As per the kinetic theory of gases, which of the following statements is/are true?
$(a)$ Temperature of a gas is a measure of the average kinetic energy of a molecule.
$(b)$ Temperature of a gas depends on the nature of the gas.
$(c)$ Heavier molecules have lower average speed.
$(d)$ Lighter molecules have lower average speed.

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$. Statement $I$: Change in internal energy of a system containing $n$ mole of ideal gas can be written as $\Delta U = nC_v(T_f - T_i) = \frac{nR}{\gamma - 1}(T_f - T_i)$, where $\gamma = C_p/C_v, T_i = $ initial temperature, $T_f = $ final temperature. Statement $II$: Relation between degree of freedom $f$ and $\gamma(= C_p/C_v)$ is $\gamma = 1 + \frac{2}{f}$. Choose the correct answer from the options given below.

The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting,movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and $1$ moles of an ideal gas,respectively. In the left compartment,the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium,the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions,if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$,then the value of $\alpha$ is:

$A$ thermally insulated vessel with nitrogen gas at $27^{\circ} C$ is moving with a velocity of $100 \ m/s$. If the vessel is stopped suddenly,then the percentage change in the pressure of the gas is nearly (assume entire loss in $KE$ of the gas is given as heat to gas and $R=8.3 \ J \ mol^{-1} \ K^{-1}$)

$A$ horizontal uniform glass tube of $100 \, cm$ length, sealed at both ends, contains a $10 \, cm$ mercury column in the middle. The temperature and pressure of the air on either side of the mercury column are $81^{\circ} C$ and $76 \, cm$ of mercury, respectively. If the air column at one end is kept at $0^{\circ} C$ and the other end at $273^{\circ} C$, the pressure of the air which is at $0^{\circ} C$ is (in $cm$ of $Hg$):

Following statements are given:
$(1)$ The average kinetic energy of a gas molecule decreases when the temperature is reduced.
$(2)$ The average kinetic energy of a gas molecule increases with increase in pressure at constant temperature.
$(3)$ The average kinetic energy of a gas molecule decreases with increase in volume.
$(4)$ Pressure of a gas increases with increase in temperature at constant volume.
$(5)$ The volume of gas decreases with increase in temperature.
Choose the correct answer from the options given below: