The average translational energy and the rms speed of molecules of a sample of oxygen gas at $300 \ K$ are $6.21 \times 10^{-21} \ J$ and $484 \ m/s$ respectively. The corresponding values at $600 \ K$ are nearly (assuming ideal gas behaviour):

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:

$A$ cylindrical tube of uniform cross-sectional area $A$ is fitted with two airtight frictionless pistons. The pistons are connected to each other by a metallic wire. Initially, the pressure of the gas is $P_0$ and the temperature is $T_0$; the atmospheric pressure is also $P_0$. Now, the temperature of the gas is increased to $2T_0$. The tension in the wire will be:

$A$ very tall vertical cylinder is filled with a gas of molar mass $M$ under isothermal conditions at temperature $T$. The density and pressure of the gas at the base of the container are $\rho_0$ and $p_0$,respectively. Choose the correct statement$(s)$ if gravity is assumed to be constant throughout the container.

Statement-$1$: Internal energy of a gas $U = nC_VT$ is due to the random motion of gas molecules.
Statement-$2$: $A$ container is moving with speed $v$. It is suddenly stopped by a force,and the temperature of the gas increases.

The density of water is $1000 \; kg \; m^{-3}$. The density of water vapour at $100 \; ^{\circ}C$ and $1 \; atm$ pressure is $0.6 \; kg \; m^{-3}$. The volume of a molecule multiplied by the total number gives what is called molecular volume. What is the average distance (in $\mathring{A}$) between atoms (interatomic distance) in water? (Radius of a water molecule $= 2 \; \mathring{A}$)