Find Temperature of Mixture of Two Ideal Gases

To find the temperature $T$ of the mixture of two ideal gases, we use the ideal gas equation $PV = nRT$.

For the first gas: $P_1 V_1 = n_1 R T_1 \implies n_1 = \frac{P_1 V_1}{R T_1}$

For the second gas: $P_2 V_2 = n_2 R T_2 \implies n_2 = \frac{P_2 V_2}{R T_2}$

When the two gases are mixed, the total number of moles $n$ in the mixture is the sum of the individual moles: $n = n_1 + n_2$

For the resulting mixture with pressure $P$, volume $V$, and temperature $T$: $P V = n R T$

Substituting $n = n_1 + n_2$ into the equation for the mixture: $P V = (n_1 + n_2) R T$

Now, substitute the expressions for $n_1$ and $n_2$: $P V = \left( \frac{P_1 V_1}{R T_1} + \frac{P_2 V_2}{R T_2} \right) R T$

Canceling the universal gas constant $R$: $P V = \left( \frac{P_1 V_1}{T_1} + \frac{P_2 V_2}{T_2} \right) T$

Taking a common denominator for the terms inside the parentheses: $P V = \left( \frac{T_2 P_1 V_1 + T_1 P_2 V_2}{T_1 T_2} \right) T$

Solving for the temperature $T$ of the mixture: $T = \frac{T_1 T_2 P V}{T_2 P_1 V_1 + T_1 P_2 V_2}$

Thus, the correct option is B.