Thermodynamics

cen84959_ch03.qxd 4/1/05 12:31 PM Page 145By performing the differentiations and eliminating v cr , the constants a and bare determined to beThe constants a and b can be determined for any substance from the criticalpointdata alone (Table A–1).The accuracy of the van der Waals equation of state is often inadequate,but it can be improved by using values of a and b that are based on theactual behavior of the gas over a wider range instead of a single point.Despite its limitations, the van der Waals equation of state has a historicalvalue in that it was one of the first attempts to model the behavior of realgases. The van der Waals equation of state can also be expressed on a unitmolebasis by replacing the v in Eq. 3–22 by v and the R in Eqs. 3–22 and3–23 by R u .Beattie-Bridgeman Equation of StateThe Beattie-Bridgeman equation, proposed in 1928, is an equation of statebased on five experimentally determined constants. It is expressed aswherea 27R 2 T 2 crandb RT cr64P crP R uTv 2 a 1 cv T 3 b1v B2 A v 2Chapter 3 | 145P(3–23)8P cr(3–24)Critical pointTcr = constantFIGURE 3–58Critical isotherm of a pure substancehas an inflection point at the criticalstate.vA A 0 a 1 a v bandB B 0 a 1 b v b(3–25)The constants appearing in the above equation are given in Table 3–4 forvarious substances. The Beattie-Bridgeman equation is known to be reasonablyaccurate for densities up to about 0.8r cr , where r cr is the density of thesubstance at the critical point.Benedict-Webb-Rubin Equation of StateBenedict, Webb, and Rubin extended the Beattie-Bridgeman equation in1940 by raising the number of constants to eight. It is expressed asP R uTv a B 0R u T A 0 C 0T b 12 v bR uT a aa2 v 3 v c6 v 3 T a 1 g 2 v b 22 eg>v(3–26)The values of the constants appearing in this equation are given inTable 3–4. This equation can handle substances at densities up to about2.5r cr . In 1962, Strobridge further extended this equation by raising thenumber of constants to 16 (Fig. 3–59).Virial Equation of StateThe equation of state of a substance can also be expressed in a series formasP RTv a 1T2 b 1T2 c 1T2 d 1T2 . . .v 2 v 3 v 4 v 5(3–27)