Consider that an ideal gas (n Moles) is expanding in a process given by P=f (V), which passes through a ponts (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 that 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 ponts (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 that 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 hrat absorbed in the PROCESS P=f(V) `dQ=dU+dW=nC_(v)dT+PdV` SINCE `T=(1//nR)PV=(1//nR)Vf(V)` `dT=(1//nR)[f(V)+Vf^(.)(V)]dV` Thus `(dQ)/(dV_(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//dVgt0` when gas EXPANDS, that is when `gammaP_(0)+V_(0)f^(.)(V_(0))gt0 "" ,f^(.)(V_(0))gt-gammaP_(0)//V_(0) `