If a gas is compressed isothermally, then the r.m.s. velocity of the molecules

Derive the equation of $v_{rms}$ in terms of the mass of a molecule.

The root mean square speed of the molecules of a gas is

$N$ molecules of gas $A$,each having mass $m$,and $2N$ molecules of gas $B$,each of mass $2m$,are contained in the same vessel at a constant temperature $T$. The mean square velocity of $B$ is $V^2$ and the mean square of the $x$-component of $A$ is $\omega^2$. The value of $\frac{\omega^2}{V^2}$ is

One mole of an ideal gas at $STP$ is heated in an insulated closed container until the average speed of its molecules is doubled. Its pressure would therefore increase by what factor?

Consider an ideal gas in a closed container at $300 \ K$. The container is then heated so that the average velocity of particles of the gas increases by a factor of $4$. What would be the final temperature (in $^{\circ} C$)?