The pressure of a gas is proportional to

The internal energy of a monoatomic ideal gas molecule is

$A$ box contains $N$ molecules at temperature $T_1$ and pressure $P_1$. If the number of molecules in the box is doubled but their total kinetic energy remains unchanged,what are the new pressure $P_2$ and temperature $T_2$?

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

Two gases $A$ and $B$ are contained in the same vessel which is at temperature $T$. The number of molecules of gas $A$ is $N$ and the mass of each molecule is $m$. The number of molecules of gas $B$ is $2N$ and the mass of each molecule is $2m$. If the mean square velocity of the $x$-component of velocity of gas $B$ is $v^2$ and the mean square velocity of the $x$-component of velocity of molecules of gas $A$ is $u_x^2$,then $\frac{u_x^2}{v^2}$ is:

$A$ balloon has $5.0$ $g$ mole of helium at $7^{\circ} C$. Calculate
$(a)$ the number of atoms of helium in the balloon,
$(b)$ the total internal energy of the system.