KTGMedium

Question

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 $\_\_\_\_$ .

Options

A.$\left( T_{1} + T_{2} \right)/2$
B.$T_{1}{\text{ }T}_{2}PV/\left( T_{2}P_{1}{\text{ }V}_{1} + T_{1}P_{2}{\text{ }V}_{2} \right)$
C.$\left( T_{2}P_{1}V_{1} + T_{1}P_{2}V_{2} \right)\left( T_{1}T_{2}PV \right)$
D.$\left| T_{1} - T_{2} \right|/2$

Solution

Sol. [Key Concept: With $n = n_{1} + n_{2}$ and $PV = nRT$, $T = \frac{PV}{P_{1}V_{1}/T_{1} + P_{2}V_{2}/T_{2}}$.]

$$n = n_{1} + n_{2}$$

$$\begin{matrix} & \ \therefore PV = nRT\ \Rightarrow n = \frac{PV}{RT} \\ & \frac{P(\text{ }V)}{T_{\text{Final}\text{~}}} = \frac{P_{1}{\text{ }V}_{1}}{{\text{ }T}_{1}} + \frac{P_{2}{\text{ }V}_{2}}{{\text{ }T}_{2}} \\ & \frac{PV}{R_{f}} = \frac{P_{1}{\text{ }V}_{1}}{{RT}_{1}} + \frac{P_{2}{\text{ }V}_{2}}{{RT}_{2}} \\ & {\text{ }T}_{f} = \frac{PV}{P_{1}{\text{ }V}_{1}/T_{1} + P_{2}{\text{ }V}_{2}/T_{2}} \end{matrix}$$

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