One mole of an ideal monoatomic gas is taken along the path \(ABCA\)  as shown in the \(PV\) diagram. The maximum temperature attained by the gas along the path \(BC\)  is given by

\(p = {p_0} - \frac{{2{P_0}}}{{{V_0}}}\left( {V - 2{V_0}} \right)\)

using=\(PV=nRT\)

Temperature,\(T = \frac{{{P_0}V - \frac{{2{P_0}{V^2}}}{{{V_0}}} + 4{P_0}V}}{{1 \times R}}\)

                                                     \(\left( {n = 1\,mole\,given} \right)\)

\(T = \frac{{{P_0}}}{R}\left[ {5V - \frac{{2{V^2}}}{{{V_0}}}} \right]\)

\(\frac{{dT}}{{dV}} = 0 \Rightarrow 5 - \frac{{4V}}{{{V_0}}} = 0 \Rightarrow V = \frac{5}{4}{V_0}\)

\(T = \frac{{{P_0}}}{R}\left[ {5 \times \frac{{5{V_0}}}{4} - \frac{2}{{{V_0}}} \times \frac{{25}}{{16}}V_0^2} \right] = \frac{{25}}{8}\frac{{{P_0}{V_0}.}}{R}\)