Question

At low temperatures, the molar heat capacity of solids at constant pressure depends on the absolute temperature $T$ according to the law $C_{T}=C_{T_{0}}\left(\frac{T}{T_{0}}\right)^{3}.$ One mole $\left(\right.40g\left.\right)$ of solid argon at temperature $8K$ is brought into thermal contact with $2kg$ of solid argon at a temperature of $1K$ and they insulated everything together. What will be the temperature (in $K$ ) of the substance when thermal equilibrium is established? Round off to nearest integer.

Answer 1

$40\,gm \rightarrow 1$ mole
$2000\,gm \rightarrow 50$ mole
$1\times C \int\limits _{T}^{8}\left(\frac{T}{T_{0}}\right)^{3}dT=50C\int \limits_{1}^{T}\left(\frac{T}{T_{0}}\right)^{3}dT$
$\frac{8^{4}}{4}-\frac{T^{4}}{4}=\frac{50}{4}\left(T^{4}-1^{4}\right)$
$8^{4}+50=51 T ^{4}$
$\quad T =\left(\frac{8^{4}+50}{51}\right)^{1 / 4} \approx 3 k$