Thermodynamic properties for real gases

The thermodynamic properties like internal energy, enthalpy, entropy for real gases can be calculated from the fundamental relations like
ΔU=Q+W,      H=U+PV,             A= U-TS,                 G=H-TS,      
dW=-PdV,       dQr = TdS
so dU=TdS-PdV like that all these are state functions and independent of path whether reversible or irrversible processes. from the differential equations of U, H, A & G, we get the Maxwell's equations applicable to ideal as well as real gases. by combining the differential equations and Maxwell's equations we can get general differential equations for the calculation of change in internal energy, enthalpy and entropy.
For pure component degree of freedom is 2. So, a property will be a function of 2 variables. for example U (T,V), H (T,P), S(T,V) or S(T,P). U is taken as a function of T & V because heat supplied at constant volume is change in internal energy and heat supplied at constant pressure is change in enthalpy.
dU =  CvdT + [T(dP/dT)v - P]dV

dH =  CpdT + [V - T(dV/dT)p]dP

dS  =  dQ/T = (dU + PdV)/T  = (Cv/T)dT  + [(dP/dT)v]dV
                    =  (dH - VdP)/T  = (Cp/T)dT  -  [(dV/dT)p]dP

Now to remember the expressions of dU & dH
at constant volume dU =  CvdT true for any gas
for any process  dU =  CvdT + [T(dP/dT)v - P]dV, the whole expression should have the units of energy i.e. PdV and the changes in other properties P & T keeping V constant.
Similarly dH =  CpdT + [V - T(dV/dT)p]dP