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# Van der Waals equation

The Van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero size and a pairwise attractive inter-particle force (such as the Van der Waals force.) It was derived by Johannes Diderik van der Waals in 1873, based on a modification of the ideal gas law. The equation accurately describes the behavior of real fluids; in particular, it exhibits a first-order phase transition between a liquid phase and a gaseous phase.

The Van der Waals equation is

$\left(P + \frac{a}{v^2}\right)\left(v-b\right) = kT$

where P is the pressure of the fluid, a is a measure of the attraction between the particles, v is the volume of the fluid per particle, b is the total volume enclosed within the particles, k is Boltzmann's constant, and T is the absolute temperature. A careful distinction must be drawn between the properties of the bulk fluid and the properties of the particles. In particular, v refers to the volume of the bulk fluid (i.e. the volume of the container) divided by the number of particles, whereas b is the volume enclosed by a single particle (i.e. the volume bounded by the atomic radius) multiplied by the number of particles.

The derivation of the Van der Waals equation begins with the equation of state of an ideal gas, which is composed of non-interacting point particles:

$P = \frac{kT}{v}$

We now stop treating the fluid's constituent particles as point particles, instead modelling them as hard spheres with a small radius (the Van der Waals radius.) Denoting the volume of each sphere by b, we modify the equation of state to

$P = \frac{kT}{v - b}$

The volume per particle, v, has been replaced by the "excluded volume" v - b, reflecting the fact that the particles cannot overlap. If the fluid is compressed, its pressure goes to infinity as the total volume approaches the volume enclosed within the particles.

Next, we introduce a pairwise attractive force between atoms. This causes the average free energy per particle to be reduced by an amount proportional to the fluid density. However, the pressure obeys the thermodynamic relation

$P = - \frac{\partial f}{\partial v}$

where f is the free energy per particle. The attraction therefore reduces the pressure by an amount proportional to 1/v². Denoting the constant of proportionality by a, we obtain

$P = \frac{kT}{v - b} - \frac{a}{v^2}$

which is the Van der Waals equation.

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