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 _____.

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