APPLIED THERMODYNAMICS SYMPOSIUM
THOMAS W. LELAND, JR. PATSY S. CHAPPELEAR
THE CORRESPOND NG STATES PRINCIPLE A Review of Current Theory and Practice
The most powerful
tool available for
.qwntitative prediction of the physical properties of pure fluids and mixtures is the .CSP
Pitzer (93) in 1939. This work shows clearly the molecular requirements which must be satisfied for a system to obey the CSP in its simplest form. These may be summarized as follows : 1. The canonical ensemble partition function for the fluid is assumed to be separable into independent translational, rotational, vibrational, and configurational factors. The translational energy states must be entirely independent of the rotational states. This requirement limits the CSP to molecules which have translational and rotational energies which are small relative to kT. This condition is usually fulfilled except at very low temperatures. Molecules which exhibit hydrogen bonding or other specific interactions between individual atoms or groups within each molecule are also exceptions. 2. I t is assumed that the internal energy states for a n individual molecule are entirely independent of the density so that only the configurational partition function is density dependent. 3. The translational and configurational portions of the partition function are assumed to be independent of any quantum effects, and Maxwell-Boltzmann statistics are considered to be applicable. Molecules with a mass equal to or less than that of neon exhibit quantum mechanical effects which become increasingly important as the mass decreases. The light molecules such as Ne, D,, H,, HD, and He are thus excluded from the simple CSP. 4. The total potential energy for the system can be expressed as the product of an energy parameter and a function of dimensionless separation distances between molecular centers. This potential can be represented by
(3) Also true a t the critical point
(4)
(5) Equations 3, 4, and 5 may be solved simultaneously to give constant values for Z,, e/kT,, and V , / N U ~ . This solution is
z,
= a
(6)
E
-I. rn = b
_ - c NU3
where the quantities a, b , and c are universal constants applicable to all fluids which obey the simple CSP. Combining Equations 6, 7, and 8 shows that P,u3 - = E
a bc
(9)
Although the simple CSP is applicable to any two-parameter potential function, the use of the Lennard-Jones potential produces the following values of a, 6 , and c: a =
0.292
b = 0.800 c = 3.14 Equations 7 and 8 mean that the constants e and a3 are directly proportional to critical constants and may be eliminated from the equation of state to obtain a new generalized function
The two-parameter CSP does not necessarily depend on pairwise additivity and it is now known that the CSP holds well for some systems at conditions for which exact pairwise additivity is not valid. These assumptions lead directly to the two-parameter CSP for simple molecules. The equation of state derived from the configurational integral using the above assumptions has the form
where f is a universal function which is valid for all fluids conforming to the assumptions. The detailed derivation is presented in various texts dealing with statistical thermodynamics of fluids, such as Hill (47) and Rowlinson ( 7 76). The two-parameter generalized equation of state in Equation 2 can be expressed alternatively in terms of critical constants instead of the force parameters e and U , Equation 2 is applied at the critical point to give 16
INDUSTRIAL A N D ENGINEERING CHEMISTRY
For fluids obeying the simple CSP, Equations 2 and 10 are entirely equivalent. A similar function can be written in terms of P / P c instead of V / V c . CSP and two-parameter equations of state. An important consequence of the CSP is that the parameters in any empirical two-parameter equation of state can always be expressed in terms of either e and u3 or critical properties. For example, any empirical twoparameter equation of state may be represented by
where a and b are two parameters independent of T and V , whose values are adjusted to fit various fluids. Simultaneous solution of Equation 11 at the critical point with Equations 4 and 5 will determine values of a and b in terms of critical constants for each fluid and, in addition, a universal constant value of Z , for all fluids. Substitution of these values into Equation 11 produces a gener-
alized equation of state which may be rearranged in the form of Equation 10. A more powerful procedure for expressing empirical equations of state in the dimensionless form of the CSP is based on the fact that any equation of state valid over the entire liquid and gaseous region must have a n odd number, equal to or greater than three, of real positive roots for the volume. The saturated vapor and liquid volumes are always, respectively, the largest and smallest roots in numerical value. The odd number is necessary so that the slopes of the isotherm (bP/bV). for both the liquid and vapor have negative values a t the saturation line. If the equation of state can be expressed as a n nth-order polynomial in the volume, then a total of (n - 1) adjustable parameters can be found in terms of the critical properties. I t is thus possible to apply the CSP to certain kinds of equations of state of greater complexity than Equation 11. As a n example, consider the equation
Z
= f(a,
b,
C,
. . ., T , V )
(12)
which can be arranged in a polynomial of nth order in V where a, b, c, . . . are equation of state constants, adjustable for various fluids. At the critical point, all real volume roots of Equation 12 must become identical and the polynomial form must approach
(V
- V,),
=
0
(13)
Equation 13 is now expressed as a binomial expansion in powers of V . The coefficients are equated to the coefficients of like powers of V in the equation of state when it is written also as an nth-order polynomial in V with coefficients evaluated at critical conditions. The equation of state in this form is
Vn
+ fi(a, b,
. . ., T,, P,,ZJ VR-'
C,
f z ( a , b,
C,
,
+
. ., T,, P,,Z,)VR-2+ ...
0
(14)
Equating coefficients of like powers of V in Equations 13 and 14 will then determine ( n - 1) parametersa, b, c,. . , in terms of critical properties. If no more than (n - 1) such parameters are present in the equation, a universal constant value will also be predicted for 2,. With this procedure one may express a n equation with a great many empirical constants in reduced form according to the CSP. Each constant, however, is ultimately determined by only two critical constants. Examples of this technique for equations of state which have polynomial forms of different orders are given in the work of Barner, Pigford, and Schreiner (3) and Kobe and von Rosenberg (57). For third-order polynomials with only two equation of state constants, this procedure gives results identical with the simultaneous solution of Equations 4, 5, and l l at the critical point. From the time of van der Waals to the present, work has continued on the development of various reduced equations of state to describe fluids obeying the simple CSP. This continuing study has been strongly motivated by the desire to find a precise two-parameter reduced equation as a starting point for further modifica-
tion to include molecules which do not conform to the simple theory. This study has proceeded along three important lines of investigation. T h e oldest of these historically is the development of empirical modifications to improve the reduced van der Waals equation. T h e next significant development chronologically was the improvement for the dilute gas region by using reduced virial expansions, obtained both from empirical correlations and by calculation from basic statistical mechanics. The most recent approach has used the theoretical treatment of the liqhid and dense gas region developed during the past 10 years in the statistical mechanics of assemblies of hard spheres. Because of the importance of these three methods to the development of reduced equations of state, a brief summary of each is presented here. Empirical modification of van der Waals equation. The development of an equation of state suitable for the generation of precise thermodynamic properties must involve much more than merely fitting the constants to give good predictions of compressibility factors. I n addition, there are many distinctive characteristics of the phase behavior of a fluid which an equation with this capability should predict. A systematic analysis of these characteristic features of a pure fluid has been presented by E. H. Brown ( 7 7) and by Rowlinson ( 7 79). These are summarized as follows : 1. The phase boundaries-liquid-solid and vaporliquid 2. The terminations of the phase boundaries-the critical point and the triple point 3. The deviation from linearity of the lines of constant volume 4. The loci of the vanishing of various derivatives: (a) The Amagat curve or Joule inversion locus where ( d Z / b T ) v and (bU/bV). are both zero (b) The Boyle curve, along which (dZ/dV). and (bZ/bP). are both zero (c) The Joule-Thompson inversion locus where ( d Z / b T ) , and (dH/bP)T are both zero (d) The loci of maxima or minima in the heat capacities where (b2P/bT2)y and (bCv/bV)T are both and (bC,/bP). are both zero or where (b2V/bT2)p zero 5. The intersections with the P = 0 axis of the curves generated by equating the derivatives in Item 4 to zero. These intersections determine some important values of the second virial coefficient and the nature of its variation with temperature Even for simple fluids, no theoretical analysis can possibly account for all the features outlined above and give accurate numerical values for the compressibility factor over the entire P-V-T region. For this reason, various empirical modifications of the basic van der Waals equation are still being studied today. The modifications to produce better agreement with a number of the criteria outlined above are discussed in a recent review by Martin (78). For fluids obeying the simple fluid CSP he recommends VOL. 6 0 NO. 7
JULY 1968
17
\
and discusses its performance. Equation 15 gives 0.335 for the universal Z , value, which is somewhat higher than the experimental average of about 0.290 for fluids obeying the simple CSP. Equation 15, however, is written in a form which can use the actual 2,of the fluid and neglect the discrepancy a t the critical. The predicted value of 2,is not a n important indicator of the over-all performance of any reduced equation of state. The reduced form of the Dieterici equation, for example, gives a better Z , value of 0.271 but, as shown in a review by Shah and Thodos (I%’), it is in all other respects much poorer than even the reduced van der Waals equation, which gives Z, = 0.333. Shah and ‘Thodos discuss and compare a number of classical and modern modifications of the original van der Waals equation. Their conclusion is that the best is that proposed by Redlich and Kwong (706) in 1949. The reduced RedlichKwong equation for simple fluids is
0.42148
(
(TR)0*6vR~?VR
6,
0.08 664y + -__Z C
The predicted Z,value is although Equation 16 like Equation 15 is written in a form which uses the actual Z , of the simple fluid, disregarding the difference at the critical. Martin reports very little difference between Equations 15 and 16. Equation 15 is reportedly better at reduced densities up through 1.6 and Equation 16 is slightly better at the higher densities. Graphical correlations of the reduced densities of the inert gases have been prepared by Hamrin and Thodos (38). Development work on empirical two-parameter reduced equations, based chiefly on various modifications of the van der Waals equation, is continuing to the present time. Examples are the equations of Benson and Golding (5) and of Yang and Yeiidall (748). The dilute gas region. The virial expansion is essential to the development of a basic reduced equation of state for simple molecules at low densities. The virial coefficients are functions of temperature only and have a direct relationship to intermolecular forces. For simple molecules the second virial coefficient can be accurately calculated from pair interaction potentials. One of the most important advantages of this approach is that the virial expansion for mixtures is known exactly. I n reduced form the virial expansion may be written
Z = 1
+
(g) (t,)+ ($) (k-
4- . .
..
(17)
The reduced second virial coefficient in the form B / (Nu9 was first calculated as a universal function of 18
INDUSTRIAL AND E N G I N E E R I N G C H E M I S T R Y
(kT/e) for the L’ennard-Jones (72) potential in 1924. Guggenheim (33) in 1945 demonstrated the conformity of reduced second virial coefficient data to the CSP for a number of simple fluids. Guggenheim and McGlashan (34) in 1951 presented a n empirical equation for the reduced second virial coefficient of molecules obeying the simple CSP. I n a n improved form developed by McGlashan and Potter (87), this equation is B _- - 0,430 - 0,886 Vc
($) - 0.694 (:>”
(18)
A similar reduced correlation for simple fluids was prepared by Pitzer and Curl (95) in 1956 in the form
Both these equations fit the available data within the limits of experimental accuracy. Both Equations 18 and 19 have been used effectively as bases for extension to molecules which deviate from the simple fluid CSP. A similar development has occurred with the third virial coeficient. The calculated reduced equation for a Lennard-Jones gas was obtained by Kihara (54)in 1948. I t was compared with available data in a reduced plot by Bird, Spotz, and Hirschfelder (6) in 1950. An empirical correlation fitting the most recent experimental data for siniple fluids has been presented by Chueh and Prausnitz (76). Their result for a simple fluid (argon) is C _ - [0.232
vcz
TR-0.26
-/- 0.468 T R - ~ ] [ I e(1-1.89rR2) I
Chueh and Prausnitz also proposed a correction for deviations from the simple CSP. As shown by Graben and Present (37), it is likely that deviations from pairwise additivity may introduce additional parameters, even for relatively simple molecules. This will seriously affect the rcduced third virial correlation. Because of this and also because of the greater uncertainties in experimental data, the scatter in the third virial correlation i s greater than in that for second virials. Except in the case of hard sphere molecules with no attractive forces, virials beyond the third are largely unknown. With three virials the gaseous region compressibility factors of simple molecules can be predicted accurately to the saturation line at reduced temperatures T/T, below about 0.9. At higher reduced temperatures the equation truncated after the third virial is accurate for V , values greater than roughly 2. With only two virials the equation is accurate at V Rvalues greater than about 8. For densities above the critical density approaching the liquid region, the virial expansion begins to diverge. An important use of the virial expansion is to incorporate it into empirical equations of state. I n this manner equations of the van der Waals type can be forced to agree with the virial expansion at low densities. A particularly promising example of this is the modification
of the reduced Redlich-Kwong equation with the PitzerCurl function for the reduced second virial coefficient. This has been developed by Barner, ,Pigford, and Schreiner (3). It provides a very effective reduced equation for simple molecules which serves as a base for the extension to more complex fluids. Dense fluid region. The use of van der Waals equation as a starting point in the development of a precise reduced equation of state for simple molecules at high densities presents some serious problems. I t has been known for a long time that the van der Waals equation is defective a t high densities. A review of its inadequacies is presented by Hirschfelder, Curtiss, and Bird (44). I t was recognized by van der Waals himself and also by Boltzmann that the [ R T / ( V - b ) ] term is seriously in error in accounting for the effect of molecular repulsion a t high densities. However, one of the first high density revisions of van der Waals equation based on a cell model by Eyring and Hirschfelder (25) showed that, although the [ R T / ( V - b ) ] term must be changed, the form of the ( a / V 2 ) term for the contribution of molecular attraction is a good approximation at high densities. This has since been confirmed theoretically by the recent work of Kac, Uhlenbeck, and Hemmer (50). Within the past 10 years, new developments in the statistical mechanics of dense fluids have produced functions which account for the molecular repulsion term almost exactly. A complete summary of these developments has been presented by Rowlinson (779). They involve the relationship of the pair distribution function and the pair correlation function to thermodynamic properties. The pair distribution function, g(r), is a dimensionless factor which measures the deviation from randomness in the probability that the molecules in a pair will have their centers separated by a distance r . The value of g ( r ) becomes unity if the probability of simultaneously observing one molecular center a t one point and a second center a t another point is independent of their separation distance. A closely related function which measures the total effect of a molecule at one position on the occupation probability of a second position is called the total correlation function and is defined as h(r) = [g(r) - 11. Its value is zero when the occupancy of one position by a molecule has no effect at all on the occupation probability of a second position at a distance r. This h(r) function was separated by Ornstein and Zernicke (88) into two parts. The first part, defined as the direct correlation function c(T), is the value which h(r) would have if no other molecules were located near enough to the pair to affect the occupation probability.
AUTHORS Thomas W. Leland, Jr., is chairman of the Department of Chemical Engineering, Rice University, Houston, Tex. Patsy S. Chappelear is Research Associate in this department. The authors wish to acknowledge that this work is part of a continuing study at Rice University of the thermodynamic properties of pure and mixed Juids made possible by assistance from the National Science Foundation and the Natural Gas Processors Association.
The second part is the contribution to h(r) caused by effects transmitted through a third molecule located near the pair. The contribution of each part is given analytically by the Ornstein-Zernicke equation
[g(d
- 11
= h(r12) = c(r12)
+ pfc(r13)h(r~)dT3 (21)
The integration is carried out over all positions of the third molecule at 73. A rigorous modern development of Equation 21, which is much easier to understand than the original article, is presented by Pearson and Rushbrooke (97). The relationship between the solution of Equation 21 for c(r) and thermodynamic properties has been developed by Rushbrooke and Scoins (722, 723). The most useful relation for equation of state development is
--!(E)
kT
3~
= 1 - pfc(r)dr
T
I n order to use Equation 22 one must solve the integral Equation 21 for c(r). For hard spheres this has been done exactly by numerical methods, and a n approximate analytical solution has been made possible by a n assumption proposed by Percus and Yevick (92). This assumption postulates a simple relationship between c(r) and g(r) for hard spheres in Equation 21. I t assumes that the ratio of c(r) tog(r) for hard spheres a t all densities is the same as the known expression for this ratio in a n ideal gas. This may be written as
c(r) = [l - e-U(‘)/”]g(r)
(23)
The Percus-Yevick approximation in Equation 23 allows a direct solution of Equation 21 for c(r) for hard-sphere fluid. This has been done analytically by Wertheim (745) and also by Thiele (735). The result of substituting this C ( T ) value into Equation 22 leads to a reduced equation of state for hard spheres in the form
where co is (N7rcr3/6) and N is Avogadro’s number. Because the solution obtained for c(r) is only approximate, its use in other thermodynamic equations leads to solutions slightly different from Equation 24. Equation 24, however, gives remarkably good agreement with Monte Carlo calculations and appears to be the best theoretically based equation for hard spheres obtained to date. I t also checks the result derived earlier by Reiss, Frisch, and Lebowitz (777) in a n entirely different manner. An important use of Equation 24 by Longuet-Higgins and Widom (75) was the development of a complete dense fluid equation of state in reduced form by coupling Equation 24 with a van der Waals attraction term to give
VOL. 6 0
NO. 7 J U L Y 1 9 6 8
19
O n e of the advantages of the CSP is that i t is not necessary to An empirical modification of the hard-sphere equation which adjusts for the slight discrepancy between the values predicted by Equation 24, which depends on the Percus-Yevick approximation, and the exact Monte Carlo calculations was developed by Ree and Hoover (707) in the form
the p2 term is small. McQuarrie and Katz also expanded the soft-sphere contribution in powers of l / n about the hard-sphere term. When ( l / n ) 2 terms are omitted the result is y
+ log,
+ 0.063507 ( b p ) + 0.017329 ( b p ) ’ ] 1 - 0.561493 ( b p ) + 0.081313 ( b ~ ) ~
“3
p*
(-)
T*
n
bZHS
-
bP*
(bp)[l +
zHs
(26) where b = (2 nNu3/3). Equation 26 is probably the most accurate simple equation currently available for the hard-sphere equation of state. An important extension of equations such as Equation 24 or Equation 26 was recently obtained by Rowlinson ( 7 78) who obtained an equation analogous to Equation 24 for “soft-sphere” molecules repelling each other with a (a/?), potential. The result has a form identical with Equation 24 but in which ca becomes temperature dependent
(,&,y’n + ;I);( 1
co = 6
7rd
F
[l
(27)
The function F ( e / k T ) has been evaluated by Rowlinson and at high temperatures approaches Euler’s constant y = 0.577216-----. For the extension of Equation 26 to soft spheres the “b” value in the equation may be replaced by 4 co with GO given in Equation 27. This procedure for soft spheres also may be extended to include attractive potentials in several ways. A theoretical method for this was developed by McQuarrie and Katz (83) who treat the attractive potential as a perturbation on the soft-sphere solution obtained from Equations 26 and 27. The method is to write the pure component configurational partition function as
where P = (l/kT). I n Equation 28 the exact pair potential has been divided into a repulsive portion zuo(rtf) plus the attractive portion,
Cut(r$,).
$
The exponential involving the at-
22.0, the value of VRiin both equations is set equal to the constant value 2.0. When VRiis 5 0 . 5 , the value of VRi in both equations is set equal to the constant value 0.5. The value of w1, the acentric factor for methane, was taken as 0.005. The equations were correlated for 0.6 5 TR 5 1.5. Data were insufficient to obtain a n accurate correlation of the shape factors a t T , values above about 1.5. The analytical form of the correlation is not valid a t high reduced temperatures and does not show the theoretically predicted constant value. This constant high temperature limit was arbitrarily assumed as the calculated shape factor at T , = 2.0. As shown in Figures 1 and 2 this limit is the value to be used at all higher temperatures. The assignment of constant values to VEiin Equation 77 correlates the two regions of density independence. These shape factors may also be related to a corresponding function +$, which modifies the critical pressure for the predicting of compressibility factors and fugacities of fluid i from the reference j
As a comparison between these shape factors and the theoretical factors based on pair interactions the factors for n-pentane relative to methane are plotted in Figures 1 and 2 as predicted by the Equations 77. Figures 1 and 2 show the density dependence by plotting the
parameters for VR = 1.5 as a comparison with the densityindependent results a t VR > 2.0 and
