An ideal gas expands adiabatically $(\gamma = 1.5)$. To reduce the root-mean-square (r.m.s.) velocity of the molecules $3$ times, the gas has to be expanded by a factor of: (in $\times$)

Read the given statements and decide which is/are correct on the basis of the kinetic theory of gases:
$(I)$ Energy of one molecule at absolute temperature $T = 0 \ K$ is zero.
$(II)$ $r.m.s.$ speeds of different gases are the same at the same temperature.
$(III)$ For one gram of all ideal gases,kinetic energy is the same at the same temperature.
$(IV)$ For one mole of all ideal gases,mean kinetic energy is the same at the same temperature.

$A$ flask is filled with $13 \, g$ of an ideal gas at $27^{\circ} C$ and its temperature is raised to $52^{\circ} C$. The mass of the gas that has to be released to maintain the pressure in the flask at the same value is ..... $g$.

One mole of an ideal diatomic gas is taken through the cycle as shown in the figure.
$1 \rightarrow 2$: isochoric process
$2 \rightarrow 3$: straight line on $P-V$ diagram
$3 \rightarrow 1$: isobaric process
The average molecular speed of the gas in the states $1, 2$ and $3$ are in the ratio

If $v_1$ is the speed of sound in a diatomic gas at $273^{\circ}C$ and $v_2$ is the r.m.s. speed of its molecules at $273 \ K$,then $\frac{v_1}{v_2}=$

One mole of an ideal gas undergoes a linear process as shown in the figure below. Its temperature expressed as a function of volume $V$ is