1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Statistical mechanics - Helmholtz free energy and Z

  1. Apr 5, 2008 #1
    1. The problem statement, all variables and given/known data
    The neutnral carbon atom has a 9-fold degeerate ground level and a 5-fold degenerate excited level at an energy 0.82 eV above the ground level. Spectroscopic measurements of a certain star show that 10% of the neutral carbon atoms are in the excited level, and that the population of higher levels is negligible. Assuming thermal equilibrium, find the temperature.

    3. The attempt at a solution

    Using the fact that P(s) = 0.1 = ..., I have found that T = 5900 K.

    Now I want to verify F = -k*T*log(Z) and P(s) = exp(-F / kt)/Z by plugging in the same values, but it doesn't make sense. This is my F:

    [tex]F = - k \cdot 5900K \cdot \ln \left( Z \right) = - k \cdot 5900K \cdot \ln \left( {9 + 5 \cdot e^{\frac{{ - 0.82}}{{k \cdot 5900K}}} } \right)[/tex]

    I insert this is in P(s) = exp(-F/kT)/Z:

    [tex]P(s) = \frac{1}{Z} \cdot \exp \left( {\frac{{ - F}}{{kT}}} \right) = \frac{1}{{9 + 5 \cdot e^{\frac{{ - 0.82}}{{k \cdot 5900K}}} }} \cdot \exp \left( {\ln \left( {9 + 5 \cdot e^{\frac{{ - 0.82}}{{k \cdot 5900K}}} } \right)} \right)[/tex]. I have NOT multiplied with 5 since we are using F.

    I can see that it gives one, since e takes the logarithm, but still - it doesn't make sense, since it should equal 0.1. Can you see where my error is?
    Last edited: Apr 5, 2008
  2. jcsd
  3. Apr 5, 2008 #2
    Is the problem that [tex]P(s) = \frac{1}{Z} \cdot \exp \left( {\frac{{ - F}}{{kT}}} \right)[/tex] only works when I write [tex]F=E-TS[/tex] and NOT [tex]F=-k\cdot T\cdot \ln(Z)[/tex]?

    What I do know is that there is something fundementally wrong in the way I use this expression. It would be great if you could point out my error.
    Last edited: Apr 5, 2008
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook