During the adiabatic expansion of $2$ moles of a gas, the internal energy of the gas is found to decrease by $2$ joules, the work done during the process on the gas will be equal to ....... $J$

A diatomic gas initially at $18^o C$ is compressed adiabatically to one-eighth of its original volume. The temperature after compression will be

In an adiabatic change, the pressure $P$ and temperature $T$ of a monoatomic gas are related by the relation $P \propto {T^C}$, where $C$ equals

A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, $C_v=2 R$. Here, $R$ is the gas constant. Initially, each side has a volume $V_0$ and temperature $T_0$. The left side has an electric heater, which is turned on at very low power to transfer heat $Q$ to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to $V_0 / 2$. Consequently, the gas temperatures on the left and the right sides become $T_L$ and $T_R$, respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

($1$) The value of $\frac{T_R}{T_0}$ is

$(A)$ $\sqrt{2}$ $(B)$ $\sqrt{3}$ $(C)$ $2$ $(D)$ $3$

($2$) The value of $\frac{Q}{R T_0}$ is

$(A)$ $4(2 \sqrt{2}+1)$ $(B)$ $4(2 \sqrt{2}-1)$ $(C)$ $(5 \sqrt{2}+1)$ $(D)$ $(5 \sqrt{2}-1)$

Give the answer or qution ($1$) and ($2$)

The work of $146\ kJ$ is performed in order to compress one kilo mole of gas adiabatically and in this process the temperature of the gas increases by $7^o  C$. The gas is $(R=8.3\ J\ mol^{-1} K^{-1})$

A gas at initial temperature $T$ undergoes sudden expansion from volume $V$ to $2 \,V$. Then,