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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 adiabat passing through (P0, V0 ).

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Slope of P = f (V), curve at (Vo, Po)

= f (Vo)

Slope of adiabat at (Vo, Po)

Now heat absorbed in the process P = f (V)

dQ = dV + dW

= nCvdT + P dV

Since T = (1/nR) PV = (1/nR) V f (V)

dT = (1/nR) [f (V) + V f ' (V)] dV

Heat is absorbed when dQ/dV > 0 when gas expands, that is when

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