A gas consisting of monatomic molecules (degrees of freedom $=\mathrm{}3$) was expanded in a polytropic process so that the rate of collisions of the molecules against the vessel's wall did not change. Find the molar heat capacity of the gas in the process, in $\mathrm{J} \mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. (Given: Gas constant $R=8.3 \mathrm{J} {\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$)

Two moles of helium gas is taken over the cycle $ABCDA$, as shown in the $P-T$ diagram.

The work done on the gas in taking it from $D$ to $A$ is

Two moles of helium gas is taken over the cycle $ABCDA$, as shown in the $P-T$ diagram.

The net work done on the gas in the cycle $ABCDA$ is

A diatomic ideal gas is used in a Carnot engine as the working substance. If during the adiabatic expansion part of the cycle, the volume of the gas increases from $V$ to $32V$, the efficiency of the engine is

A thermally insulated vessel contains an ideal gas of molecular mass $M$ and a ratio of specific heats $\gamma$. It is moving with speed $v$ and is suddenly brought to rest. Assuming no heat is lost to the surroundings, its temperature increases by,