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Thermal Physics, energy and temperature

  1. May 20, 2013 #1
    1. The problem statement, all variables and given/known data
    In section 2.5 I quoted a theorum on the multiplicity of any system with only quadratic degrees of freedom: In the high-temperature limit where the number of units of energy is much larger than the degrees of freedom, the multiplicity of any such system is proportional to ## U^{Nf/2} ##, where Nf is the total number of degrees of freedom. Find an expression for the energy of such a system in terms of its temperature, and comment on the result. How can you tell that this formula for ## \Omega ## cannot be valid when the total energy is very small?

    2. Relevant equations

    ## S = k \ln (\Omega) ##

    ## \dfrac{1}{T} = \left( \dfrac{\partial S}{\partial U} \right) ##

    3. The attempt at a solution

    So they said that the multiplicity is proportional to ## U^{Nf/2} ##

    And so I'm assuming ## \Omega = A \cdot U^{Nf/2} ## for some constant A

    Putting that into the Entropy formula I get

    ## S = k \ln (A \cdot U^{Nf/2}) ##

    ## S = k \ln (A) + \dfrac{Nfk}{2} \ln(U) ##

    Partial differentiating with respect to U I get:

    ## \dfrac{1}{T} = \left( \dfrac{\partial S}{\partial U} \right) = \dfrac{Nfk}{2U} ##

    And so: ## T = \dfrac{2U}{Nfk} ##
    which tells us that T is proportional to U and inversely proportional to f.

    Is that right? And what about the last part of the question? If U is very small, then T is very small as well, so why can it not be valid? Is it because the temperature is measured in Kelvin starting from -273.15 and so no amount of U will give you that?
     
  2. jcsd
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