The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is P V = nRT, where R is the gas constant. Show that

Question

a) The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV=nRT, where R is the gas constant. Show that (∂P/∂V)(∂V/∂T)(∂T/∂P) = -1

b) Show that for an ideal gas, (T)(∂P/∂T)(∂V/∂T) = mR

Answer 1

and to prove not all differentials can be canceled (some can like chain rule) see it seems dP/dP=-1

have PV=nRT

dP/dV=-nRT/(V²)

dV/dT=nRT/P

dT/dP=V/(nR)

(dP/dV)(dV/dT)(dT/dP)=

[-nRT/(V²)][nRT/P][V/(nR)]=-nRT/(PV)=-nRT/(nRT)=-1

(dP/dT)(dV/dT)=n²R²/(PV)=n²R²/(nRT)=nR/T