Simple Entropy/Multiplicity:

  • Thread starter HalfManHalfAmazing
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In summary, the conversation discusses the distribution of two distinguishable particles between 3 energy levels with a total energy of 2e. The entropy of the system is calculated using the formula S = k\ln\Omega. The addition of a third distinguishable particle with zero energy increases the entropy by a factor of 1.63. When considering indistinguishable particles, the number of microstates changes and the factor of increase in entropy is 1.29. The formula for calculating the entropy of n particles in n states involves factorials.
  • #1

Homework Statement

Two distinguishable particles are to be distributed between 3 energy levels: 0, 1e and 2e such that the total energy level E = 2e.

a) What is the entropy of the system?
b) if a third distinguishable particle with zero energy were added to the system show that the entropy increases by a factor of 1.63.
c) Reevaulate a) with undistinguishable particles. does b) still hold if this time the added particle is undistinguistable?

Homework Equations

[tex] S = k\ln\Omega[/tex]
Various methods of determining multiplicity...

The Attempt at a Solution

Okay. First up this system has 3 possible states per particle and we've got 2 particles. We want the multiplicity of the macrostate 2e. Thus we need to determine the number of microstates that this macrostate has. Looks to me like we have the following states: 0/2, 2/0, 1/1, 1/1. Thus 4 microstates? so we plug 4 into the omega from the entropy formula and we're done?

Next, adding the third 0 energy level particle confuses me. It won't change the number of combinations required to reach 2e, will it? or do we do something like this: 0/2/0 0/0/2 2/0/0 0/1/1, 1/1/0, 1/0/1? Also, what formula could i use to make this a general expression such as what is the entropy of n particles in n states? I know it involves factorials...

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  • #3
You're right on both counts. Adding the 3rd particle changes the number of combinations in just the way you surmised.

As for n particles in n states, I'll leave that for someone else. :biggrin:
  • #4
I find a multiplicity of 6 for b. in which case the when i check the factor of increase i only find a factor of 1.29, not 1.63. So that can't be right?

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