$A$ monoatomic gas $\left( \gamma = \frac{5}{3} \right)$ is suddenly compressed to $\frac{1}{8}$ of its original volume. What will be the final pressure of the gas in terms of its initial pressure?

$A$ 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,$C_v = 2R$. 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 $\frac{Q}{RT_0}$ is
$(A)$ $4(2\sqrt{2}+1)$ $(B)$ $4(2\sqrt{2}-1)$ $(C)$ $(5\sqrt{2}+1)$ $(D)$ $(5\sqrt{2}-1)$

$A$ 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?

During an adiabatic process,the pressure of a gas is found to be proportional to the cube of its temperature. The ratio of $\frac{C_P}{C_V}$ for the gas is

Can a change in the internal energy of an ideal gas be considered an adiabatic process?

Identify the characteristics of an adiabatic process in a monoatomic gas.
$(A)$ Internal energy is constant.
$(B)$ Work done in the process is equal to the change in internal energy (in magnitude).
$(C)$ The product of temperature and volume is a constant.
$(D)$ The product of pressure and volume is a constant.
$(E)$ The work done to change the temperature from $T_1$ to $T_2$ is proportional to $(T_2 - T_1)$.
Choose the correct answer from the options given below.