3. A cyclic process $1-2-3-4-1$ is carried out with an ideal gas, consisting of two isotherms (2-3 and 4-1) and two isochors ($1-2$ and $3-4$; see figure). It is known that the ratio of temperatures on the isotherms 2-3 and 4-1 is $T_{2-3} / T_{4-1}=3$, and that during the isothermal expansion, the gas received twice as much heat as during the isochoric heating. Find the efficiency of the cycle.

![](https://cdn.mathpix.com/cropped/2024_05_06_2fcbf33eea2a4c6aa801g-1.jpg?height=309&width=448&top_left_y=979&top_left_x=1478)
3. Let the gas receive an amount of heat $Q$ during the isothermal expansion segment. Then, according to the condition, the gas receives an amount of heat $Q / 2$ during the isochoric heating segment. Therefore, the total amount of heat received by the gas from the heater during the cycle is

$$
Q_{n}=\frac{3}{2} Q
$$

Now, let's find the work done by the gas. Work is done only in the processes 2-3 and 4-1, and the magnitude of the work done by the gas in these processes differs by a factor of three (since at the same volume, the pressure in process 2-3 is three times greater than the pressure in process 4-1, and work is the area under the process curve in the $p-V$ coordinates). Therefore, the work done by the gas during the cycle is

$$
A=Q-\frac{1}{3} Q=\frac{2}{3} Q
$$

From this, we can find the efficiency of the cycle

$$
\eta=\frac{A}{Q_{u}}=\frac{4}{9}
$$