Heat Capacity Power Series Approximation

[Fuzzletop]

Homework Statement

"Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through x^{3} in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than (\frac{ε}{kT})^{2} in the final answer. When the smoke clears, you should find C = Nk(1 - \frac{1}{12}(\frac{ε}{kT})^{2})"

Homework Equations

In addition to the above:
The "exact" formula for the heat capacity was found in an earlier part to the question:
C = \frac{Nε^{2}e^{\frac{ε}{kT}}}{kT^{2}(e^{\frac{ε}{kT}} - 1)^{2}}

The Attempt at a Solution

I used the Power Series expansion for small x:

e^{x} ≈ 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6}

where x = \frac{ε}{kT}

I expanded all this out, factoring out x^{2} on bottom to cancel the same on top, removed powers of x greater than 2:

C = N k \frac{x^{2}(1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6})}{(x + \frac{x^{2}}{2} + \frac{x^{3}}{6})^{2}}

C = N k \frac{x^{2}(1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6})}{x^{2} + x^{3} + \frac{7 x^{4}}{12} + \frac{x^{5}}{6} + \frac{x^{6}}{36}}

C = Nk\frac{1 + x + \frac{x^{2}}{2}}{1 + x + \frac{7 x^{2}}{12}}

I'm not sure how to take this any further. I have gotten something that almost resembles what I want:

C (1 + x + \frac{7x^{2}}{12}) = N k (1 + x +\frac{7x^{2}}{12} - \frac{x^{2}}{12})

But clearly, I can't just divide by the part attached to C and call it a day. Any tips to help get to the target answer?

 

[I like Serena]

Welcome to PF, Fuzzletop!

You can use: {1 \over 1 - x}=1+x+x^2+x^3+...

But before you do so, you should expand your denominator with one extra term in the first step.
And then you need to keep all powers up to power 3 (power 5 in the denominator since that reduces to power 3).

 

[Fuzzletop]

Thanks!

I did manage to get something of an answer. I did so while still ignoring the powers of 3 (I tried to use them at first, but I couldn't get it to work, so I switched back to just up to powers of 2, and I tried this below; I'm sure including the powers of 3 would work equally as well):

C = Nk\frac{(1 + x + x^{2} - \frac{5x^{2}}{12} - \frac{x^{2}}{12})}{(1 + x + x^{2} - \frac{5x^{2}}{12})}<br /> <br /> = Nk\frac{(\frac{1}{1 - x} - \frac{5x^{2}}{12} - \frac{x^{2}}{12})}{(\frac{1}{1 - x} - \frac{5x^{2}}{12})}

Which eventually comes very close to the goal equation, but for a term \frac{5x^{2}}{12} which I removed as it's "smaller than (\frac{ε}{kT})^{2}" (a bit less than half), and I got the answer, C = Nk(1 - \frac{1}{12}(\frac{ε}{kT})^{2})

Thanks a lot! I'm not usually one to think of using series expansions unless I've been told to do so explicitly, so that was a great help!

 

[I like Serena]

You're welcome!

Actually, I intended (but with terms up to the 3rd power):
C = Nk\frac{1 + x + \frac{x^{2}}{2}}{1 + x + \frac{7 x^{2}}{12}}<br /> = Nk(1 + x + \frac{x^{2}}{2})(1 - (x + \frac{7 x^{2}}{12}) + (x + \frac{7 x^{2}}{12})^2 - ...)

 

[I like Serena]

Btw, you could also have consulted wolframalpha.com:
http://www.wolframalpha.com/input/?i=x^2e^x%2F%28e^x-1%29^2

which gives you a series expansion that is exactly the one you're looking for!