$A$ thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $1.4$. The vessel is moving with speed $v$ and is suddenly brought to rest. Assuming no heat is lost to the surroundings,the temperature of the gas increases by ... ($R =$ universal gas constant)

An ideal gas at pressure $P_0$ undergoes an isothermal expansion until its volume is $8.0$ times its initial volume. The gas is slowly and adiabatically compressed back to its original volume. If the adiabatic constant of the gas is $\gamma = 4/3$, then the ratio of the average kinetic energy per molecule in this final state to that in the initial state is

An ideal gas is compressed isothermally until its pressure is doubled and then allowed to expand adiabatically to regain its original volume ($\gamma = 1.4$ and $2^{-1.4} = 0.38$). The ratio of the final to initial pressure is (in $: 1$)

One mole of a monoatomic ideal gas $\left(c_{V} = \frac{3}{2} R\right)$ undergoes a cycle where it first goes isochorically from the state $\left(\frac{3}{2} P_{0}, V_{0}\right)$ to $\left(P_{0}, V_{0}\right)$,and then is isobarically contracted to the volume $\frac{1}{2} V_{0}$. It is then taken back to the initial state by a path which is a quarter ellipse on the $P-V$ diagram. The efficiency of this cycle is

The change in internal energy of a given mass of a gas,when its volume changes from $V$ to $3V$ at constant pressure $P$,is (where $\gamma$ is the ratio of the specific heat capacities of the gas).

In a certain thermodynamical process,the pressure of a gas depends on its volume as $P = kV^{3}$. The work done when the temperature changes from $100^{\circ}C$ to $300^{\circ}C$ will be .......... $nR$,where $n$ denotes the number of moles of a gas.