Three Parameter Equations of State

8. Three Parameter Equations of State

If v = v c (i.e., along the critical isochore), employing the Van der Waals equation, P = RT/(v

c – b) – a/v c ,

which indicates that the pressure is linearly dependent on the temperature along that isochore. Likewise, the RK equation also indicates a linear expression of the form

P = RT/(v c – b) – a/(T 1/2 v c (v c + b)).

However, experiments yield a different relation for most gases. Simple fluids, such as argon, krypton and xenon, are exceptions. The compressibility factors calculated from either the VW or RK equations (that are two parameter equations) are also not in favorable agreement with experiments. One solution is to increase the number of parameters.

a. Critical Compressibility Factor (Z c ) Based Equations

Clausius developed a three parameter equation of state which makes use of experi- mentally measured values of Z c to determine the three parameters, namely

(66) where the constants can be obtained from two inflection conditions and experimentally known

P= R T/( v – b )– a /(T ( v + c ) 2 ).

value of Z C , critical compressibility factor.

b. Pitzer Factor The polarity of a molecule is a measure of the distribution of its charge. If the charge it

carries is evenly or symmetrically distributed, the molecule is non–polar. However, for some chemical species, such as water, octane, toluene, and freon, the charge is separated across the carries is evenly or symmetrically distributed, the molecule is non–polar. However, for some chemical species, such as water, octane, toluene, and freon, the charge is separated across the

pirical values correspond with those obtained from experiments. This factor was developed as

a measure of the structural difference between the molecule and a spherically symmetric gas (e.g., a simple fluid, such as argon) for which the force–distance relation is uniform around the molecule. In case of the saturation pressure, all simple fluids exhibit universal relations for P sat R with respect to T R (as illustrated in Figure 14 ). In Chapter 7 we can derive such a relation using a two parameter equation of state. For instance, when T R = 0.7, all simple fluids yield P sat R ≈ 0.1, but polar fluids do not. The greater the polarity of a molecule, the larger will be its

deviation from the behavior of simple fluids. Figure 14 could also be drawn for log

10 P r vs. 1/T R as illustrated in Figure 12 . The acentric factor ω is defined as

sat

(67) Table A-1 lists experimental values of ω” for various substances. In case they are not listed, it

ω = –1.0 – log

10 (P R ) TR=0.7 = –1 – 0.4343 ln (P sat R ) TR=0.7 .

sat

is possible to use Eq. (68).

i. Comments The vapor pressure of a fluid at T R = 0.7, and its critical properties are required in or- der to calculate ω. For simple fluids ω = 0.

For non-spherical or polar fluids, a correction method can be developed. If the com- pressibility factor for a simple fluid is Z (0) , for polar fluids Z ≠Z (0) at the same values of T R and P R .

We assume that the degree of polarity is proportional to ω. In general, the difference (Z – Z (0) ) at any specified T R and P R increases as ω becomes larger (as illustrated by the line SAB in Figure 13 ).

With these observations, we are able to establish the following relation, namely.

R ,P R )– Z (0) (P R ,T R )) = ωZ (T R ,P R ). (68) Evaluation of Z( ω,T R ,P R ) requires a knowledge of Z (1) , w and Z (0) (P R ,T R ).

(Z ( ω,T

c. Evaluation of Pitzer factor, ω

i. Saturation Pressure Correlations The function ln(P sat ) varies linearly with T –1 , i.e.,

ln P sat =A–BT –1 . (69) Using the condition T = T c ,P=P c , if another boiling point T ref is known at a pressure P ref , then

the two unknown parameters in Eq. (70) can be determined. Therefore, the saturation pressure at T = 0.7T c can be ascertained and used in Eq. (69) to determine ω.

ii. Empirical Relations Empirical relations are also available, e.g.,

ω = (ln P sat R – 5.92714 + 6.0964/T R,BP +1.28862 ln T R,BP – 0.16935 T R,BP )/ (15.2578 – 15.6875/T R,NBP + 0.43577 T R,NBP ),

(70) where P R denotes the reduced vapor pressure at normal boiling point (at P = 1 bar), and T R,NBP

the reduced normal boiling point.

P R2 ,T R2

P R1 ,T R1

Z (1)

Z (P R ,T R )

ref

w ref

Figure 13: An illustration of the variation in the compressibility factor with respect to the acentric factor.

An alternative expression involves the critical compressibility factor, i.e.,

(71) Another such relation has the form

ω = 3.6375 – 12.5 Z c .

ω = 0.78125/Z c – 2.6646.