One mole of monoatomic gas and three moles of diatomic gas are put together in a container. The molar specific heat (in $J\,K^{-1}\,mol^{-1}$) at constant volume is $(R = 8.3\,J\,K^{-1}\,mol^{-1})$.

Five moles of helium are mixed with two moles of hydrogen to form a mixture. Take the molar mass of helium $M_1 = 4 \ g/mol$ and that of hydrogen $M_2 = 2 \ g/mol$. The equivalent value of $\gamma$ for the mixture is:

Three closed vessels $A, B$ and $C$ are at the same temperature $T$ and contain gases which obey the Maxwellian distribution of velocities. Vessel $A$ contains only $O_2$,$B$ only $N_2$ and $C$ a mixture of equal quantities of $O_2$ and $N_2$. If the average speed of the $O_2$ molecules in vessel $A$ is $V_1$,and that of the $N_2$ molecules in vessel $B$ is $V_2$,what is the average speed of the $O_2$ molecules in vessel $C$?

Two gases have the same pressure $P$,temperature $T$,and volume $V$. If they are mixed and the final volume and temperature remain $V$ and $T$ respectively,what will be the final pressure?

$2$ moles of $O_2$ and $4$ moles of $Ar$ are filled in a container at temperature $T$. What is the total internal energy (in $, RT$)?

Two thermally insulated vessels $1$ and $2$ are filled with air at temperatures $T_1, T_2$,pressures $P_1, P_2$,and volumes $V_1, V_2$ respectively. If the valve connecting them is opened,what is the temperature of the equilibrium mixture?