A thermally insulated vessel is divided into two parts by a heat-insulating piston which can move in the vessel without friction. The left part of the vessel contain one mole of an ideal monoatomic gas, & the right part is vacuum. The piston is connected to the right wall of the vessel through a spring whose length in free state is equal to the length of the vessel as shown in the figure. Determine the heat capacity C of the system, neglecting the heat capacities of the vessel, piston and spring.

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Solution

$C = \frac{2 d W}{Δ T} = \frac{2 n R Δ T}{Δ T} [∵ C = \frac{d Q}{Δ T}, a n d d Q = d U + d W = d W + d W = 2 d W]$
Now, in this case $n = 1$ mole $∴ C = 2 R$

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