For an ideal gas,the instantaneous change in pressure $p$ with volume $v$ is given by the equation $\frac{dp}{dv} = -ap$. If $p = p_{0}$ at $v = 0$ is the given boundary condition,then the maximum temperature one mole of gas can attain is: (Here $R$ is the gas constant)

When the gas is heated,in which direction will the piston move?

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?

The average translational kinetic energy of $N$ molecules in a gas is $E_1$. The kinetic energy of the electron $(e)$ accelerated from rest through potential difference $V$ volt is $E_2$. The temperature at which $E_1=E_2$ is possible is ( $R=$ gas constant,$N=$ number of molecules).

An ideal gas is expanding such that $PT^{3} = \text{constant}$. The coefficient of volume expansion of the gas is: (in $/T$)

The average translational kinetic energy of ${N}_{2}$ gas molecules at $... {}^{\circ} {C}$ becomes equal to the ${K.E.}$ of an electron accelerated from rest through a potential difference of $0.1 \ V$. (Given ${k}_{B} = 1.38 \times 10^{-23} \ J/K$). (Fill the nearest integer).