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Equation with exponential

  1. Jan 15, 2007 #1
    1. The problem statement, all variables and given/known data

    This regards a question on heat capacity; I'm trying to find the temperature at which the heat capacity is maximum. After differentiating the heat capacity expression, equating to zero, and rearranging (all of which I have omitted), my problem boils down to:

    exp(x) = (x + 2)/(x - 2)

    Solve for x. (x in this context is theta/T, but I don't think it's relevant since I'm trying to find their ratio)

    2. Relevant equations

    See above

    3. The attempt at a solution

    I calculated x = 2.4 by trial-and-error, and the maximum heat hapacity for this value is in agreement with the my textbook. I just don't know how to solve it properly!
     
  2. jcsd
  3. Jan 15, 2007 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    What do you mean by "solve it properly"? A numerical solution is perfectly valid. In this equation, with x both in and outside of the exponential, there is no "elementary" solution. You could try Lambert's W function which is defined as the inverse of the function xex but that is no more "proper" than a numerical solution.
     
  4. Jan 15, 2007 #3
    Ah, sorry, I just thought there was a "neater" solution that was eluding me. Thanks for clearing that up.
     
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