Problem 2

plumleyj 10 Jan 2007 17:06

In order to obtain the high temperature limiting form of the heat capacity according to the Einstein model, a A Taylor series expansion of the denominator, truncated at the first order, must be performed;

(1)\begin{align} C_v = 3Nk\left(\frac {\Theta_E} {T}\right)^2 \frac {e^{\frac {-\Theta_E} {T}}} {\left(1-{e^{\frac {-\Theta_E} {T}}}\right)} \end{align}

(2)
\begin{align} \begin {flalign*} \lim_{T\to \infty} C_v & = 3Nk\left(\frac {\Theta_E} {T}\right)^2 \frac {e^{\frac {-\Theta_E} {T}}} {\left(1-{1+ \frac {\Theta_E} {T} }\right)^2} \\ & = 3Nk \end {flalign*} \end{align}

The low temperature limiting form of the heat capacity according to the Einstein model is as follows;

(3)\begin{align} \begin {flalign*} \lim_{T\to 0} C_v & = 3Nk\left(\frac {\Theta_E} {T}\right)^2 \frac {e^{\frac {-\Theta_E} {T}}} {1-0} \\ & = 3Nk \left(\frac {\Theta_E} {T}\right)^2 e^{\frac {-\Theta_E} {T}} \end {flalign*} \end{align}