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 ($2$)

A sample of an ideal gas is contained in a cylinder. The volume of the gas is suddenly decreased. A student makes the following statements to explain the change in pressure of the gas
$I.$ The average kinetic energy of the gas atoms increases
$II.$ The atoms of the gas hit the walls of the cylinder more frequently
$III.$ Temperature of the gas remains unchanged
Which of these statements is true?

Figure shows, the adiabatic curve on a $\log T$ and log $V$ scale performed on ideal gas. The gas is …………

The adiabatic Bulk modulus of a perfect gas at pressure is given by