From Eqs. (2.154) and (2.155),

(2.156)

Experimental values of Z_c for most substances fall within a narrow range of 0.20-0.30. Therefore, Z_c may be taken to a constant.

Eq.(2.156) reduces to

Z = f (pr, Tr)

(2.157)

where T_r is plotted as a function of reduced pressure and Z, a single plot, known as the generalized compressibility chart, is found to be satisfactory for a great variety of substances. Although necessarily approximate, the plots are extremely useful in situations where detailed data on a particular gas are lacking but its critical properties are available.

Law of corresponding states for van der Waal's gas

For a van der Waals gas,

(2.158)

where a, b, and R are the characteristic constants of the particular gas.

(2.159)

(2.160)

It is therefore a cubic profile in v and for given values of p and T has three roots of which only one need be real.

• For low temperatures, three positive real roots exist for a certain range of pressure.

• As the temperature increases the three real roots approach one another

• At the critical temperature all the roots become equal. Above this temperature only one real root exists for all values of p.

• the critical isotherm T_c at critical state on the p-v plane (fig.10.4),where the three real roots of the van der Waals Equation coincide, not only has a zero slope, but also its slope changes at the critical state (point of inflection), so that the first and second derivatives of p with respect to v at T = T_c are each equal to zero. Therefore,

(2.161)

(2.162)