Consider that an ideal gas is expanding
Question:

Consider that an ideal gas is expanding in a process given by P = f(V), which passes through a point (V0, P0). Show that the gas is absorbing heat at (P0, V0) if the slope of the curve P = f(V) is larger than the slope of the adiabat passing through (P0, V0).

Solution:

Slope of the graph,

$\left(V_{0}, P_{0}\right)=\left(\frac{\mathrm{d} P}{\mathrm{~d} V}\right)_{V_{0}, P_{0}}$

Using the above relation, we can find that

$V_{0} f^{\prime}\left(V_{0}\right)>-\gamma P_{0}$

$f^{\prime}\left(V_{0}\right)>\frac{-\gamma P_{0}}{V_{0}}$