Question

Sol. [Key Concept: Mixing gives $T_{mix} = \frac{n_{1}C_{v}T_{1} + n_{2}C_{v}T_{2}}{(n_{1} + n_{2})C_{v}} = \frac{2T + 12T}{8} = \frac{7T}{4}$.]

$$T_{mix} = \frac{n_{1}C_{1}{\text{ }T}_{1} + n_{2}C_{2}{\text{ }T}_{2}}{n_{1}C_{1} + n_{2}C_{2}}$$

$$\begin{matrix} & \ = \frac{\left( 2 \times \frac{3}{2}R \times T \right) + \left( 6 \times \frac{3}{2}R \times 2\text{ }T \right)}{\left( 2 \times \frac{3}{2}R \right) + \left( 6 \times \frac{3}{2}R \right)} \\ & T_{mix} = \frac{2\text{ }T + 12\text{ }T}{8} = \frac{7}{4}\text{ }T \end{matrix}$$