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 $(40g)$ 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

Answer: 3

$40gm\to 1$ mole
$2000gm\to 50$ mole
$1\times C\underset{T}{\overset{8}{\int }}{\left(\frac{T}{T_0}\right)}^{3}dT=50C\underset{1}{\overset{T}{\int }}{\left(\frac{T}{T_0}\right)}^{3}dT$
$\frac{8^4}{4}-\frac{T^4}{4}=\frac{50}{4}\left(T^4-1^4\right)$
$8^4+50=51T^4$
$T={\left(\frac{8^4+50}{51}\right)}^{1/4}\approx 3k$