# Derivation of the van der Waals equation

I’m trying to rationalize the van der Waals equation for a non-ideal gas. I understand it conceptually, but I cannot finish the algebra:

Begin with the ideal gas law: P V = n R T

This is really: P(ideal) V(ideal) = n R T (1)

Now consider the particle-particle interactions. This has the effect of decreasing the pressure, relative to the ideal condition. It also decreases the real, or effective volume occupied by the gas.

Express this as: P(real, or observed) = P(ideal) – P(due to particle-particle interactions)

Rearranging: P(ideal) = P(real) + P(interactions) (2)

We also have: V(real) = V(ideal, or measured) - V(due to particle-particle interactions)

Rearrange this to: V(ideal) = V(real) + V(interactions) (3)

Now consider again the volume occupied by the gas, and start with the understanding that the expression “volume of gas” is not literally correct. The “P”, “n”, and “T” are intrinsic properties of the gas particles, but “V” refers to the volume that the gas occupies, not the sum of the volume of the individual gas particles.

In the ideal condition, the measured volume of the container is equal to the volume occupied by the gas. Given that a real gas has finite volume, the volume available for the gas particles to occupy decreases by the volume of the gas particles themselves.

Express this as: V(occupied by the real gas) = V(total, of the container) – V(of the gas particles)

I consider that the measured volume of the container is the theoretical maximum volume that the gas particles may occupy (maybe here is where I’m going off the rails, but what is the correct argument?)

Rearrange this to: V(ideal) = V(occupied by the real gas) + V(of the gas particles) (4)

Substitute equation (2) into equation (1): [P(real) + P(interactions)] V(ideal) = n R T (5)

But now, substitute equations (3) and (4) (they express the same concept; V(real) < V(ideal)) into (5), and I get:

[P(real) + P(interactions)] [V(occupied by the real gas) + V(of the gas particles)] = n R T (6)

The volume expression is incorrect.

I’m thinking that the logic error is somehow related to the fact that “volume” is the volume occupied by the gas, and not an intensive property of the gas particles themselves, but I can’t find how that translates to an error in my argument above.

So, let’s cheat. Logically, at any rate. Let’s assume the van der Waals equation, and see if I can go backwards, without logical error:

The volume term in the van der Waals equation is: (V – nb)

Since it arises by substitution into the ideal gas law, it follows that: V(ideal) = V – nb.

This makes the “V” on the right a real volume. If it was ideal, then either “n” is zero (no gas), or “b” is zero (no volume), or both. Logically impossible.

It must be that: V(ideal) = V(real) – nb

This implies that the volume that may be occupied by a real gas is always greater than the volume that may be occupied by an ideal gas. But the volume occupied by an ideal gas is equal to the volume of the container. Therefore, the volume of a real gas is greater than the volume of the container it occupies. This cannot be.

Where am I going wrong?

I appear to be left with the notion that deriving the van der Waals equation by substitution into the ideal gas law is not correct, but why?

And if it is correct, is there a rationale for the volume expression like that for the pressure expression? In other words, what is the logical argument, that makes sense when implemented algebraically?

• This webpage may be of interest to you. It gives two derivations of the equation. The first is most similar to the method you are attempting. – jheindel Sep 15 '17 at 3:48