Consider that an ideal gas (n moles) 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 adiabatic passing through (P0, V0).
If an adiabatic process is passing through the point P0, V0 we can write its equation as
As 
Slope at P0, V0= 
Now, coming to the process given,
P = f (V)
We can write the equation,
Since, 
Now, as we know PV=nRT by rearranging it we get,
T=
and P = f (V)
T=
Now,
dQ=ncv
we know,
=
and 
And P=f(V)
Now, the volume is increasing.
dV=+ve
dQ should also be +ve if heat is given to the system.
at point (P0, V0) for heat to be absorbed.
Since, 
Hence proved.
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