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

[jrevill]

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!

 

[HallsofIvy]

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 xe^x but that is no more "proper" than a numerical solution.

 

[jrevill]

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