To conclude this chapter, we will not attempt a direct evaluation of the partition function, but rather examine its structure in terms of scaling procedures. Such a method is useful for quickly obtaining fairly approximate information on completely unknown thermodynamic data for one fluid from known data relating to another suitable related fluid. The two fluids are in corresponding states: The principle of corresponding states has played a very valuable role in empirical thermodynamics in the past and is still useful in the initial study of complicated fluids. Many methods [4,5,6] have been proposed to estimate specific volumes or densities of pure liquids. All are based on the law of the corresponding states. They are algebraically complex and therefore require the help of a computer for many calculations. Appendix A1 gives tabular values for correlation constants A and B for certain chemical compounds. The PROG21 computer program uses equation 2-1 to estimate the density of liquids. Typical results for water are presented in Table 2-1. Figure 2-1 shows the density of water in the range of 0°C to a critical value of Tc = 374.2°C. For an ideal gas, the compressibility factor Z = 1 {displaystyle Z=1} by definition.

In many real-world applications, precision requirements require that deviations from the ideal gas behaviour, i.e.dem the actual behaviour of the gas, be taken into account. The value of Z {displaystyle Z} usually increases with pressure and decreases with temperature. At high pressure, molecules collide more frequently. As a result, the repulsive forces between molecules can act noticeably, the molar volume of the real gas (V m {displaystyle V_{mathrm {m} }}) becomes larger than the molar volume of the corresponding ideal gas ( ( V m ) ideal gas = R T / p {displaystyle (V_{mathrm {m} })_{text{ideal gas}}=RT/p} ), where Z {displaystyle Z} exceeds one. [3] When the pressure is lower, molecules can move freely. In this case, the forces of attraction dominate, making Z < 1 {displaystyle Z<1}. The closer the gas is to its critical point or boiling point, the more Z {displaystyle Z} deviates from the ideal case. This dimensionless approach to the empirical study of fluids is of great value in practice, but its scope is ill-defined and should be used with caution for all but the simplest substances, such as noble gases. An important difficulty is that the interparticle force laws theoretically envisaged are only approximations of those that apply in practice, and it is important to ensure that each group of liquids that are discussed collectively actually falls under the fact that they have essentially the same two-parameter force law, and that no critical force characteristic is overlooked in such a representation. However, it turns out that the statistical theory of fluids is able to provide a theoretical basis for the law of corresponding states and give clear indications regarding the general conditions for its safe application. The equations of the form shown above as equations (13.1), (13.2) and (13.4) are called power law relations. This leads to behavior somewhat similar to that which van der Waals discovered many years ago and formulated as the "law of the corresponding states".

In the law of the corresponding states, the reduced temperature is a ratio of absolute temperatures. In critical phenomena, however, reduced temperature is a temperature difference, relative to the critical temperature and scaled by it. In continuous phase transitions, there are several sets of curves that describe the different properties, each determined by a critical exponent called α, β, γ, and so on. Unlike the theory of corresponding states, the behavior of power laws is observed only within the limit as T → Tc. The temperature range over which the power law is followed depends on the particular property and some other characteristics of the transition, which we will discuss below. Table 13.2 defines the critical exponents we will examine here, t and Table 13.3 summarizes a number of transitions and systems for which these exponents have been measured.