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Number of microstates in multi-particle system

  1. Nov 16, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the number of accessible microstates for a multi-particle system whose energy depends on temperature as ##U = aT^n## where a is a positive constant and ##n>1##. Use the fact that
    ##S = \int \frac{C_v}{T}dT##

    2. Relevant equations


    3. The attempt at a solution
    ##U = nC_vdT ##

    So ##\frac{U}{n} = C_v dT = \frac{aT^n}{n}##

    ##S = \frac{a}{n} \int T^{n-1}dT = \frac{a(n-1)^2}{n}T^{n-2}##

    ##=kln(\Omega)##
    Rearranging and raising both sides to the power of e gives
    ##\Omega = e^{\frac{a}{k}(n-1)^2T^{n-2}}##

    I'm slightly suspicious of that answer and in particular of whether the internal energy U in the equation ##U=nC_vdT## is the same as the U in the question, and the n in the equation ##U=nC_vdT## is the same n as the one in the equation for energy. Because the n in the given energy equation isn't defined. Is what I've done ok?
     
    Last edited: Nov 16, 2016
  2. jcsd
  3. Nov 16, 2016 #2

    DrClaude

    User Avatar

    Staff: Mentor

    Your starting point should be the definition of ##C_v##:
    $$
    C_v = \frac{dU}{dT}
    $$
    Also, you should check that integration you did.
     
  4. Nov 16, 2016 #3
    Because I differentiated. How ridiculous. Ok, I'll try again!
     
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