The two first partial derivatives of the vdW equation are
The first equation is , while the second is , where , the isothermal compressibility, is a measure of the relative increase of volume from an increase of pressure, at constant temperature, while , the coefficient of thermal expansion, is a measure of the relative increase of volume from an increase of temperature, at constant pressure. Therefore
In the limit   while . Since the vdW equation in this limit becomes , finally . Both of these are the ideal gas values, which is consistent because, as noted earlier, the vdW fluid behaves like an ideal gas in this limit.
The specific heat at constant pressure,  is defined as the partial derivative . However, it is not independent of , they are related by the Mayer equation, . Then the two partials of the vdW equation can be used to express  as
Here in the limit , , which is also the ideal gas result as expected; however the limit  gives the same result, which does not agree with experiments on liquids.
In this liquid limit we also find , namely that the vdW liquid is incompressible. Moreover, since , it is also mechanically incompressible, that is  faster than .
Finally , and  are all infinite on the curve . This curve, called the spinodal curve, is defined by , and is discussed at length in the next section.