Two containers of equal volume contain the same gas at pressures $P_1$ and $P_2$ and absolute temperatures $T_1$ and $T_2$ respectively. On joining the vessels,the gas reaches a common pressure $P$ and common temperature $T$. The ratio $P/T$ is equal to

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

When the temperature of an ideal gas is increased by $600 \ K$,the velocity of sound in the gas becomes $\sqrt{3}$ times the initial velocity in it. The initial temperature of the gas is .... $^oC$

$A$ 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 $R=8.3 \ J \ mol^{-1} \ K^{-1}$)

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio $3:4$. The ratio of their partial pressures when enclosed in a vessel kept at a constant temperature is $2:3$. The ratio of their densities is

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: