3. (6 points) In a horizontal stationary cylindrical vessel closed by a piston of mass $M$, a gas is located. The gas is heated for a time $\tau$, during which the piston moves from rest with a constant acceleration $a$. Find the average power of the heater. The internal energy of one mole of gas is $U=c T$. Neglect the heat capacity of the vessel and the piston. There is a vacuum on the outside of the piston.

Possible solution. Since the piston moves with constant acceleration, the process is isobaric. According to the first law of thermodynamics,

$$
Q=p \Delta V+v c \Delta T
$$

Here $p$ is the pressure of the gas, $V$ is its volume, and $v$ is the amount of substance of the gas.

According to the Clapeyron-Mendeleev equation,

$$
p \Delta V=v R \Delta T
$$

By the work-energy theorem,

$$
p \Delta V=\frac{M(a \tau)^{2}}{2}
$$

Thus,

$$
v \Delta T=\frac{M(a \tau)^{2}}{2 R}
$$

The amount of heat transferred is:

$$
Q=\frac{M(a \tau)^{2}}{2}+\frac{c M(a \tau)^{2}}{2 R}=\frac{M(a \tau)^{2}}{2}\left(1+\frac{c}{R}\right) .
$$

In the final form, we get

$$
P=\frac{Q}{\tau}=\frac{M a^{2} \tau}{2}\left(1+\frac{c}{R}\right)
$$
Answer: $P=\frac{M a^{2} \tau}{2}\left(1+\frac{c}{R}\right)$.

## Evaluation Criteria

| Participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| :--- | :---: |
| The expression for the first law of thermodynamics is written | $\mathbf{1}$ |
| The Clapeyron-Mendeleev equation is written | $\mathbf{1}$ |
| The theorem on the change in kinetic energy is written | $\mathbf{2}$ |
| Necessary transformations are performed, the final answer is obtained | $\mathbf{2}$ |
| Total points | $\mathbf{6}$ |