3. $v_{1} C_{V}\left(t_{1}-t\right)=v_{2} C_{V}\left(t-t_{2}\right) ; \quad v_{1} t_{1}-v_{1} t=v_{2} t-v_{2} t_{2}$

$t=\frac{v_{1} t_{1}+v_{2} t_{2}}{v_{1}+v_{2}}=\frac{0.2 \cdot 127+0.8 \cdot 7}{0.2+0.8}=31^{\circ} \mathrm{C} \quad(304 \mathrm{~K})$

$\frac{P_{1}^{\prime}}{P_{1}}=\frac{T}{T_{1}}=\frac{v_{1} T_{1}+v_{2} T_{2}}{\left(v_{1}+v_{2}\right) T_{1}}=\frac{0.2 \cdot 400+0.8 \cdot 280}{400}=0.76$

![](https://cdn.mathpix.com/cropped/2024_05_06_4d7aec0c8bbef904a8ecg-3.jpg?height=200&width=349&top_left_y=1670&top_left_x=1479)
3. A thermally insulated vessel is divided by a heat-conducting partition into two parts of different volumes. In the first part, there is helium at a temperature of $127{ }^{\circ} \mathrm{C}$ in an amount of $v_{1}=0.2$ moles. In the second part, there is helium at a temperature of $7{ }^{\circ} \mathrm{C}$ in an amount of $v_{2}=0.8$ moles.
1) What temperature (in degrees Celsius) will be established in the vessel after the thermodynamic equilibrium is reached?
2) Find the ratio of the final to the initial pressure in the first part of the vessel.