7. (15 points) Inside a cylinder, there are two pairs of identical supports. The distance between the lower supports and the bottom $l_{1}=10 \mathrm{~cm}$, between the lower and upper supports $l_{2}=15 \mathrm{~cm}$. On the lower supports lies a piston with the maximum possible mass $M=10$ kg that they can withstand. The base area of the cylinder $S=10 \mathrm{~cm}^{2}$. What is the minimum amount of heat $Q$ that should be transferred to a monoatomic ideal gas under the piston so that the piston can jump out of the cylinder? The amount of gas under the piston $v=1$ mole, its initial pressure is equal to atmospheric pressure $p_{0}=10^{5} \Pi a$. Neglect the thickness of the piston. The acceleration due to gravity $g=10 \mathrm{M} / \mathrm{c}^{2}$.

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Answer: 127.5 D

Solution. During the heat transfer to the gas, three stages can be distinguished.

First, the isochoric increase in pressure to a value sufficient to move the piston from its position: $p_{1}=p_{0}+\frac{M g}{S}$.

The amount of heat received by the gas during this stage:

$$
Q_{1}=\Delta U_{1}=\frac{3}{2} v R \Delta T=\frac{3}{2} V \Delta p=\frac{3}{2} l_{1} S \frac{M g}{S}=\frac{3}{2} M g l_{1}=15 \text { D }
$$

The second stage is the isobaric increase in the volume of the gas, during which the piston moves to the upper stops.

The amount of heat received by the gas during this stage:

$$
Q_{2}=\Delta U_{2}+A_{2}=\frac{5}{2} p_{1} \Delta V=\frac{5}{2}\left(p_{0}+\frac{M g}{S}\right) S l_{2}=75 \text { J }
$$

The third stage is the isochoric increase in pressure to a value sufficient to break the upper stop: $p_{2}=p_{0}+\frac{M g}{S}+\frac{M g}{S}$.

The amount of heat received by the gas during this stage:

$$
Q_{3}=\Delta U_{3}=\frac{3}{2} v R \Delta T=\frac{3}{2} V \Delta p=\frac{3}{2}\left(l_{1}+l_{2}\right) S \frac{M g}{S}=\frac{3}{2} M g\left(l_{1}+l_{2}\right)=37.5 \text { J . }
$$

Final answer: $Q=Q_{1}+Q_{2}+Q_{3}=15+75+37.5=127.5$ D.