4. Condition One mole of an ideal gas undergoes a cyclic process, which in terms of $P, T$ is described by the equation

$$
\left(\frac{P}{P_{0}}-a\right)^{2}+\left(\frac{T}{T_{0}}-b\right)^{2}=c^{2}
$$

where $P_{0}, T_{0}, a, b, c-$ are some constants, $c^{2}<a^{2}+b^{2}$. Determine the maximum volume occupied by the gas.
Answer: $V_{\text {max }}=\frac{R T_{0} a \sqrt{a^{2}+b^{2}-c^{2}}+b c}{b \sqrt{a^{2}+b^{2}-c^{2}}-a c}$

Solution In the axes $P / P_{0}, T / T_{0}$, the process is represented by a circle with center at point $(a, b)$ and radius $c$. For each point, the volume occupied by the gas is $R \frac{T}{P}$, which is proportional to the tangent of the angle of inclination of the line connecting the given point to the origin. It is clear that the extreme values are achieved when this line is tangent. From geometric considerations,

$$
\left(\frac{T / T_{0}}{P / P_{0}}\right)_{\max }=\frac{a \sqrt{a^{2}+b^{2}-c^{2}}+b c}{b \sqrt{a^{2}+b^{2}-c^{2}}-a c}
$$

that is,

$$
V_{\max }=\frac{R T_{0}}{P_{0}} \frac{a \sqrt{a^{2}+b^{2}-c^{2}}+b c}{b \sqrt{a^{2}+b^{2}-c^{2}}-a c}
$$

## Criteria: