Thermal Physics: Evaluating Limits at Zero Temperature

[Niles]

Homework Statement

Hi all.

Please take a look at this expression, where T is our variable (it represents temperature):

$$C_v = 2\left( {\frac{{\hbar \omega }}{T}} \right)^2 \frac{{\exp \left( {\frac{{\hbar \omega }}{T}} \right)}}{{\left( {\exp \left( {\frac{{\hbar \omega }}{T}} \right) - 1} \right)^2 }}.$$

I have to evaluate this for $T \rightarrow 0$. I would use L'Hopital, but isn't there an easier way? Because when I differentiate the nominator (the top), then I will end up with an expression like the original nominator, which won't help me.

Thanks in advance.


Niles.

 

[MathematicalPhysicist]

Thermal physics correct?

I had some sort of this question, it goes like this:
change variables, to dimensionless i.e x=hbar*w/T
so you now evaluate:
$$lim_{x\leftarrow \infty} 2x^2(\frac{e^x}{(e^x-1)^2})$$
Other than L'hopital twice there isn't any other approach.

 

[Niles]

Yeah, thermal physics

Thanks!