Cubic Equations of State in Software

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35 Cubic Equations of State
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most suitable is elimination of Vc to yield expressions relating a(Tc) and b to PCand T, The reason is that PCand Tc are usually more accurately known than V, An equivalent, but more straightforward, procedure is illustrated for the van der Waals equation Since V = Vcfor each of the three roots at the critical point,
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Equation (340) expanded in polynomial form becomes:
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Recall that for a particular substance parameter a in the van der Waals equation is a constant, independent of temperature Term-by-term comparison of Eqs (A) and (B) provides three equations:
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Solving Eq (D) for a , combining the result with Eq (E), and solving for b gives:
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Substitution for b in Eq (C) allows solution for V,, which can then be eliminated from the equations for a and b:
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Although these equations may not yield the best possible results, they provide reasonable values which can almost always be determined, because critical temperatures and pressures (in contrast to extensive PV T data) are often known, or can be reliably estimated Substitution for V, in the equation for the critical compressibility factor reduces it immediately to: 3 z = -pcvc - " - RT, 8 A single value for Z,, applicable alike to all substances, results whenever the parameters of a two-parameter equation of state are found by imposition of the critical constraints Different values are found for different equations of state, as indicated in Table 31, p 93 Unfortunately, the values so obtained do not in general agree with those calculated from experimental values of Tc, PC,and Vc;each chemical species in fact has its own value of Z, Moreover, the values
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CHAPTER 3 Volumetric Properties of Pure Fluids
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given in Table Bl of App B for various substances are almost all smaller than any of the equation values given in Table 31 An analogous procedure may be applied to the generic cubic, Eq (341), yielding expressions for parameters a(Tc) and b For the former,
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This result may be extended to temperatures other than the critical by introduction of a dimensionless function a(T,) that becomes unity at the critical temperature Thus
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Function a(Tr) is an empirical expression, specific to a particular equation of state Parameter b is given by:
In these equations C2 and Q are pure numbers, independent of substance and determined for a particular equation of state from the values assigned to t and a The modern development of cubic equations of state was initiated in 1949 by publication of the RedlicWKwong (RK) equation:'
where, in Eq (342), a(T,) = T,-''~
Theorem of Corresponding States; Acentric Factor
Experimental observation shows that compressibility factors Z for different fluids exhibit similar behavior when correlated as a function of reduced temperature T, and reducedpressure P,; by dejinition, T'I = - T Tc and
This is the basis for the two-parameter theorem of corresponding states:
All fluids, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor, and all deviate from ideal-gas behavior to about the same degree
Although this theorem is very nearly exact for the simple fluids (argon, krypton, and xenon) systematic deviations are observed for more complex fluids Appreciable improvement
80tto Redlich and J N S Kwong, Chem Rev,vol 44, pp 233-244, 1949
35 Cubic Eauations o f State
results from introduction of a third corresponding-states parameter, characteristic of molecular structure; the most popular such parameter is the acentric factor w , introduced by K S Pitzer and coworker^^
Figure 313 Approximate temperature dependence of the reduced vapor pressure
The acentric factor for a pure chemical species is defined with reference to its vapor pressure Since the logarithm of the vapor pressure of a pure fluid is approximately linear in the reciprocal of absolute temperature,
d log P,Sat d(llTr)
where P T t is the reduced vapor pressure, T, is the reduced temperature, and S is the slope of a plot of log P,Sat vs 1/T, Note that "log" denotes a logarithm to the base 10 If the two-parameter theorem of corresponding states were generally valid, the slope S would be the same for all pure fluids This is observed not to be true; each fluid has its own characteristic value of S, which could in principle serve as a third corresponding-states parameter However, Pitzer noted that all vapor-pressure data for the simple fluids (Ar, Kr, Xe) lie on the same line when plotted as log P,Sat vs 1/T, and that the line passes through log P T t = -10 at T, = 07 This is illustrated in Fig 313 Data for other fluids define other lines whose locations can be fixed in relation to the line for the simple fluids (SF) by the difference: log p , S a t ( s ~ ) log p,sat The acentric factor is defined as this difference evaluated at T, = 07:
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