What is the Maximum Temperature for a Given Heat Capacity Equation?

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SUMMARY

The maximum temperature for a given heat capacity equation is determined by solving the equation exp(x) = (x + 2)/(x - 2). The value of x, representing theta/T, was calculated to be 2.4 through trial-and-error, which aligns with the results in the textbook. A numerical solution is valid for this equation, as it lacks an elementary solution due to the presence of x both inside and outside the exponential function. The Lambert's W function can be utilized for a more formal approach, although it does not provide a simpler solution than numerical methods.

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Homework Statement



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)

Homework Equations



See above

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!
 
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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.
 
Ah, sorry, I just thought there was a "neater" solution that was eluding me. Thanks for clearing that up.
 

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