At the top of a mountain,a thermometer reads $7^{\circ}C$ and a barometer reads $70 \ cm-Hg$. At the base of the mountain,the thermometer reads $27^{\circ}C$ and the barometer reads $76 \ cm-Hg$. The ratio of the density of air at the top of the mountain to the density of air at the base is:

$125 \, ml$ of gas $A$ at $0.60 \, atm$ and $150 \, ml$ of gas $B$ at $0.80 \, atm$ pressure at the same temperature are filled in a vessel of $1 \, L$ volume. What will be the total pressure of the mixture at the same temperature in $atm$?

Consider the following statements for air molecules in an air-tight container:
$(I)$ The average speed of molecules is larger than root mean square speed.
$(II)$ Mean free path of molecules is larger than the mean distance between molecules.
$(III)$ Mean free path of molecules increases with temperature.
$(IV)$ The rms speed of nitrogen is smaller than oxygen molecule.
Which of the above statements are correct?

$A$ 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:

As per the kinetic theory of gases, which of the following statements is/are true?
$(a)$ Temperature of a gas is a measure of the average kinetic energy of a molecule.
$(b)$ Temperature of a gas depends on the nature of the gas.
$(c)$ Heavier molecules have lower average speed.
$(d)$ Lighter molecules have lower average speed.

The temperature $(T)$ of one mole of an ideal gas varies with its volume $(V)$ as $T = -\alpha V^3 + \beta V^2$,where $\alpha$ and $\beta$ are positive constants. The maximum pressure of the gas during this process is ............