3. (4 points) In an ideal monatomic gas, a process occurs where the pressure is directly proportional to the volume. Determine the work done by the gas if the change in internal energy $\Delta U$ of the gas in this process is known.

## Possible solution.

According to the first law of thermodynamics, the amount of heat $Q$ supplied to the gas is equal to the sum of the work done by the gas and the change in internal energy:

$$
Q=A+\Delta U
$$

When transitioning from state 1 to state 2 under the condition that the pressure is directly proportional to the volume with a proportionality constant $k$, the work done by the gas is determined as

$$
A=0.5\left(p_{1}+p_{2}\right)\left(V_{2}-V_{1}\right)=0.5 k\left(V_{2}^{2}-V_{1}^{2}\right)=0.5 v R \Delta T .
$$

Here, $v$ is the amount of gas, and we have used the fact that

$$
p V=\alpha V^{2}=v R T
$$

Considering that

$$
\Delta U=1.5 v R \Delta T
$$

We obtain

$$
A=\frac{\Delta U}{3}
$$
Answer: $A=\frac{\Delta U}{3}$.

## Grading Criteria

| Completion | Score |
| :---: | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | 0 |
| The expression for the first law of thermodynamics is written | 1 |
| The expression for the work of the gas is written | 1 |
| The expression for internal energy is written | 1 |
| Calculations are performed, and the final answer is obtained | 1 |
| Total points | 4 |