• Support PF! Buy your school textbooks, materials and every day products Here!

Statistical Mechanics question: calculate energy difference

  • Thread starter trelek2
  • Start date
  • #1
88
0

Homework Statement


In a system of N weakly interacting particles each particle can be in one of M energy states:
E_1 < E_2 < ... < E_M
At T=300K there are 3 times as many particles in E_2 as in E_1.
Calculate E_2 - E_1

Homework Equations


This is not my homework, just a tutorial question, I'm revising and I'm not sure if I'm understanding this stuff. Let me know if this is correct and if not please tell me how it should be done.


The Attempt at a Solution


I consider only particles of Energies E_1 and E_2.

Probability of particle in state E_1=
[tex]P(E_{1})=Z^{-1}exp(- \frac{E_{1}}{kT})=0.25 [/tex]

Probability of particle in state E_2=
[tex]P(E_{2})=Z^{-1}exp(- \frac{E_{2}}{kT})=0.75 [/tex]

[tex]Z=exp(- \frac{E_{1}+E_{2}}{kT}) [/tex]
Hence:
[tex]E_{1}-E_{2}=kT(ln(1/3))[/tex]
 

Answers and Replies

  • #2
674
2
How do you know the true probabilities are 0.25 and 0.75 without knowing the partition function. The particle also has a probability of being in a higher state.

The only thing you know is the ratio of the probabilities:

[tex]\frac{P(E_2)}{P(E_1)} = exp(-\frac{E_2-E_1}{kT}) = 3[/tex]

Which would make E_2 < E_1 for the single particle (opposite of what you said initially).

Edit: Dealing with N particles can be tricky since you have a lot of combinations. You can have 1 particle in E_1, 3 particles in E_2, and the N-4 particles can occupy any of the higher states. So you might want to limit yourself to a single particle, or a set number of particles.
 
Last edited:

Related Threads for: Statistical Mechanics question: calculate energy difference

  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
1K
Replies
4
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
1K
Replies
3
Views
2K
Replies
7
Views
1K
Top