Discussion
The Debye temperature of diamond is 2230 K. The molar heat capacity of diamond at 10 K is :
(R = 8.314 Jmol\(^{-1}\)K\(^{-1}\))
Answer: B
The molar heat capacity is 0.175 Jkmol\(^{-1}\)K\(^{-1}\).
At 10 K, far below the Debye temperature, the Debye \(T^{3}\) law applies:
\[ \begin{aligned} C &= \frac{12\pi^{4}}{5}R\left(\frac{T}{\theta_{D}}\right)^{3} \\ &= 1944 \times \left(\frac{10}{2230}\right)^{3} \\ &= 1944 \times 9.02 \times 10^{-8} = 1.75 \times 10^{-4} \end{aligned} \]
in J mol\(^{-1}\) K\(^{-1}\), which is 0.175 J per kilomole per kelvin, the unit used in the options. The tiny value shows how completely the vibrations of stiff diamond are frozen out at low temperature; the other options do not follow from the \(T^{3}\) law with these numbers.
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