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Homework Help: Blackbody Radiation - Entropy and Internal Energy

  1. Feb 25, 2013 #1
    1. The problem statement, all variables and given/known data

    Expression for the entropy and internal energy of black body radiation.

    Using the below relations:

    2. Relevant equations

    Total free energy for black body:
    $$ F = (k_b TV/\pi^2) \int k^2 ln[1-exp(-\hbar ck/k_b T)]dk $$
    Relationship between partition function and internal energy:
    $$ E = -\partial ln(z)/ \partial \beta $$

    Where ##\beta## is the inverse temperature given by:
    $$ \beta = (1/k_b T) $$
    Relationship between the free energy, internal energy and entropy:
    $$ F = E - TS $$

    3. The attempt at a solution

    If I use ## F = E - TS## rearranged to

    $$ S = (E-F)/T $$

    Then substitute the relations in and calculate.

    I make a little progress until I hit the ## F ## part, the integral gives me some problems as I am having trouble calculating it, I tried using Wolfram Alpha as a guide but it won't actually give me an answer which suggested to me that I'm going about it the wrong way.
    Last edited: Feb 25, 2013
  2. jcsd
  3. Feb 25, 2013 #2


    User Avatar
    Gold Member

    Use a u-substitution, like

    [tex] u = \frac{\hbar c k}{k_B T} [/tex]

    then you get

    [tex]dk = \frac{k_B T}{\hbar c} du[/tex]

    and the integral becomes

    [tex] F = \left( \frac{V}{\pi^2}\right) \left(\frac{(k_B T)^4}{(\hbar c)^3} \right) \int u^2 \ln{\left(1-e^{-u}\right)}du[/tex]

    which you can easily use Wolfram Alpha to solve
  4. Feb 25, 2013 #3


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    Science Advisor
    Homework Helper

    Don't forget the limits of integration. The variables are independent of photon's (angular) frequency.
  5. Feb 25, 2013 #4
    Oh man, I really should have seen that...

    Yeah I have the limits wrote down, I just didn't know how to show them in the post.
  6. Feb 25, 2013 #5


    User Avatar
    Gold Member

    Code (Text):
    \int_{-\infty}^{\infty} x^2 dx
    [tex]=\int_{-\infty}^{\infty} x^2 dx[/tex]

    You can also right click any TeX equation to see the code that produced it.
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