4. One mole of a monatomic ideal gas undergoes a cyclic process $a b c a$. The diagram of this process in the $P-T$ axes represents a curvilinear triangle, the side $a b$ of which is parallel to the $T$ axis, the side $b c$ - a segment of a straight line passing through the origin, and the side $c a$ - an arc of a parabola passing through the origin, the axis of which is parallel to the $T$ axis. At points $a$ and $c$, the temperature of the gas is the same and equal to $T_{0}=320 \mathrm{~K}$, and the pressure at point $a$ is half the pressure at point $c$. Determine the work done by the gas over the cycle.
Answer: 664 J. Solution. Process $a b$ is an isobar, process $b c$ is an isochore, process $c a$ is described by the equation $T=P(d-k P)$, where $d, k$ are some constants. It is not difficult to see that in such a process, the volume turns out to be a linear function of pressure, that is, in the $P V$ axes, this cyclic process is represented by a right triangle with legs $a b$ and $b c$.

Let $P_{0}, V_{0}$ be the parameters of the gas at point $b$. Then the parameters of the gas at point $a: P_{0}, 2 V_{0}$, at point $c$: $2 P_{0}, V_{0}$. From this, we find: $A=\frac{1}{2} P_{0} V_{0}=\frac{1}{4} R T_{0}=664$ J.

Answer to variant 162: $\frac{2}{3} R T_{0}=1662$ J.

Answer to variant 163: $\frac{2}{3} R T_{0}=1994$ J.

Answer to variant 164: $\frac{1}{4} R T_{0}=582$ J.