ln n + n - (N - n) \ln (N - n) + (N - n) = k_B ( N \ln N - n \ln n - (N - n) \ln (N - n)$$

This change will result in some cancellations that help simplify your expression. Rewriting

$$(N-n)\ln(N-n) = \left(1-\frac{n}{N}\right)N\ln N + (N-n)\ln\left(1-\frac{n}{N}\right)$$

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.

Thread Tools

|                                                          |                               |         |
|----------------------------------------------------------|-------------------------------|---------|
| Similar Threads for: Statistical mechanics: multiplicity |                               |         |
| Thread                                                   | Forum                         | Replies |
|                                                          | Quantum Physics               | 2       |
|                                                          | Introductory Physics Homework | 3       |
|                                                          | Academic Guidance             | 5       |
|                                                          | Classical Physics             | 0       |
|                                                          | Classical Physics             | 0       |
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 16, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9191608428955078, "perplexity_flag": "middle"}