The classical virial expansion expresses the pressure P of a many-particle system in equilibrium as a power series in the density: where is called the compressibility factor. This is the virial equation of state, the most general function relating pressure, , density, , and temperature, , of fluids. It was first proposed by Kamerlingh Onnes. The compressibility factor is a dimensionless quantity, indicating how much a real fluid deviates from an ideal gas. A is the first virial coefficient, which has a constant value of 1. It makes the statement that at low density, all fluids behave like ideal gases. The virial coefficients , , , etc., are temperature-dependent, and are generally presented as Taylor series in terms of . Second and third virial coefficients The second, , and third, , virial coefficients have been studied extensively and tabulated for many fluids for more than a century. Two of the most extensive compilations are in the books by Dymond. and NIST Thermo Data Engine Database and its Web Thermo Tables. Tables of second and third virial coefficients of many fluids are included in these compilations. The second and third virial coefficients as functions of temperature are shown in the following figure for argon. Reduced temperature and reduced virial coefficients, scaled by respective critical properties, are all dimensionless. Notice in the figure that the second virial coefficient decreases monotonically as temperature is lowered. However, the third virial coefficient has a bell shape. It increases as temperature is lowered to the critical temperature, then it passes through a peak and decreases rapidly to zero as temperature is lowered from the critical point to the triple point. It is physically unreasonable for it to decrease below the critical temperature. This is because the third virial coefficient theoretically represents the repulsive force among molecules, which is expected to increase at a lower temperature, as molecules are pressed together. The behaviors just described are similar to many other gases, as shown in the figure below. The reason why the third virial coefficient decreases below the critical temperature is as follows. Generally, the PρT isotherms are measured in the gaseous phase. Below the critical temperature, the gaseous phase condenses and coexists with the liquid phase, and the PρT isotherm becomes flat. Saturation pressure does not change until the gas condenses completely to liquid, and then pressure rises as density increases. There is a large gap between