In the derivation of the ideal gas equation and kinetic gas theory, it was implicitly assumed that the gas particles are infinitely small and that interactions between them can be neglected.
For real gases, however, this is no longer the case.
Instead, the finite volume of the gas particles must be subtracted from the total volume of the gas, since this volume is no longer available to the particles.
Thus, in the ideal gas equation, the volume \(V\) must be replaced for \(n\) mol of particles by the difference \(V - nb\), where \(b\) is a substance-specific constant of the gas determined experimentally.
To account for the attractive forces between the gas particles, we assume that each particle in the given gas volume exerts cohesive forces on the others.
The force acting on all gas particles must therefore be proportional to the square of their density, and thus inversely proportional to the gas volume itself.
As a result, the gas pressure \(p\) is reduced by the value \(na/V^2\), which is called the cohesion pressure or internal pressure.
The parameter \(a\) is also substance-specific and must be determined experimentally.
Altogether, this leads to the so-called Van der Waals equation for the state of real gases:
$$
\left(p + \frac{na}{V^2}\right)(V - nb) = nRT
$$
If the equation is rearranged for \(p\), the pressure can be expressed as a function of the volume for a fixed temperature.
For carbon dioxide (CO\(_2\)) at a temperature of 273.15 K (isotherm), this curve can be plotted.
The constants taken from the literature are typically converted so that the volume can be expressed in cm\(^3\) and the pressure in bar.
For large volumes and low pressures, the curve is nearly identical to that of the ideal gas equation.
Below a certain volume, however, an inflection point is reached, followed by a pressure drop, which then turns into a steeper rise at even smaller volumes.
These inflection points cannot be observed in nature, since the entire system is in an unstable state there.
Instead, a line, the so-called Maxwell construction, must be drawn through the graph such that the two areas between the line and the graph are equal.
When approaching from large volumes and reaching the right intersection, the gas begins to condense into a liquid.
The pressure remains nearly constant during this process until the left intersection is reached.
At this point, the entire gas volume has ideally transitioned into the liquid phase.
The steep increase in pressure at even smaller volumes corresponds to the observed incompressibility of liquids.