One gas of n_1 mole of molecules at temperature T_1 , volume V_1 ,...

From the ideal gas equation, the number of moles of the first gas is $n_1 = \dfrac{P_1 V_1}{R T_1}$ and the number of moles of the second gas is $n_2 = \dfrac{P_2 V_2}{R T_2}$.

For the mixture, the total number of moles is $n = n_1 + n_2$.

Applying the ideal gas equation for the mixture:

$P V = n R T$

$T = \dfrac{P V}{n R}$

Substituting the values of $n_1$ and $n_2$:

$T = \dfrac{P V}{\left( \dfrac{P_1 V_1}{R T_1} + \dfrac{P_2 V_2}{R T_2} \right) R}$

$T = \dfrac{P V}{\dfrac{P_1 V_1}{T_1} + \dfrac{P_2 V_2}{T_2}}$

$T = \dfrac{T_1 T_2 P V}{P_1 V_1 T_2 + P_2 V_2 T_1}$

Answer: $T_1 T_2 PV/(T_2 P_1 V_1 + T_1 P_2 V_2)$