Problem 2
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Problem 2
plumleyjplumleyj 1168448799|%e %b %Y, %H:%M %Z|agohover

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

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

(3)
\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*}
last edited on 1170786743|%e %b %Y, %H:%M %Z|agohover by plumleyj + show more
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