Statistical mechanics: multiplicity

Then you can use the fact that the average number of adsorbed atoms is given by the derivative of the entropy with respect to energy.
  • #1
SoggyBottoms
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Homework Statement


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.

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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  • #2
SoggyBottoms said:

Homework Statement


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.

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?

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:

[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](N-n)\ln(N-n) = \left(1-\frac{n}{N}\right)N\ln N + (N-n)\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.
 

FAQ: Statistical mechanics: multiplicity

What is statistical mechanics?

Statistical mechanics is a branch of physics that studies the behavior of systems with a large number of particles, such as gases, liquids, and solids. It uses statistical methods to understand the macroscopic properties of these systems based on the microscopic behavior of their constituent particles.

What is multiplicity in statistical mechanics?

In statistical mechanics, multiplicity refers to the number of ways in which a particular state of a system can be achieved. This includes the number of different arrangements of particles that can produce the same macroscopic properties, such as energy or volume.

How is multiplicity related to entropy?

Entropy is a measure of the disorder or randomness of a system. In statistical mechanics, entropy is directly related to the multiplicity of a system. The higher the multiplicity, the higher the entropy, as there are more ways for the particles to arrange themselves and produce the same macroscopic properties.

What is the significance of multiplicity in statistical mechanics?

Multiplicity is a key concept in statistical mechanics as it allows us to calculate the probabilities of different states of a system and predict its behavior. By understanding the multiplicity of a system, we can determine the most likely state and make predictions about its thermodynamic properties.

How is the concept of multiplicity used in real-world applications?

Multiplicity has various applications in fields such as thermodynamics, chemistry, and materials science. It is used to predict the behavior of gases, liquids, and solids, as well as the phase transitions that occur in these systems. Multiplicity is also used in the design of new materials and in the study of chemical reactions.

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