Applying the Critical Conditions to Equations of State J. G. Eberharl University of Colorado, Colorado Springs, CO 80933 This paper examines some of the important features of using the three critical conditions as a means of determining the parameters in some of the simpler and more popul& fluid equations of state. The equations considered here are those of vau der Waals. Berthelot. Redlich-Kwone. Claumodisius, a Berthelot modifidation, and ~edlich-KWOG fication. These various eauations of state contain from two to four parameters. In relating the parameters in these equations of state to the critical constants of the fluid of interest, a number of important principles will be illustrated. (1) Considerable simplification of the mathematics of the problem is possible if dimensionless parameters and critical conditions are employed. (2) In all of the two-parameter equations of state considered, the critical conditions overdetermine the parameters. and the narameters can he found from anv two of the three'critical c h a n t s . These two-parameter equations have a "build-in" theoretical value for the critical compressihility factor, Z,, which differs considerably from the experimental value. (3) With some three-parameter equations of state the three parameters can be determined from the three critical conditions and all three of the critical constants. The experimental value of Z, is thus imposed on these equations of state. (4) With other three-parameter equations of state the three parameters cannot be determined through the three critical conditions and the critical constants because the three equations are not independent in the three parameters. This situation can be remedied by introducing an "auxiliarv" critical constant that is the tanpent to the vanor pressure curve a t the critical point, along with an accompanvine " ., auxiliarv critical condition. ( 5 ) With four-narameter equations of siate the four parameters are underdktermined bv the threecritical conditions.'I'hissituationcan alsosometimes be remedied by including the same auxiliary critical condition and constant. (6)Finally, it will be illustrated that finding the relationship between equation-of-state parameters and fluid critical properties sometimes involves relatively simple algebraic manipulation, while with the more complicated equations of state numerical solutions of nonlinear equations may be required.
a
-
The Crltlcal Conditions
An equation of state for a pure substance provides a relationship between the intensive variablesp, u, and T, wherep is pressure, v is molar volume, and T is absolute temperature. The critical conditions place three constraints on this
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p-u-T relationship in the neighborhood of the critical point (u,, T,, p,). Stated in terms of the critical isotherm curve (where T = TJ,these three constraints are: (1)the critical isotherm passes through the point (u,, p,), (2) the critical isotherm has a horizontal tangent a t (u,, p,), and (3) the critical isotherm has an inflection point a t (u,, p,). If the p-u-T relationship is represented as p = f(u, T ) , then the three critical conditions can be stated mathematically as
where f,and fi, are the first and second partial derivatives of f with respect to u. Equations 1-3 are usually called the critical conditions. For a particular equation of state they provide three relationships among the parameters in the equation of state and are used in the calculation of these parameters from the critical constants of the fluid of intera-t
It will be shown that forsome equationsofstate there may be parameters that are not independent in the critical equations. In such cases an auxiliar; critical condition invol;ine the tangent to the vapor-pressure curve at the critical point, s, = (dp.ldT),, where p, is the equilibrium vapor pressure, can provide the additional information needed. According to Gihbs (I) this critical property is related to the equation of state p = f(u, T ) through the equation
-
where f ~ ithe s partial derivative of f with respect to T. This relationship has been used by Riedel(2), Eberhart ( 3 , 4 ) ,and others in equation-of-state formulation. Convenient tabulations are available of the critical constants of a wide variety of inorganic (4) and organic (5) fluids. In addition a brief listing of values of the reduced tangent t o the vapor-pressure curve a t the critical point, a, = (T,lp,)(dp,ldT),, is also available (3). The specifics of how to relate the constants u,, T,, p,, and s, t o the parameters in a particular equation of state will be considered next, beginning with the simplest two-parameter equations of state.
The van der Waals Equatlon
The most famous of all equations of state, and the first to provide a fluid model containing both the gaseous and the liquidstate,is that of van der Waals (7).His equation has the form where the constant b is a measure of molecular size and the constant a reflects the strength of the intermolecular attractive forces. The equation is familiar to most students of chemistry and in many cases is the only equation of state with which the student has gone through the exercise of relating parameters to critical constants. This is unfortunate because, as will be illustrated, the two-parameter equations have a number of important features unique to themselves. To apply the critical conditions the appropriate function and derivatives are first required. They are
these results into eqs 14, 18, and 19 gives, for the van der Waals equation,
With the selection of T, andp, as the only critical constants used in finding band a, eq 20 can be viewed as the means of finding the predicted value of u,. The Berthelot Equation
The Berthelot equation of state (9) is slightly more complicated than that of van der Waals. I t has an attractive term that depends on T as well as on u, and bas the form Finding the appropriate derivatives of the right-hand side of eq 23 and applying the critical conditions yields
Then, setting u = u, and T = T, in eqs 6-8, the critical conditions, eqs 1-3, yield Again dimensionless variables are introduced, based on the form of eq 23, via the definitions
Clearly these three equations overdetermine the two parameters band a (8). Next, eqs 9-11 will be translated into a form involving dimensionless parameters. If dimensional arguments are applied to eq 5, it is clear that b and a can be written as b = Bu,
(12)
where p and a are dimensionless. A third dimensionless unknown, the critical compressibility factor, Z,, is also introduced as Z, = p,uJRTC
(14)
Substituting eqs 12-14 into eqs 9-11 yields
This system of three equations is simpler in appearance than eqs 9-11. Equations 15-17 provide the means of relating b and a to the critical constants (through 0 and a), and also yield the theoretical value of Z, which relates the three critical constants. Since p,, u,, and T, are related by this value of Z, only two of the three critical constants can be used in calculatine b and a (8). The constants T,and .D,.are most commonly selecwd fur this purpose, and that practlce will he followed here. This selection can be accomdished bv eliminating u, from eqs 12-14, yielding
(Note that the second member of eq 29 does not imply that all three critical constants are required to find a, because the constants are again related through eq 27.) Substituting eqs 27-29 intoeqs 24-26 yields aset of equations that is identical to those obtained from the van der Waals model, eqs 15-17. This is in spite of the fact that eqs 9-11 are clearly different from eqs 24-26. Thus, another advantage of a dimensionless formulation of the problem of relating parameters to critical constants is that sometimes the dimensionless problem has been previously solved with another fluid model. For the Berthelot equation of state, then, it follows that 0 = 113, a = 3, and Z, = 318, just as with the van der Wads equation. Substituting these results into eqs 27-29 yields
Again, with the selection of T,and p, as the experimental critical constants to be employed, eq 30 can be viewed as the means of predicting u,. The Redllch-Kwong Equation
The last and most complicated of the two-parameter equations of state to he considered here is the RedlichKwong equation (10). It also has an attractive term that depends on T and u, and has the form p = RTl(u - b)
Equation 15-17 are easily solved algebraically. Dividing eq 16 by eq 17 gives 1- 0 = 213 or 8 = 113. Dividing eq 15 by eq 16 yields 1- p = (1 a)/2 and a = 1/(1- 28) = 3. Solving eq 15 then providesz, = 1/(1- 0)(1+ a) = 318. Substituting
+
- alT1"u(u + b)
(33)
This equation is highly regarded because it has a quantitative accuracy that is surprisingly good for a two-parameter equation of state (11, 12). The equation is also sufficiently complex that numerical methods are very helpful in finding the relationship between the parameters and the critical constants. Volume 66 Number I2 December 1989
99 1
The application of eqs 1-3 to eq 33 yields the three critical conditions Substituting eqs 50-53 into eqs 4 7 4 9 yields l/Zc(l - 8) = 1+ al(1+ 7)'
Again dimensionless unknowns are introduced through the equations Z, = p,u,lRT, (37)
Substituting eqs 37-39 into eqs 34-36 yields the critical equations in dimensionless form
In these equations%, is the known experimental value ofthe critical compresnibility factor and not (an in the case ot'the two-parameter equations) an unknown to be determined. Dividing eq 55 by eq 56 and solving for -f gives Substituting eq 57 into eqs 54 and 55 yields a system of two equations in @anda
Solving eq 59 for a yields where the three unknowns are, as usual, Z,, 0, and a. Dividing eq 41 by eq 42 eliminates Z, and u, and gives, after rearrangement, p3 3p2 38 - 1 = 0. This equation is most readily solved by numerical methods, such as Newton's method (13),which yield 0 = 0.259921. Dividing eq 40 by eq 41 eliminatesZ,and yields, after solving, a = (1 + p)2/(1 - 28 - p2) = 3.84732. Finally, solving eq 40 gives& = (1 p)l(l/3)(1+ 0 + a ) = 0.333333. The additional parameters needed in the third members of eqs 38 and 39 are BZ, = 0.0866403 and aZC2= 0.427480. It is also possible to solve the cubic polynomial in 6 algebraically, although the manipulations involved are quite lengthy and complicated. If this task is carried out, however, it is found that p = Z1'3 - 1,and, as a result, eqs 37-39 yield
+ +
Substituting eq 60 into eq 58 and solving for p produces 8 = 1- 1/42,
(61)
Substituting eq 61 into eq 57 and 60 gives, in addition,
+
for Z, and the two parameters. The Clausius Equation The simplest three-parameter equation of state is probably that of Clausius (14) where both b and c are constants related to the size of the fluid molecules. Applying the critical conditions yields three equations in the three unknowns, b, c, and a
Clearly, eqs 4 7 4 9 do not overdetermine the three unknowns, but provide a unique value for each parameter, requiring the use of all three critical constants. Thus, it is expected that the experimental value of Z, can be imposed on this equation of state. Based on dimensional arguments the dimensionless parameters p, r , a,and Z,are defined through the equations
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Journal of Chemical Education
Thus, the final relationships between the parameters and the critical constants are, from eqs 50-53, b = (1- 1/4Z,)u,
(64)
It is noted here that if Z, is set equal to 318 in eqs 6466, then the expressions reduce to those for the Berthelot equation (where c = 0). The Modified Berthelot Equation Unlike the Clausius equation, some three-parameter equations of state are not amenable to calculating the three parameters from T,, u,, and p,. A case in point is what might be called the modified Berthelot equation of state [a special case of an equation employed by Eberhart (3)]
where b, a, and mare sought from the critical constants. Applying the critical conditions to eq 67 yields the three equations
I t is clear that eqs 69 and 70 can easily yield b by division. However, a and m cannot be obtained from eqs 68-70 because they enter into each of the three equations via the Thus, eqs 68-70 are not same functional form, namely alTCm. independent in a and m. The situation is clearer when the critical conditions are made dimensionless through the relationships
eqs 82-84. These three equations are simplified further by the variable changes S=1-8 (87) where m is already dimensionless. Substituting eqs 71-73 into eqs 6E-70 gives the same set of three dimensionless critical conditions as were obtained with the van der Wads and the Berthelot fluid models, i.e., eqs 15-17. These equations do not contain m a s an unknown. The three unknowns are Z,, 8, and a,where (unlike the Clausius model) Z, does not have its experimental value. It has already been shown that for these three simultaneous equations Z, = 318, B = 113, and a = 3. Examining eqs 72 and 73, it is seen that knowing 8 permits the calculation of b but that a cannot be found without m (even though a is available). Clearly another relationship involving critical properties is required to find the parameter m, and that relationship is eq 4. From eqs 67 and 4 the tangent to the vapor-pressure curve at the critical point is, for the modified Berthelot equation, s, = R/(u, - b)
+ arnlTcmf'u:
where eq 71 was used to simplify the results. Substituting the previously derived values of Z,, 8, and a,and solving for m gives (76)
With eq 76 now available (as well as the values of Z, 8, and a) it is possible to calculate the value of a from eq 73 and complete the determination of parameters for the modified Berthelot equation. The Modlfled Redllch-Kwong Equatlon The original Redlich-Kwong equation can he improved by the introduction of two additional adiustable Darameters. c and m, (15) p = RTIb
- b ) - alT"u(u + c)
The solution of these three equations gives a cubic polynomial in E which is most readily solved by numerical methods. From its solution p and 6 follow from p = (1
(74)
Substituting eqs 72 and 73 into eq 74 gives a reduced tangent to the vapor-pressure curve at the critical point, a, = (T,/ ~,)(d~.ldT),= T ~ J P , ,of
rn = (cc - 4)/3
which transform eqs 82-84 into
(77)
yielding a four-parameter equation of state. This eouation will reouire all four of the critical conditions dis~ussed~reviously (eqs 1-4). Introducing the usual dimensionless parameters through the equations
and applying the critical conditions to eq 77 yields
+ r + c2)/(1 + e)
(94)
while p,?, and a can he found from @=1-S
y=r-1
Then. the orieinal four Darameters can be found from eos 79-8i and 86.It is clear that as the number of Darameters and the complexity of an equation of state po&, the task of calculating the parameters from the critical constants becomes more difficult. It is also apparent that as the number of parameters aoes bevond three or four. thereis aneed toem~lovu-uT measureients other than &tical properties to deterkine the parameters. Virial coefficients are an obvious choice for thistask. Conclusions A variety of two., three-, and four-parameter equations of state have been considered here. The problem of relating the parameters to the critical constants of the fluid of interest has been examined in detail for the exam~lesselected. In some cases the equations sought could be soived by algebraic mani~ulation-in others numerical methods were required. The advantages of dimensionless critical conditions and parameters were amply illustrated, both through the algebraic simplification achieved and the repetition of the same dimensionless equations in several fluid models. Finally, the pattern has been illustrated of the overdetermination of parameters in two-parameter equations of state and the change to full determination or underdetermination of parameters as the number of parameters is increased to three or four (or even more).
LReiature Cited Gibba. J. W. TheSciontificPopersofJ. Willard Gibb3: Dover: NewYork, 1961: Vol. 1, p 108. 2. Riadol, L. Chemie-lng.-Tech". 1951.26.83-89. 3. Eberhart. J. G.J. Colloidlnforfoce Sci. 1976.56,262269. 4. Eberhart. J. G.: Pinks. V., 11. J.ColloidIni~r/oceSci.1985.107. 57-75. 5. Mathews, J. F. Chrm.Reu. l972.72.71-1W. 6. Kudchadkcr.A.P.:Alsni,G. H.;Zwolimki. 8. J-ChemRou. 1968.68.659-735. 7. van dor wsals. J. D. over de Continuiraii "on den ~ o a - o nvloer~o(toastond. u~esin, Uniu. Leiden, 1873. 8. Guggenheim.E.A. Thermodynomies.3rded.: NorthHolland: Amsterdam, 1951:p165. 9. Berthelnt. 0.J. Physique, 111 1699.8.265-274. 10. Redlich. 0.;Kwong, J. N.S. Chem.Ra8. 194'3.44.233-243. 11. Ott. J. B.;Goates. J. R.;Hall, H.T., Jr. J . Chern.Educ. 1971,48,515-517. 12. Ksmp, M. K.;Thompson, R. E.; Zigrang, D. J. J. Chem. Educ. 1975.52,802. 13. Mortimer. R. G.. Mothomtie$ for Physical Chemistry: M a c m i l h : New York, 1981ip + *.d".o 11. Cleusius. R. Ann. Physik 1880,169,337-357 15. Eberhsrt, J. G.,unpublished work. 1.
where Z, is the experimental value of the critical compressibility factor. Eq 85 can be simpified by elimination of 11ZJ1 - 8) with eq 82. The result is Thus, m can be found once 8,?, and a are determined from
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