Consider that an ideal gas (n moles) is expanding in a process given by P=f(V), which passes through a point(V_(0),P_(0)). Show that the gas is absorbing heat at (P_(0),V_(0))if the slope of the curve P=f (V) is larger than the slope of the adiabat passing through (P_(0), V_(0)).

Consider that an ideal gas (n moles) is expanding in a process given b

1.	Consider that an ideal gas (n moles) is expanding in a process given by P=f(V), which passes through a point(V_(0),P_(0)). Show that the gas is absorbing heat at (P_(0),V_(0))if the slope of the curve P=f (V) is larger than the slope of the adiabat passing through (P_(0), V_(0)).
Answer» Solution :Slope of `P=f(V)`, curve at `(V_(0) , P_(0))= f(V_(0))` Slope of ADIABAT at `(V_(0) P_(0))` `=k(-gamma) V_(0)^(-1-gamma)= -gammaP_(0)//V_(0)` Now heat absorbed in the process P = f(V) `dQ = dU +dW = nC_(v)dT +PdV` Since `T=(1//nR) PV=(1//nR) V f(V)` `dT = (1//nR) [f(V)+Vf.(V)]dV` Thus `(dQ)/(dV)underset(V=V_(0))("" ) =(CV)/(R )[f(V_(0))+V_(0)f.(V_(0))]+f(V_(0))` `=[(1)/(gamma-1)+1]f(V_(0)) +(V_(0)f.(V_(0)))/(gamma-1)` `=(gamma)/(gamma-1)P_(0) +(V_(0))/(gamma-1)f.(V_(0))` Heat is absorbed when `dQ//dV GT 0` when gas EXPANDS, that is when `gammaP_(0) +V_(0) f.(V_(0)) gt 0` `f.(V_(0)) gt -gammaP_(0)//V_(0)`