Proceeding under conditions (7.4.16) rather than (7.4.13), show that [ C_{V}(T)=N kleft{Dleft(x_{0, L}ight)+2 Dleft(x_{0, T}ight)ight} ] where

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Proceeding under conditions (7.4.16) rather than (7.4.13), show that

\[
C_{V}(T)=N k\left\{D\left(x_{0, L}ight)+2 D\left(x_{0, T}ight)ight\}
\]

where \(x_{0, L}=\left(\hbar \omega_{D, L} / k Tight)\) and \(x_{0, T}=\left(\hbar \omega_{D, T} / k Tight)\). Compare this result with equation (7.4.17), and estimate the nature and the magnitude of the error involved in the latter.

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