If $E$ is the kinetic energy per mole of an ideal gas and $T$ is the absolute temperature,then the universal gas constant is given as

$10^{23}$ molecules of a gas, each having a mass of $3 \times 10^{-27} \,kg$, strike per second per $cm^2$ of a rigid wall at an angle of $60^{\circ}$ with the normal and rebound with a velocity of $500 \,m/s$. The pressure exerted by the gas molecules on the wall is: (in $\,N/m^2$)

At $27^{\circ} C$ temperature,the mean kinetic energy of the atoms of an ideal gas is $E_1$. If the temperature is increased to $327^{\circ} C$,then the mean kinetic energy of the atoms will be

The temperature at which the kinetic energy of oxygen molecules becomes double its value at $27^{\circ}\,C$ is $............^{\circ}\,C$.

$Assertion$: The total translational kinetic energy of all the molecules of a given mass of an ideal gas is $1.5$ times the product of its pressure and its volume.
$Reason$: The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.

If the momentum transferred by gas molecules to the walls of a container is $p_1 = nmAv_x^2 \Delta t$,then derive the equation for the pressure of the gas.