(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

(a)

The formula for the multiplicity of an Einstein solid in the “high-temperature” limit,

q >> N, was derived in one of the lectures. Use the same methods to show that the multiplicity of an Einstein solid in the “low-temperature” limit, q << N, is

Ω(N,q)=(eN/q)^q (When q≪N)

(b)

Find a formula for the temperature of an Einstein solid in the limit q << N. Solve for the energy as a function of temperature to obtain U=Nϵe^(-ϵ/(k_B T)), where ε is the size of an energy unit.

2. Relevant equations

N/A

3. The attempt at a solution

Okay, I have started (a):

[tex] \Omega (N,q) = (\frac{(N - 1 + q)!}{(n - 1)!q!}) [/tex]

N large:

(N - 1)! approx = N!

[tex] \Omega (N,q) = (\frac{(N + q)!}{N! q!}) [/tex]

Take logs

[tex] ln(\Omega (N,q)) = ln(N + q)! - ln(N!) - ln(q!) [/tex]

Use Stirling aprrox:

[tex] ln N! \approx N ln(N) - N [/tex]

[tex] ln(\Omega (N,q)) = (N + q)ln(N + q) - (N + q) - (Nln(N) - N) - (qln(q) - q) [/tex]

Cancels down to:

[tex] ln(\Omega (N,q)) = (N + q)ln(N + q) - Nln(N) - qln(q) [/tex]

Now I have to use the Taylor Expnasion for q << N, but I got slightly confused here.

Could anyone please offer some assistane what I need to do from here?

Many thanks in advance,

TFM

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# Homework Help: Multiplicity of a Einstein Solid, Low Temperature Limit

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