Calculating the Limit of an Einstein Solid Specific Heat

[Amok]

So, I can't really find this limit:

\lim_{T \to \infty} \ 3Nk {(\epsilon/kT)}^2 \frac{e^{(\epsilon/kT)}}{{(e^{(\epsilon/kT)}-1)}^2}

This is actually the formula for the specific heat of an Einstein solid, which is pretty easy to derive but I haven't been able to calculate the limit to show it becomes Dulong-Petit at hight temperatures. Maybe I've been missing something simple... I don't see how you could use l'Hopital's rule.

note: I wasn't sure whether I should post this in one of the physics forums, but I figured it's more of a calculus problem at this point.

 

[spamiam]

I think you can use L'Hôpital's rule. First of all, I made the change of variables
<br /> y = \frac{\epsilon}{kT}<br />
so that y \to 0 as T \to \infty. This yields
<br /> \lim_{y \to 0} 3Nk\frac{y^2 e^y}{(e^y - 1)^2}<br />
which is of the form \frac{0}{0}. Using L'Hôpital's rule, this equals
<br /> \lim_{y \to 0} 3Nk \frac{2y e^y + y^2 e^y}{2(e^y -1)e^y} = 3Nk \lim_{y \to 0} \frac{2y + y^2}{2e^y -2}<br />
which is once again of the form \frac{0}{0}. Applying LH again, we get
<br /> 3Nk \lim_{y \to 0} \frac{2 + 2y}{2 e^y} = 3Nk \lim_{y \to 0} \frac{y +1}{e^y} = 3Nk \frac{0 +1}{1} = 3Nk<br />

I hope this helps: hopefully I didn't make any mistakes!

 

[Amok]

I think you can use L'Hôpital's rule. First of all, I made the change of variables
<br /> y = \frac{\epsilon}{kT}<br />
so that y \to 0 as T \to \infty. This yields
<br /> \lim_{y \to 0} 3Nk\frac{y^2 e^y}{(e^y - 1)^2}<br />
which is of the form \frac{0}{0}. Using L'Hôpital's rule, this equals
<br /> \lim_{y \to 0} 3Nk \frac{2y e^y + y^2 e^y}{2(e^y -1)e^y} = 3Nk \lim_{y \to 0} \frac{2y + y^2}{2e^y -2}<br />
which is once again of the form \frac{0}{0}. Applying LH again, we get
<br /> 3Nk \lim_{y \to 0} \frac{2 + 2y}{2 e^y} = 3Nk \lim_{y \to 0} \frac{y +1}{e^y} = 3Nk \frac{0 +1}{1} = 3Nk<br />

I hope this helps: hopefully I didn't make any mistakes!

That makes perfect sense. For some reason I forgot that there is also an inverse exponential in the numerator (I was thinking it was just an exponential). Thanks a bunch dude.