The Van der Waals equation is often written as
(P + a/V2)(V-b) = RT
Here V refers to the specific volume not to the total volume.
If you wish to write the equation in terms of extensive quantities, and get n into the picture the equation then it's
(P + a n2/V2)(V-nb) = nRT
The Van der Waals equation is used very commonly not because it provides a very good description but because it's easy to use and provides an improvement over the ideal gas equation at least.
It's easy to derive the VDW equation of state. The equation is derived by relaxing two of the constraints imposed for an ideal gas:
- Allowing molecules to have a finite size
- Allowing for intermolecular forces
Let's correct for finite size first. Each molecule takes up some space. Thus each molecule is surrounded by an exclusion zone which is a sphere of radius d (d is the diameter of the molecule). If the volume of one such sphere is V the excluded volume due to n moles of the gas is n*NaV where Na is Avogadro's number. So we get:
P(V-nb) = nRT
where b is a constant depending on the gas. Now each molecule near the wall experiences an attractive force due to other molecules. Thus whenever a molecule collides with a wall, the force exerted on the wall is weakened due to this attractive force. The frequency of collisions is proportional to n, and the force exerted on the walls is also proportional to n. Thus this means that the pressure at the walls is reduced by a factor of n2/V2. So finally we get:
(P + a n2/V2)(V-nb) = nRT
Please msg me if some of the arguments in this writeup are unclear.