A Four-Parameter Extension of the Theorem of Corresponding States Eugene A. Harlacher’ and Walter G. Braun Department of Chemical Engineering, The Pennsylvania State University, University Park, Pa.
16802
The usefulness of the theorem of corresponding states has been extended to include polar substances. The parachor is used as a fourth parameter to characterize molecular size-shape; the acentric factor, used as the third parameter, is a measure of molecular size-shape, polarity, quantum influence, and hydrogen bonding. These two parameters were combined empirically in a polynomial expansion to separate molecular size-shape effects from the association effects. The effectiveness of this separation was demonstrated by using this four-parameter theorem to generalize the Frost-Kalkwarf vapor pressure equation. Vapor pressures may be predicted from a knowledge of only the critical temperature, critical pressure, acentric factor, and parachor. An average deviation of 4.92% was obtained for 5952 vapor pressure data points representing 242 substances.
CHEMICAL
ENGINEERS must deal with many types of fluids in the development, design, and operation of chemical plants. I t is essential to know the physical and thermodynamic properties of these fluids as accurately as possible in order to determine equipment sizes, energy requirements, equilibrium yields, and separation ratios. Furthermore, this information can be best utilized in the areas of optimization and process control if it is represented as an analytical function of a process variable. For these reasons accurate correlations of these properties are becoming increasingly important. Although many correlations are available, there is very little uniformity among them. Different correlating parameters are required depending on the property being correlated and, in some instances, the temperature or pressure range represented. Many of these correlations were developed specifically for one substance by empirically determining specific coefficients for a given substance. The generalized correlations which use the same characterizing parameters for all properties and conditions are based on the theorem of corresponding states.
Corresponding State Extensions
The early work of van der Waals used the critical temperature and pressure to characterize a substance. According to the simple theorem of corresponding states, two substances a t equal reduced conditions should behave identically. This has been shown to be valid only for spherical, noninteracting (simple) substances; however, Riedel (1954), Pitzer et al. (1955), Su (1946), and Lydersen et al. (1955) added a third parameter to the theorem I Present address, Research and Development Department, Continental Oil Co., Ponca City, Okla. 74601
to correct it for nonidealities due to molecular size-shape. The result was a considerable improvement in the theorem for nonpolar substances. The most popular third parameter was the Pitzer et al. acentric factor, W . The acentric factor is used to correct the value of a physical property for a simple fluid:
G = G ” +wG’
(1)
where G is any correlatable property, G o is a simple fluid property, and G is a correction term. Basically, the acentric factor is a measure of vapor pressure deviations from the simple fluid, caused by sizeshape, quantum, bonding, and association effects. Because the acentric factor approach works so well for nonpolar substances (where the interaction effects are negligible in contrast to size-shape effects), the next logical extension of the theorem should be to add a fourth parameter to correct only for the interaction effects. This fourth parameter should be used in an extended form of Equation 1. This approach was followed by Eubank and Smith (1962), who defined a fourth parameter as a function of dipole moment; such a parameter has been previously criticized by Curl and Pitzer (1958) and Riedel (1956). Equation 1, with an additional term, was used in their approach. The appropriate acentric factor for a polar substance was that of its hydrocarbon homomorph. This restricted their method essentially to organic substances. Halm and Stiel (1967) recognized these restrictions and presented a correlation for vapor pressure and entropy of vaporization of polar fluids by adding a fourth parameter based on a vapor pressure measurement. Their method improves predictions only for the substances for which their fourth parameter has been empirically determined. Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970 479
S
Their technique lacks a theoretical basis and does not separate size-shape and interaction effects. A most noteworthy contribution was made by Thompson (1966), who used the fundamentally satisfying approach of divorcing the size-shape effects from the polarity effects. He defined an association factor, 7, and "true" acentric factor, 0, such that -
w =
0 4
(2)
w + 7
2-
0 2 a 0 >
w.
+
w
G'
I
4
The "true" size-shape parameter, 0,was correlated with the molecular radius of gyration and the association factor was then obtained as the difference between w and The theorem as extended by Thompson was expressed as -
G = G'O
I
I
e n-PARAFFINS A I-OLEFINS
+ TG" + W7G3
BENZENE
3
AEyE "/"I
h. 0
-
u)
3
(3)
9
K
The only drawback to the use of this extended form is the difficulty involved in obtaining the characterizing parameters, 0 and 7. The radius of gyration must first be obtained, either from the literature or from a tedious calculation, then 0 and 7 must be obtained from a graphical correlation. Because of the good results obtained by Thompson and his fundamentally sound approach, his method was used as a springboard for this work.
2
Mao7 YeNHz
I
%.IC1
I
I
I
PARACHOR, P
Correlational Development
A characterizing parameter was sought that would be a good indicator of molecular size-shape, but would not require lengthy, difficult calculations. The parameter used here was the parachor, which can be shown to be a measure of comparison between liquid molar volumes at equal surface tensions. The liquid molar volume has been shown (Harlacher, 1968) to be a function of the radius of gyration; hence, the parachor should be a good characterizing parameter. Furthermore, the value for the parachor for any substance may be easily obtained from an accurate group contribution method formulated by Quayle (1953). Figure 1 shows the excellent correlation between the radius of gyration, taken from Thompson, and the parachor for several substances. Consequently, the parachor was plotted against the acentric factor, according to the method used by Thompson, for several substances (Figure 2). The straight line was arbitrarily drawn on Figure 2 as the locus of zero association effects, 7 = 0. If this line is stated analytically
Figure 1. Relation of parachor to radius of gyration
e n- PARAFFINS A I-OLEFINS 0.5
-
0.4
-
2
0.3
-
0.2
-
3
a 0 u I-
uK !E U
a
I
IO0
0.0
w=aP+b
(4)
-_ T = W -
(5)
w
and substituting Equations 4 and 5 into 3, the following is obtained:
G = G o + aPG'
+
(W
- U P )G'
+ aP(w - uP)G'
(6)
which can be simplified to
G = G o + PG"+wG2
+ PwG3'-
P2G3" (7)
where G ' = G ' - G 2 , Gi3 = a G 3 , and G 3 = a L G 3 . This work, then, uses a polynomial expansion of parachor and the acentric factor to predict the correlatable properties. The recommended values of the four correlating parameters (pc,T,, W, and P ) , for all substances used in this work, are listed in Table I. (Tables I and I1 have been deposited with the ASIS.) 480
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970
I 300
400
PARACHOR, P
-
The intercept, b , is zero because the line passes through the origin. Recalling the definition of tau:
200
Figure 2. Relation of parachor to acentric factor
The advantage of this form is that w does not have to be actually separated into 0 and T ; the separation is accomplished internally in the expansion. The fact that may not the relationship between the parachor and be linear, as implied by Equation 4, does not invalidate this reasoning, so long as the relationship can be expressed in an analytical series. The correlation which follows uses the polynomial expansion to predict the coefficients to a nongeneralized correlation. This in effect generalizes the correlation in terms of the four-parameter theorem of corresponding states presented above. Vapor Pressure
Vapor pressures, using the theorem of corresponding states as extended in this work, were correlated to test
the validity of this extension. Many equations have been given relating the vapor pressure to the absolute temperature, all of which can be traced back to the rigorous Clapeyron equation.
Coefficient Z I has been found by Reynes and Thodos (1962a,b) and Thompson to be the same for all substances; its value can be determined by recalling Frost and Kalkwarf’s definition:
D‘ = a R2 The relationship between I)‘ and D is The simplest vapor pressure equation can be obtained by integrating the Clapeyron equation using three assumptions: A H > a p is a constant, independent of temperature, the liquid volume is negligible, and the vapor obeys the ideal gas law. The resulting equation is the familiar ClausiusClapeyron equation
lnp* = A - B / T
(9)
which predicts a linear relationship between the logarithm of the vapor pressure and the reciprocal of the absolute temperature. Thodos (1950) has shown, however, that real substances exhibit a slightly S-shaped curve when the logarithm of the vapor pressure is plotted against the reciprocal temperature. Any vapor pressure correlation which is to represent real substances must take this S-curvature into account (Reid and Shenvood, 1966). An excellent vapor pressure correlation was developed by Frost and Kalkwarf (1953) by assuming that the heat of vaporization is a linear function of temperature, the volumetric behavior of the gas is adequately described by van der Waals equation of state, and the molar volume of the liquid is approximated by van der Waals constant, b . When these assumptions are introduced into the Clapeyron equation (Equation 8), the following form of the FrostKalkwarf equation results after integration:
By using the relation between the critical properties and van der Waals’ constant, a (Hougen et al., 1959):
the value of D is obtained:
p,a D=-T,‘R2
B‘ - C’ In T T
+ D’
P*
-T
T
(-T1 - -)T1 ,
-
64
-- 0.4218
A=B-D
(18)
The values of B , C, and T , result as the independent variables which define the reduced vapor pressure. Before a polynomial expansion of parachor and acentric factor could be fitted to coefficients B and C, accurate values of these coefficients were required. These values were obtained for 242 substances by fitting vapor pressure data to the Frost-Kalkwarf equation by a least-squares procedure. This was facilitated by linearizing the FrostKalkwarf equation by substituting Equation 18 into Equation 13 and rearranging to get .
E)
Pr*
=
B
(4
- 1) In T,
+c
(19)
Equation 19 is now in the form of a straight line where B is the slope and C the intercept of the line when X‘ equals zero:
(11)
where A ’ , B’, and C’ are empirical constants unique for each substance and D’ = a / R 2 . The Frost-Kalkwarf equation predicts the S-curvature of real substances. The Frost-Kalkwarf equation may be put into reduced form by writing Equation 11 twice-once for any vapor pressure and once for the critical point --and subtracting one from the other with the following results:
l n p * - l n p , = -B’
27 _ -
The value of A may be determined from the other coefficients by applying Equation 13 a t the critical point:
In T,
In p ” = A’ -
27 R2T? 64 p.
a=
- In pr* - D ( 1 or, in the more familiar form
(14)
1
The value of X‘ does not approach zero until very high values of the reduced temperature are attained ( X ’ -+ 0 as T , + m ) . Because the concept of vapor pressure vanishes at the critical point, the data used to determine the intercept will necessarily stop at T , = 1.0. Thus, a long. undesirable extrapolation of the data would be required to determine the value of C by this method. It would be much more appropriate to define the intercept at the critical point. By applying L’Hospital’s rule, the value of X ’ is determined at the critical point to be -1.0. Equation 19 may be changed simply by adding 1.00 to X‘ to yield
Rearranging:
B - Cln T,+ D P: lnp: = A - T,
T12
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 3, 1970 481
This new parameter, X ,goes to zero a t the critical point, and Equation 19 becomes
or
Y=BX+E
(23)
where E is the value of Y a t the critical point (X = 0) and is related to the Frost-Kalkwarf coefficient, C, by
E=C-B
(24)
Y and X are defined by Equations 20 and 21, respectively. The final form of the Frost-Kalkwarf vapor pressure equation used in this work for the correlation of vapor pressure is
2
In pr* = B 1 - - - ( E + B ) In T , +
i
0.4218
- 1)
(25)
T o generalize the vapor pressure correlation, the values of B and E , determined according to the procedure described above, were fitted to a polynomial expansion of the parachor and acentric factor by means of a least squares technique. The following equations are the result of the generalization:
+ 8.30918 x P+ 8.87139~- 5.586785 x P2+ 5.173538 x lo-* PO- 3.06570’
= 6.545986
(26)
and = -4.14342
+ 1.13843 x
1.223881 x
P - 4.0796~P2+ 1.2214605 x lo-’ PW- 1.348060’
(27)
Bond and Thodos (1960), Pasek and Thodos (1962), Perry and Thodos (1952), Reynes and Thodos (1962a,b), Smith and Thodos (1960), and Sondak and Thodos (1956) have published tables of the values of B and C obtained by direct fit of vapor pressure data, but Table I1 is the most comprehensive list available. Two values each are listed for B and E in Table 11; the column labeled “best” refers to the values obtained by fitting vapor pressure data to Equation 22, whereas the “predicted” values are obtained from Equations 26 and 27. To evaluate the vapor pressure correlation, coefficients B and E were used in Equation 25 to calculate values of vapor pressure which were then compared to actual data. Table I11 summarizes the results of this comparison according to families of substances. A more detailed comparison is given by Harlacher (1968). These results show the truly excellent fit of the Frost-Kalkwarf equation to almost 6000 data points, with an average deviation of only 0.61%. The large deviations obtained for the aromatics when using the calculated coefficients are due to poor results of only six of the aromatic substances: 1,2,3-, 1,2,4-, 482
Table Ill. Evaluation of Vapor Pressure Correlation by Families No. of Substances
In T ,
E
~
__ ____
-- . - -
B
~~
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970
Hydrocarbons Paraffin s N aphthenes Olefins Diolefins-alkynes Aroma tics Nonh ydrocarbons Over-all
Average Deviation”
Data Points
Direct fit
Predicted
144
1488 208 571 87 604 2994
0.20 0.18 0.27 0.41 1.02 0.77
2.65 9.76 3.29 9.94 18.28 3.71
242
5952
0.61
4.92
43 9 23 5
18
calculated value - experimental value x 100% Deviation = ___--experimental value Av. deviation =
2 1 deviation I No. of points
and 1,3,5-trimethylbenzene and 1,2,3,4-, 1,2,3,5-, and 1,2,4,5-tetramethylbenzene.These deviations should be attributed to poor critical constants for these six substances. An indication of the improvement in accuracy gained by the addition of a fourth parameter to the theorem of corresponding states is given in Table IV, where the results, as given by Thompson, of comparisons between two existing correlations and the vapor pressure data are shown. The Reynes-Thodos method uses the FrostKalkwarf equation but requires a vapor pressure data point (usually the normal boiling point) as an input parameter. The Riedel (1954) method uses a third parameter, ( Y k , which is related to the acentric factor. The results of these comparisons are given according to a range of tau values. The range of tau = 0 to 0.2 includes all hydrocarbons (adequately correlated by three-parameter methods), whereas substances having a value of tau greater than 0.2 are the associative and polar substances. These results show that the correlation presented here improves predictions of vapor pressure for substances exhibiting associative tendencies; no improvement was made for nonassociative substances. This simply shows that the extended theorem presented here properly reverts back to the three-parameter method when dealing with a nonassociative substance. Even though tau is not used explicitly in this work, it is a convenient concept. I t aids in classifying substances according to their association tendencies. I n the early stages of development it was found that the slope of the straight line in Figure 1
Table I V . Vapor Pressure Correlation Results Compared to Other Correlations Average Deviationn
~-
Tau
Substances
Points
This work
ReynesThompson Thodos
0-0.2 0.2-0.3 0.3-0.4 > 0.4
210 13 7 12
5028’ 297 321 306
4.87 6.10 4.89 4.58
4.51 5.18 4.27 6.11
4.44 6.95 8.41 5.71
4.63 7.88 8.93 5.70
All
242
5952‘
4.92
4.61
7.04
7.52
Riedel
“Defined in Table 111. ‘5024 for Thompson (1966), Reynes and Thodos (1962a,b), and Riedel (1954, 1956). ‘5948 for Thompson (1966), Reynes and Thodos (1962a,b), and Riedel (1954, 1956).
(7 = 0) was approximately 0.001. Thus, a rough approximation of tau may be obtained by combining Equations 4 and 5, with this slope, to obtain
7
= w - 0.001
P
(28)
For example, a rough approximation of the value of tau for ethanol would be T
= 0.625 - 0.001(127) = 0.498
which clearly indicates the strong associative tendencies of this substance. Conclusions
The correlational approach used here separates the molecular size-shape effects from the association effects according to a method discussed by Thompson and Braun (1968). This approach offers an improvement over Thompson’s fourth parameter (radius of gyration) by using a much more tractable parameter, the parachor. I t also eliminates the graphical techniques required in Thompson’s method. The four-parameter extension to the theorem of corresponding states substantially improves vapor pressure correlations for associative substances. Expectedly, it gives the same results as .three-parameter correlations for normal substances. Because of the fundamentally sound approach of this extension, it should give equally good results when used for correlations of other physical and thermodynamic properties. Nomenclature
G = m.,p
=
P = p* = pc = pr =
R T T, T,
= = = =
x = v = Y = ffh
=
7
=
w = w =
constants in vapor pressure equation van der Waals constants; slope and interceDt of straight line property correlatable within the framework of the theorem of corresponding states latent heat of vaporization, Btu/ Ib parachor vapor pressure, psia critical pressure, psia reduced pressure gas law constant, consistent units absolute temperature, R critical temperature, R reduced temperature defined by Equation 21 volume defined by Equation 20 Riedel third parameter association factor Pitzer acentric factor size-shape factor
literature Cited
Bond, D. L., Thodos, G., J . Chem. Eng. Data 5 , 289 (1960). Curl, R. F., Jr., Pitzer, K. S.,Ind. Eng. Chem. 50, 265 (1958). Eubank, P. T., Smith, J. M., A.I.Ch.E. J . 8, 117 (1962). Frost, A. A., Kalkwarf, D. R., J . Chem. Phys. 21, 264 (1953). Halm, R . L., Stiel, L. I., A.I.Ch.E. J . 13, 351 (1967). Harlacher, E . A., “A Four-Parameter Extension of the Theorem of Corresponding States,” Ph.D. thesis, The Pennsylvania State University, 1968. Hougen, 0. A., Watson, K. M., Ragatz, R. A., “Chemical Process Principles,” 2nd ed., p. 561, Part 11, Wiley, Kew York, 1959. Lydersen, A. L., Greenkorn, R. A,, Hougen, 0. A., “Generalized Thermodynamic Properties of Pure Fluids,” University of Wisconsin, Madison, Wis., Eng. Expt. Sta., Rept. 4 (October 1955). Pasek, G. J., Thodos, G., J . Chem. Eng. Data 7, 21 (1962). Perry, R. E., Thodos, G., Id. Eng. Chern. 44, 1649 (1952). Pitzer, K. S., Lippmann, D. Z . , Curl, R. F., Jr., Huggins, C. H., Petersen, D. E., J . Amer. Chem. Soc. 77, 3433 (1955). Quayle, 0. R., Chem. Reus. 53, 439 (1953). Reid, R . C., Sherwood, T . K., “The Properties of Gases and Liquids,” 2nd ed., p. 115, McGraw-Hill, New York, 1966. Reynes, E . G., Thodos, G., A.I.Ch.E. J . 8, 357 (1962a). Reynes, E . G., Thodos, G., Ind. Eng. Chem. Fundam. 1, 127 (1962b). Riedel, L., Chem.-Ing.-Tech. 26, 83 (1954). Riedel, L., Chem.-lng.-Tech. 28, 557 (1956). Smith, C. H., Thodos, G., A.I.Ch.E. J . 6, 569 (1960). Sondak, N. E., Thodos, G., A.I.Ch.E. J . 2, 347 (1956). Su, G. J., Ind. Eng. Chem. 38, 803 (1946). Thodos, G., Ind. Eng. Chem. 42, 1514 (1950). Thompson, W. H., “A Molecular Association Factor for Use in the Extended Theorem of Corresponding States,” Ph.D. thesis, The Pennsylvania State University, 1966. Thompson, W. H., Braun, W. G., Proc. A P I 48, 477 (111) (1968). RECEIVED for review June 26, 1969 ACCEPTED March 13, 1970
For supplementary material, order NAPS Document 00936 from ASIS National Auxiliary Publications Service, c/o CCM Information Sciences, Inc., 22 West 34th St., New York, N.Y., 10001, remitting $1.00 for microfiche and $3.00 for photocopies. Financial support provided by the National Aeronautics and Space Administration in the form of a predoctoral traineeship for E. A. Harlacher.
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 3, 1970 483
