$A$ light container having a diatomic gas enclosed within is moving with velocity $v$. The mass of the gas is $M$ and the number of moles is $n$. The kinetic energy of the gas with respect to the ground is:

$A$ jar contains a gas and a few drops of water at $T \, K.$ The pressure in the jar is $830 \, mm$ of mercury. The temperature of the jar is reduced by $1\%.$ The saturated vapour pressure of water at the two temperatures are $30 \, mm$ and $25 \, mm$ of mercury. Then the new pressure in the jar will be ..... $mm$ of $Hg$.

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

During an experiment,an ideal gas is found to obey a condition $\frac{P^2}{\rho} = \text{constant}$ [$\rho = \text{density of the gas}$]. The gas is initially at temperature $T$,pressure $P$,and density $\rho$. The gas expands such that density changes to $\rho/2$.

Speed of sound in a gas is $v$ and $r.m.s.$ velocity of the gas molecules is $c.$ The ratio of $v$ to $c$ is

$(a)$ When a molecule (or an elastic ball) hits a massive wall,it rebounds with the same speed. When a ball hits a massive bat held firmly,the same thing happens. However,when the bat is moving towards the ball,the ball rebounds with a different speed. Does the ball move faster or slower?
$(b)$ When gas in a cylinder is compressed by pushing in a piston,its temperature rises. Guess at an explanation of this in terms of kinetic theory using $(a)$ above.
$(c)$ What happens when a compressed gas pushes a piston out and expands? What would you observe?
$(d)$ Sachin Tendulkar used a heavy cricket bat while playing. Did it help him in any way?