One gas of $n_1$ mole of molecules at temperature $T_1$,volume $V_1$,and pressure $P_1$,and another gas of $n_2$ mole of molecules at temperature $T_2$,volume $V_2$,and pressure $P_2$,are mixed resulting in pressure $P$ and volume $V$ of the mixture. The temperature of the mixture is . . . . . . .

$1 \, \text{mole}$ of a gas having $\gamma = \frac{7}{5}$ is mixed with $1 \, \text{mole}$ of a gas having $\gamma = \frac{4}{3}$. What will be the $\gamma$ for the mixture?

Considering the gases to be ideal,the value of $\gamma = \frac{C_P}{C_V}$ for a gaseous mixture consisting of $3$ moles of carbon dioxide and $2$ moles of oxygen will be $(\gamma_{O_2} = 1.4, \gamma_{CO_2} = 1.3)$.

Five moles of helium are mixed with two moles of hydrogen to form a mixture. Take molar mass of helium $M_1=4\ g$ and that of hydrogen $M_2=2\ g$. If the internal energy of the $He$ sample is $100\ J$ and that of the hydrogen sample is $200\ J$,then the internal energy of the mixture is ..... $J$.

$A$ container is divided into two chambers by a partition. The volume of the first chamber is $4.5 \, L$ and the second chamber is $5.5 \, L$. The first chamber contains $3.0 \, \text{moles}$ of gas at a pressure of $2.0 \, \text{atm}$, and the second chamber contains $4.0 \, \text{moles}$ of gas at a pressure of $3.0 \, \text{atm}$. After the partition is removed and the mixture attains equilibrium, the common equilibrium pressure existing in the mixture is $x \times 10^{-1} \, \text{atm}$. The value of $x$ is......... (in $.5$)

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?