text: The internal energy is given by the energetic equation of state,
  where  is an arbitrary constant of integration.
  Now in order for  to be an exact differential, namely that  be continuous with continuous partial derivatives, its second mixed partial derivatives must also be equal, . Then with  this condition can be written simply as . Differentiating  for the vdW equation gives , so . Consequently  for a vdW fluid exactly as it is for an ideal gas. To keep things simple it is regarded as a constant here,  with  a number. Then both integrals can be easily evaluated and the result is
  This is the energetic equation of state for a perfect vdW fluid. By making a dimensional analysis (what might be called extending the principle of corresponding states to other thermodynamic properties) it can be written simply in reduced form as
  where  and C is a dimensionless constant.