One kg of a diatomic gas is at a pressure of $8 \times 10^4 \; N/m^2$. The density of the gas is $4 \; kg/m^3$. What is the energy (in $\times 10^4 \; J$) of the gas due to its thermal motion?

Assertion $(A):$ The total translational kinetic energy of all the molecules of a given mass of an ideal gas is $1.5$ times the product of its pressure and volume.
Reason $(R):$ The molecules of gas collide with each other and the velocities of the molecules change due to the collision.

$2 \text{ moles}$ of a monatomic gas requires heat energy $Q$ to be heated from $30^{\circ} C$ to $40^{\circ} C$ at constant volume. The heat energy required to raise the temperature of $4 \text{ moles}$ of a diatomic gas from $28^{\circ} C$ to $33^{\circ} C$ at constant volume is

As shown schematically in the figure,two vessels contain water solutions (at temperature $T$) of potassium permanganate $(KMnO_4)$ of different concentrations $n_1$ and $n_2$ $(n_1 > n_2)$ molecules per unit volume with $\Delta n = (n_1 - n_2) \ll n_1$. When they are connected by a tube of small length $\ell$ and cross-sectional area $S$,$KMnO_4$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule,where $\beta$ is a constant. Neglecting all terms of the order $(\Delta n)^2$,which of the following is/are correct? ($k_B$ is the Boltzmann constant)
$(A)$ the force causing the molecules to move across the tube is $\Delta n k_B T S$
$(B)$ force balance implies $n_1 \beta v \ell = \Delta n k_B T$
$(C)$ total number of molecules going across the tube per sec is $\left(\frac{\Delta n}{\ell}\right)\left(\frac{k_B T}{\beta}\right) S$
$(D)$ rate of molecules getting transferred through the tube does not change with time

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):

According to the kinetic theory of gases,which of the following statements are correct?
$A$. The motion of the gas molecules freezes at $0^{\circ} C$.
$B$. The mean free path of gas molecules decreases if the density of molecules is increased.
$C$. The mean free path of gas molecules increases if temperature is increased keeping pressure constant.
$D$. Average kinetic energy per molecule per degree of freedom is $\frac{3}{2} k_{B} T$ (for monoatomic gases).
Choose the most appropriate answer from the options given below: