Question

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 ).

Answer 1

Slope of P = f (V), curve at (V_o, P_o)

= f (V_o)

Slope of adiabat at (V_o, P_o)

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

dQ = dV + dW

= nC_vdT + 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