Equation of state of gases

The Equation of State of gases' combines the basic thermodynamic quantities describing the behavior of a gas. The equation of state, or rather equations of state, connect the basic state quantities of a thermodynamic system. In the case of gases, such state quantities are usually pressure, volume, temperature and substance quantity, or quantities derived from them.

Ideal Gas Equation of State
An ideal gas is the simplest gas model. The model assumes that gas molecules are essentially point particles without their own volume, and their only force action occurs during elastic collisions. Such a gas, for example, does not exhibit internal friction, so an ideal gas has zero viscosity.

For an ideal gas, a simple equation of state applies that binds pressure (p), volume (V), amount of matter (n) and temperature (T). The gas constant R [J.mol-1.K-1] also appears in the equation.
 * $$pV = nRT$$

Real gases differ from ideal gases in that there are other force interactions between gas molecules than just elastic collisions. These interactions are short-range because their essence is the electric forces between molecules (see nonbonding interactions). For obvious reasons, noble gases will be sufficiently accurately described by the ideal gas equation of state at higher pressures than molecular gases.

Van der Waals equation of state
The equation of state is usually not very suitable for describing the behavior of a real gas, because for most interesting problems the behavior of a real gas is significantly different from the behavior of an ideal gas. The physical essence of such behavior is, in particular, that one cannot neglect the mutual interactions of gas molecules or their finite volume, more precisely, repulsion when too close (see chemical bond). The first successful equation of state of a real gas was constructed at the end of the 19th century by van der Waals, who introduced two correction factors into the equation of state of an ideal gas.

He expressed the equation of state of an ideal gas using the molar volume (volume related to the amount of substance):
 * $$pV_m = RT$$

In the first step, he started from the idea that one mole of gas molecules occupies a finite volume, which he designated as the excluded volume (b). For compressing one mole of gas, only the volume reduced by the excluded volume is available, i.e.:
 * $$V_m - b$$

Mutual attractive forces between molecules cause the molecules to be more attracted to each other. This reduces the speed at which the molecules hit the container wall, the macroscopic effect of which is that less pressure is exerted on the container wall. By analyzing the forces acting between molecules, van der Waals determined that the pressure drop is inversely proportional to the square of the molar volume and depends on the constant a.

The equation of state after including these two corrections has the form
 * $$\left(p + \frac {a}{V_m^2} \right) \cdot \left(V_m - b \right) = RT $$

Unlike the ideal gas equation, the van der Waals equation is also capable of describing gas condensation. Even the van der Waals equation is not a very accurate model, for a number of tasks it is more important as a tool for qualitative description of gas behavior. However, several equations of state of real gas are still used in practice, which are based on the van der Waals equation.