$N$ molecules each of mass $m$ of gas $A$ and $2N$ molecules each of mass $2m$ of gas $B$ are contained in the same vessel at temperature $T$. The mean square of the velocity of molecules of gas $B$ is $v^2$ and the mean square of $x$ component of the velocity of molecules of gas $A$ is $w^2$. The ratio $\frac{w^2}{v^2}$ is

For two different gases $X$ and $Y$, having degrees of freedom $f_1$ and $f_2$ and molar heat capacities at constant volume $C_{V1}$ and $C_{V2}$ respectively, the $\ln P$ versus $\ln V$ graph is plotted for an adiabatic process, as shown.

$A$ closed vessel contains a mixture of two diatomic gases $A$ and $B$. Molar mass of $A$ is $16$ times that of $B$ $(M_A = 16 M_B)$ and mass of gas $A$ contained in the vessel is $2$ times that of $B$ $(m_A = 2 m_B)$. Which of the following statements are true?
$(i)$ Average kinetic energy per molecule of $A$ is equal to that of $B$.
$(ii)$ Root mean square value of translational velocity of $B$ is four times that of $A$.
$(iii)$ Pressure exerted by $B$ is eight times that exerted by $A$.
$(iv)$ Number of molecules of $B$ in the cylinder is eight times that of $A$.

Two identical bulbs are filled with gas at $N.T.P.$ One bulb is placed in ice and the other in a hot bath, such that the pressure becomes $1.5$ times the initial pressure. What is the temperature of the hot bath in $^\circ C$?

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. Estimate the ratio (or fraction) of the molecular volume to the total volume occupied by the water vapour under the above conditions of temperature and pressure.

Given below are two statements:
Statement $I$: The temperature of a gas is $-73^{\circ} C$. When the gas is heated to $527^{\circ} C$,the root mean square speed of the molecules is doubled.
Statement $II$: The product of pressure and volume of an ideal gas is equal to the translational kinetic energy of the molecules.
In the light of the above statements,choose the correct answer from the options given below: