$0.056 \, kg$ of Nitrogen is enclosed in a vessel at a temperature of $127 \, ^{\circ}C$. The amount of heat required to double the speed of its molecules is $k \, cal$. (Take $R = 2 \, cal \, mole^{-1} K^{-1}$)

In the kinetic theory of gases,which of the following statements regarding elastic collisions of the molecules is wrong?

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.

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?

An ideal gas $(\gamma = 1.5)$ is expanded adiabatically. How many times must the gas be expanded to reduce the root mean square velocity of the molecules $2.0$ times?

An ideal gas at pressure $P$ and temperature $T$ is expanding such that $PT^3 = \text{constant}$. The coefficient of volume expansion of the gas is . . . . . . .