4.2.1 An ideal gas is used as the working substance of a cyclic heat engine. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and return to the initial state, where the process in the $P / P_{0}, \rho / \rho_{0}$ axes is represented by a quarter circle centered at the point $(1,1)$. Determine the efficiency of this cycle, knowing that it is 8 times less than the maximum possible efficiency for the same minimum and maximum gas temperatures as in this cycle.
Solution. Let's redraw the cycle in the $p v$ diagram. There, the line first goes down, then to the left, and then closes along some curve. Since the pressure at all points does not exceed $n P_{0}$, and the volume $V_{0}=m / \rho_{0}$, where $m$ is the mass of the gas, the maximum temperature is $T_{1}=n P_{0} V_{0} / R$. Similarly, the minimum temperature of the gas is $T_{3}=P_{0} V_{0} /(n R)$. The highest efficiency for such a range of temperatures is $1-T_{2} / T_{1}$, therefore, the efficiency of the considered cycle is $(1 / \alpha) \cdot\left(1-1 / n^{2}\right)$.

Answer. $1 / 9$