Ind. Eng. Chem. Res. 1993,32, 241-244
241
CORRELATIONS
New Acentric Factor Correlation Based on the Antoine Equation
Daniel H.Chen*
Department of Chemical Engineering, Lamar University, Beaumont, Texas 77710
Murty V. Dinivahi
Chemtex Environmental Lab, Port Arthur, Texas 77642
Chang-Y uan Jeng
Enertech Engineering Go., Taipei, Taiwan, R.O.C.
The Edmister method for the estimation of acentric factor, a widely used parameter in thermodynamic
and transport property correlations, is modified. The new method requires only two vapor pressure
data points for each compound (the normal boiling point and the critical point) as in the methods
of Edmister and Lee-Kesler while employing the three-parameter Antoine vapor preasure correlation.
The average absolute error is 3.69% based on the most recent literature data (498 compounds). This
compares favorably with the Edmister method (5.10%) and the LeeKesler method (7.09%). A
correlation is also developed to relate the third parameter C/T, in the Antoine equation with 8 (i.e.,
Tb/ Tc)*
Daubert, 1983); (h) equations of state (Soave, 1972; Peng
and Robinson, 1976).
A fair amount of acentric factor data have been accuThe acentric factor (Pitzer, 1955; Pitzer et al., 1955) is
mulated over the years, especially for hydrocarbons (Paasut
an important physical constant used in calculating the
and Danner, 1973; Henry and Danner, 1978; Ambrose,
physical properties of pure compounds as well as mixtures,
1980). However, there are many instances where experiespecially in the corresponding-states type correlations.
mentally determined acentric factor values are not availThe acentric factor, w , is defined as
able. For example, the vapor pressure has not been
(1)
0 = -log(Pvp)rlTr=O.,
- 1.00
measured near/at T,= 0.7 or the critical constants are
simply not available. Under these circumstances, the
where (P,), = P,/P,; P, = vapor pressure, atm; T,=
acentric
factor can not be obtained from eq 1. As a result,
T/T,,
reduced temperature; T = absolute temperature, K
the estimation methods must be used to facilitate the
T,= critical temperature, K; and P, = critical pressure,
above-mentioned calculations (a)-(h).
atm. As can be seen, critical constants and the vapor
Most literature methods for estimating the acentric
pressure data at T,= 0.7 are needed to determine the
factor require the normal boiling point (Tb),critical conacentric factor.
stanta (T,,
PJ, molecular weight (MW),and specific gravity
The acentric factor is supposed to represent the acen(SG).The method of Hoshino et al. (1982) based on group
tricity or nonsphericity of a molecule. For example, the
contributions is an exception. Table I provides more inw values of argon, methane, and ethane are very small
formation (input data needed, percent deviation (% dev),
(0.001,0.012, and 0,099, respectively). The value of w
number of compounds used, and comments) about these
increases with carbon chain length (0.907 for n-eicosane)
predictive methods. All the % dev values in Table I were
and generally rises with increasing polarity (0.644 for
given by the references listed under the comment column.
ethanol). Actually some large w values (e.g., for alcohols)
Note that the methods based on SG, Tb,and MW (Lin and
are more closely related to polarity than acentricity.
Chao, 1984; Watanasiri et al., 1985; Roman et al., 1986)
The acentric factor values are widely used in estimating
can be used to predict w for coal-liquid or petroleum
thermodynamic and transport properties for gases and
fractions. Because the input values can be estimated by
liquids. Examples of these applications are given as folgroup contributions, the methods based on Tb,T,,and P,
lows: (a) compressibility factor (Pitzer, 1955; Pitzer et d.,
are still applicable even when some or all of the input data
1955; Edmister, 1958; Lee and Kesler, 1975); (b) heat ca(experimental) are missing.
pacity of real gases (Lee and Kesler, 1975); (c) enthalpy
In this work an empirical correlation based on the Anof vaporization (Nath, 1979); (d) aaturated density or molar
toine equation is developed (eq 15; see later derivation).
volume of liquids (Rackett, 1970; Thompson et al., 1982);
The new correlation can predict the acentric factor more
(e) vapor pressure of pure liquids (Lee and Kesler, 1975);
accurately than the Edmister and the LeeKesler methods
(f) liquid heat capacity (Rowlinson, 1969); (g) saturated
while utilizing essentially the same input information.
liquid viscosity (Letsou and Stiel, 1973; Danner and
0SSS-5SS5/93/2632-0241~~4.O0/0
0 1993 American Chemical Society
Introduction
242 Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993
Development of Correlation
Edmister (1958) proposed an equation to correlate the
acentric factor with the normal boiling point and critical
constants:
w=--
e
log P,- 1
7 1-8
where 8 = Tb/Tc. Tb = normal boiling point, K; T,, P, =
as defined in eq 1. Equation 2 is based on the Clapeyron
equation:
In Pv = A - B / T
(3)
in which two data points are needed to specify the constants A and B.
A similar relationship was reported by Lee and Kesler
(1975):
w = (-ln P, - 5.92714 + 6.096486' + 1.28862 In 8 0.16934786)/(15.2518 - 15.68758-1 - 13.4721 In 8 +
0.4357786) (4)
where P,, T,, 8 = as defined in eqs 1 and 2. Equation 4
is obtained from the following vapor pressure relation (eqs
5-7):
the Pitzer expansion:
ln(P,),
= po)(Tr)+ wfc*)(T,)(5)
The Lee and Kesler functions are
Po'(Tr) =
5.92714 - (6.O9648/Tr) - 1.28862 In T, +0.169347T,6
(6)
<
o.60
1
0.40
1
0.20
4
* 0 8 * * Exp.
Calc
~
0.00 i
-0.20
"
1
4
-0.40 4
-0.60 A
- 0 80
-
- 1 00 f
0.55
I
I
0.60
0.65
0.70
I
I
0.75
0.80
0.85
Theta, Tb/Tc
Figure 1. Linear relationship between C / T , and 8.
1.00
0 90
1
/
1
"
/
080
0
70
0060
c
u 030
fc')(Tr) =
15.2518 - (15.6875/Tr) - 13.4721 In T, +0.435777',6
(7)
by setting P = 1 atm at T = T,,.
Antoine (&8) proposed a vapor pressure correlation
similar to eq 3,
B'
In Pvp= A'- (8)
T+C
which can be written in a slightly different form,
B
log(P,), = A - (9)
T+C
where A = A'/2.3026 - log P,; B = B'12.3026. Usually, at
least three vapor pressure data points are needed to completely determine the parameters A, B,and C.
Q
/(
0.10
t
X
Figure 2. Acentric factor values as calculated by the proposed
correlation vs experimental data.
(eq 2) is a special case of eq 13 when CIT, is set to 0.
Given the acentric factor, critical temperature, critical
pressure, and the boiling temperature, C / T , can be determined for a compound by eq 13. Thus obtained C/T,
was regressed against the known 8 for each compound to
determine a functional relationship,
C/T, = 0.2803 - 0.52iie
By substituting eq 12 into eq 1, the acentric factor can be
expressed as
0.3(e + c/T,) log P,
w =
(13)
(1- 8 ~ 0 . 7+ C/TJ
In this work, the remaining parameter C/T, is determined
by a linear function of 8. Note that the Edmister method
(14)
based on 217 compounds over the widest possible range
of 8. Quite a few compounds (e.g., many with 8 close to
0.7) were outliers and were excluded from the regression.
Equation 14 represents the best general trend for C / T ,
versus 8 regardless of compound classes. As can be seen
in Figure 1,there is a good agreement between the C/T,
values obtained from Tb, T,, P,, w data and the ones
predicted by eq 14.
In eq 13, C/T, is eliminated in favor of 8 to obtain the
following empirical correlation:
0"
0.3(0.2803 + 0.47898) log P,
-1
(1 - e)(0.9803 - 0.52iie)
(15)
where 8,P, = as defined in eqs 1 and 2. If P,is given in
kilopascala, the conversion factor needed is 1 kPa =
0.009869 atm (or 1 atm = 101.325 kPa).
Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993 243
Table I. Literature Information on the Acentric Factor Prediction Methods
no. of
comments
method
input data 9% d e 9 compds
based on a wide variety of compounds and a vapor pressure correlation; Edmister
Edmister
Tb,Tc,pc
6.0
277
(1958), Yaws et al. (1989)
based on a wide variety of compounds; Lee and Kesler (1975), Lin and Chao (1584)
20.5
133
Lee and Kesler Tb,Tc,Pc
based on hydrccarbons and heterocyclic compounds; Watanasiri et al. (1985)
94
Watanasiri et al. Tb,MW, SG 11.8
based on hydrocarbons and derivatives; Lin and Chao (1984)
MW, SG, Tb
8.5
133
Lin and Chao
based on hydrocarbons and heterocyclic compounds; Roman et al. (1986)
SG, Tb
14.0
140
Roman et al.
applicable to alkanes only; Hoshino et al. (1982)
molec struct
2.2
59
Hoshino et al.
90’ dev: percent average absolute relative deviation.
Table 11. Comparison of Acentric Factor Predictions for
Different ComDound Classes
av abs. 7
’0 error
no. of
Leeclass
compds Edmister Kesler proposed
acids
5
4.77
6.16
5.00
20
1.11
3.11
0.50
alcohols
6.31
4.27
11
4.28
aldehydes
24
5.34
6.86
3.88
amines
31
2.45
4.67
1.63
esters
26
3.10
5.10
1.60
ethers and epoxides
86
4.50
6.24
2.69
halides
207
6.18
8.58
4.63
hydrocarbons
14
5.39
6.61
5.20
ketones
4.00
4.52
0.47
10
mercaptans
5
1.70
3.31
0.99
nitriles
12
2.34
6.22
3.77
silaiies
47
7.28
8.21
5.38
heterocyclic compds
~~
total
~~
498
5.10
7.09
3.69
Data Sources
The acentric factor data from literature (Passut and
Danner, 1973; Henry m d Damer, 1978; Nath, 1979; Kesler
et al., 1979; Ambrose, 1980; Lin et al., 1980a,b; Daubert
and Danner, 1985; WaeanaSiri et al., 1985; Reid et al., 1987)
were screened and collected in an ASCII data base. The
data base includes acentric factor vaules for 498 organic
chemicals, covering hydrocarbons, acids, alcohols, aldehydes, amines, esters, ethers and epoxides, halides, ketones,
nitriles, silanes, and heterocyclics. The critical constants
data used in comparison were obtained from Ambrose
(1980),Daubert and Danner (1985),Simmrock et al. (1986),
Reid et al. (1987), and Yaws et al. (1989).
Results and Discussion
The acentric factor values calculated by eq 15 have been
compared to the literature acentric factor data, Figure 2.
An excellent agreement can be noted. Comparison of each
compound class is also given in Table 11. The average
absolute error is 3.69% based on 498 compounds as compared to 5.10% by Edmister’s and 7.09% by LeeKesler’s
methods.
Even though the Clapeyron equation involves assumptions of constant enthalpy of vaporization (AHv) and
constant compreasibility factor change of vaporization
(A&), the Edmister method is surprisingly robust. Our
experience has been that the Edmister method is more
reliable than the Lee-Kesler method in terms of extrapolating to new compounds. The dependence of C / T , on
t9 (eq 14) is indispensable and contributes to a genuine
modification to the Edmister method. Other vapor pressure correlations which involve more parameters, e.g., the
Wagner (1973) equation, may give better correlations than
eq 15 but will require more input data.
In cases where experimental values of t9 (i-e., Tb/Tc)and
P,are not available, the proposed method is still applicable
by estimating Tb, T,,and P, from group contribution
methods such as the Joback (1984) method.
Conclusion
A new correlation bared on the Anmiiie wpx pitai+,.ie
equation has been developed for the eu’cmaxion of aca,imc
factor. The new correlation couipwes fafoiaoiy vrith tne
Edmister and the Lee-Kesler methods w-hile miiizi.ig me
same input information: the noriial boiling pomt, critical
temperature, and critical pressme. A cor,elatiun is aiao
developed to relate the third parainecer C/T,iii tile AItoine equation with t9 (Le., Tb/Tc). A floppy dabs ai&
containing acentric factor, critical C O M ’ ~ & T ~n, o r a d b o i i x
point, and specific gravity data in ASCII forlnat for 498
compounds is available from the authors.
Literature Cited
Ambrose, D. “Vapor-Liquid Critical Properties”; NTL I i e p ~ ~CEcm.
r.
107; National Physical Laboratory, Division of Chexical Stmdards: Teddington, England, February 1980.
Antoine, C. Compt. Rend. 1888,107, 681, 836.
Danner, R. P.; Daubert, T. E. Data Prediction Manual; DI??X,
AIChE: New York, 1983.
Daubert, T. E.; Danner, R. P. Data Compilation: Tabies of i + o p
erties of Pure Compounds;American Institute of Chemical Engineers: New York, 1985.
Edmister, W. C. Applied Hydrocaibon Thermodpanics, Pa-t 4:
Compressibility Factors and Equations of State. Pet. Refin. 1938,
37, 173.
Henry, W. P.; Danner, R. P. Revised Acextric Factor Vaices. Ind.
Eng. Chem. Process Des. Deu. 1978, I7 (3), 373.
Hoshino, D.; Nagahama, K.; Hiiata, M. Prediction of Acentric Factor
of Alkanes by the Group Contribution Method. J. Chcm. Eng.
Jpn. 1982,15 (21, 153.
Joback, K. G. A Unified Approach to Physical Prope5y Estixa’mn
Using Multivariate Statistical Techniques. M.S. Thesis in Chamical Engineering, Massachusetts Institute of Technology, 1984.
Kesler, M.; Lee, B. I.; Sandler, S. I. A Third Parameter for Use in
Generalized Thermodynamic Correlations. Ind. Eng. Chem.
Fundam. 1979, 18 (l),49.
Lee, B. I.; Kesler, M. G. A Generalized Thermodj-naxic Cor:eiatwn
Based on Three-Parameter Corresponding States. AIChE J. 1975,
21 (31, 510.
Letsou, A.; Stiel, L. I. Viscosity of Saturated Nonpolar Liquids at
Elevated Pressure. AZChE J . 1973, 19, 409.
Lin, C. T.; Brule, M. R.; Young, F. K.; Starling, K. E.; Chao, J. Data
Bank for Synthetic Fuels Part 2-Properties for 161 One- and
Two Ring Coal Chemicals. Hydrocarbon Process. 19SOa, Aug,
117.
Lin, C. T.; Brule, M. R.; Young, F. K.; Starling, K. E.; Chao, J. Data
Bank for Synthetic Fuels Part 3-Properties’ Constants for 137
More Coal Chemicals. Hydrocarbon Process. 1980b,Nov, 225.
Lin, H. M.; Chao, K. C. Correlation of Critical Properties and
Acentric Factor of Hydrocarbons and Derivatives. AIChE J . 1984,
30 (6), 981.
Nath, J. Acentric Factor and the Heats of Vaporization for Unnssociated Polar and Nonpolar Organic Liquids. Ind. Eng. Chem.
Fundam 1979,8 (3), 297.
Pasaut, C. A,; Danner, R. P. Acentric Factor. A Valuable Correlating
Parameter for the Properties of Hydrocarbons. Ind. Eng. Chem.
Process Des. Deu. 1973, 13 (3), 365.
244 Ind. Eng. Chem. Rea., Vol. 32, No. 1, 1993
Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of
State. Ind. Eng. Chem. Fundam. 1976,15,59.
Pitzer, K. S. The Volumetric and Thermodynamic Properties of
Fluids. I. Theoretical Basic and Virial Coefficients. J. Am.
Chem. SOC.1955, 77,3427.
Pitzer, K. S.; Lippmann, D. M.; Curl, R. F.; Husgins, C. M.; Peterson,
D. E. The Volumetric and Thermodynamic Properties of Fluids.
11. Compressibility Factor, Vapor Pressure and Entropy of Vaporization. J. Am. Chem. SOC.1956, 77, 3433.
Rackett, H.G. Equation of State for Saturated Liquids. J. Chem.
Eng. Data 1970, 15, 514.
Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases
& Liquids, 4th ed.; McGraw-Hill: New York, NY, 1987.
Roman, C. E.; Luther, J. E.; Wetzel, D. M. Estimation of Properties
of Coal Liquids. Energy Prog. 1986,4 (4), 235.
Rowlinson, J. S. Liquids and Liquid Mixture, 2nd ed.; Butterworth
London, 1969.
Simmrock, K. H.;Janowaky, R.; Ohnsorge, A. Critical Data of Pure
Substances. In DECHEMA Chemistry Data Series: Frank-
a
furt/Main, Germany, 1988; Vol. 11, Parte 1 and 2.
Soave, G. Equilibrium Constante from a Modified Redlich-Kwong
Equation of State. Chem. Eng. Sci. 1972,27,1197.
Thompson, G. H.;Brobst, K. R.; Hankinson, R. W. An Improved
Correlation for Densities of Compressed Liquids and Liquid
Mixtures. AZChE J . 1982,28,671.
Wagner, W. Cryogenics 1973,13, 470.
Watanaeiri, S.; Owens, V. H.;Starling, K. E. Correlations for Estimating Critical Constante, Acentric Factor, and Dipole Moment
for Undefined Coal-Fluid Fractions. Znd. Eng. Chem. Process
Des. Deu. 1985, 24 (21, 294.
Yaws, C. L.; Chen, D. H.;Yang, H.C.; Tan, L.; Nico, D. Critical
Properties of Chemicals. Hydrocarbon Process. 1989, July, 61.
Received for review June 8, 1992
Revised manuscript receiued October 19, 1992
Accepted October 28, 1992