Van der Waals equation

The van der Waals equation is a cubic equation of state, proposed by Dutch physicist Johannes van der Waals in 1873, quantitatively describing the relationship between pressure, volume and temperature of a substance.

The van der Waals equation was a historic landmark since the derivation of the Ideal Gas Law, because the model included quantitative terms for the volumes of the molecules, and the attractive/repulsive forces between them. Also, the model qualitatively explains the coexistence of liquid and vapor phases, and the occurence of the critical state.

Van der Waals received the Nobel Prize in 1910 for his research on the gaseous and liquid states of matter. The van der Waals equation is the basis for many equations of state that are currently in use.

The van der Waals equation is given as:

P = nRT / (V-nb) - an²/V²

where P= Pressure, V= Volume, T= Absolute Temperature, n= number of moles, a= attraction parameter, b= repulsion parameter (= effective molecular volume).

For a=b=0, the van der Waals equation reduces to the Ideal Gas Law.

Also note that the van de Waals equation can be written in a virial form:

Z = 1 +(b - a/RT)(1/V) + (b/V)² + (b/V)³ + ...

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:

1. Allowing molecules to have a finite size
2. 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*N_aV where N_a 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.