$A$ box contains $N$ molecules of a perfect gas at temperature ${T_1}$ and pressure ${P_1}$. The number of molecules in the box is doubled while keeping the total kinetic energy of the gas the same as before. If the new pressure is ${P_2}$ and temperature is ${T_2}$,then:

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$. Assertion $A$: If the average kinetic energy of $H_2$ and $O_2$ molecules, kept in two different sized containers are same, then their temperatures will be same. Reason $R$: The r.m.s. speed of $H_2$ and $O_2$ molecules are same at same temperature. Choose the correct answer from the options given below.

If $\Delta Q$ is the amount of heat supplied to $n$ moles of a rigid diatomic gas at constant pressure, $\Delta U$ is the change in internal energy, and $\Delta W$ is the work done, then $\Delta W : \Delta U : \Delta Q$ is

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.

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:

An ideal gas has $N$ molecules in a closed box at temperature $T_1$ and pressure $P_1$. If the number of molecules in the box is doubled and the total kinetic energy is kept the same,what will be the new pressure $P_2$ and temperature $T_2$?