4. A vessel is divided by a partition into 2 unequal parts. In one part of the vessel, there is $1 / 3$ mole of a monatomic ideal gas with a temperature of $T_{1}=300 \mathrm{~K}$. In the other part, there is $1 / 5$ mole of another monatomic ideal gas with a temperature of $T_{2}=500$ K. The walls of the vessel and the partition do not conduct heat. The pressures of the gases in the parts of the vessel are the same. The partition is removed, and the gases in the vessel mix.
1) Find the temperature in the vessel after thermal equilibrium is established.
2) Find the ratio of the final pressure in the vessel to the initial pressure in the parts of the vessel.
4. Since p-const, then $\frac{v_{1} T_{1}}{V_{1}}=\frac{v_{2} T_{2}}{V_{2}} ; \quad \frac{V_{2}}{V_{1}}=\frac{v_{2} T_{2}}{v_{1} T_{1}}$

$3 C E: v_{1} C_{\mathrm{V}} T_{1}+v_{2} C_{V} T_{2}=\left(v_{1}+v_{2}\right) C_{v} T \Rightarrow$ $\Rightarrow T=\frac{v_{1} T_{1}+v_{2} T_{2}}{v_{1}+v_{2}}=\frac{\frac{1}{3} \cdot 300+\frac{1}{5} \cdot 500}{\frac{1}{3}+\frac{1}{5}}=375 K$

![](https://cdn.mathpix.com/cropped/2024_05_06_556368b0f812d0b7a511g-3.jpg?height=245&width=415&top_left_y=2379&top_left_x=1363)

$$
\begin{aligned}
& P^{\prime}=\frac{\left(v_{1}+v_{2}\right) R T}{V_{1}+V_{2}} ; P=\frac{v_{1} R T_{1}}{V_{1}} \\
& \frac{P^{\prime}}{P}=\frac{\left(v_{1}+v_{2}\right)}{v_{1}} \cdot \frac{V_{1}}{V_{1}+V_{2}} \cdot \frac{T}{T_{1}}=\frac{v_{1}+v_{2}}{v_{1}} \cdot \frac{1}{1+V_{2} / V_{1}} \cdot \frac{T}{T_{1}}=\frac{\left(v_{1}+v_{2}\right)}{v_{1}} \cdot \frac{1}{1+\frac{v_{2} T_{2}}{v_{1} T_{1}}} \cdot \frac{v_{1} T_{1}+v_{2} T_{2}}{\left(v_{1}+v_{2}\right) T_{1}}=1
\end{aligned}
$$