- #1

- 219

- 0

I have trouble with this task

## Homework Statement

the specific heat cv is given by

[tex] c_v =\frac {N_A k_b \hbar^2}{{\Omega_D}^3 {k_b}^2 T^2} \int \limits_{0}^{\Omega_D} \! \frac {\Omega^4 exp\frac{\hbar \Omega}{k_b T}}{{(exp\frac{\hbar \Omega}{k_b T}-1})^2} \, d\Omega [/tex]

I shall show that the approximation for high temperatures leads to:

[tex] c_v=3 N_A k_b [1-0.05 (\frac{\Theta_D}{T})^2][/tex]

## Homework Equations

[tex] k_b \Theta_D= \hbar \Omega_D[/tex]

## The Attempt at a Solution

I tried to approximate my exponential function with the taylor seris, so I get for the exponential function something like:

[tex] exp(x) \approx 1+x[/tex] whereas x is equal to (h*omega)/(kbT)

I will only have x^2 in my denominator and x^2 in my numerator (because I neglect the h*omega/(kbT) in the numerator because T is very high).

But this only leads me to the Dulong Petit law. I have no idea where the minus aswell as the 0.05 should come from.

I also tried to approach the whole fraction via taylor but that didn't work out. Furthermore I tried to approximate via taylor until x^2, didn't work out either.

Can anyone help me out with this?

Thanks for your help in advance