Question

Write van der Waals’ equation of state for one mole of a gas. What do the

symbols a and b signify? Under what conditions does this equation reduce to

ideal gas equation?

Answer 1

The Van der Waals equation for one mole of gas is:

$(p+\frac{a}{V^2})(V-b)=RT$

The motivation of introducing this equation as a correction to the ideal gas state equation $pV=RT$ gives us directly the signification of the constants $a,b$ :

• The constant $b$ is introduced to take into the account the fact that the molecules are not the material points, i.e. they have a non-zero volume, thus the constant $b$ provides a correction to the volume due to the fact, that not all the space is "permitted", as molecules themselves occupy a part of it.
• The constant $a$ is introduced to take into the account the fact that the molecules interact with electromagnetic forces (as the ideal gas equation treats molecules as interacting only during the collisions). Smaller is volume, closer are molecules to each other, and thus their interaction becomes more important, so that's why the term added to the pressure is of a form $a/V^2$.

Comparing this equation to the ideal gas state equation we see, that the Van der Waals equation reduces to the ideal gas state equation when the terms $a/V^2, b$ are negligible compared to respectively $p, V$, so the conditions of reduction are: $V\gg b, p\gg a/V^2$.