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Calculation of thermodynamic properties from a modified Redlich-Kwong equation of state
The Chemical E&irzeering Journd, II (1976) 183-190 @ Elsevier Sequoia S. A., Lausanne. Printed in the Netherlands
Calculation of Thermodynamic Properties from a Modified Redlich-Kwong Equation of State H. HEDERER,
S. PETER and H. WENZEL
Institute fiir Technische Chemie II, Universitiit Erlangen-Niirnberg, Egerlandstrasse 3, 8520 Erlangen (F. R. G.) (Received 30 October 1975)
Abstract A method is presented to improve the calculation of thermodynamic properties and vapour-liquid equilibrium data for pure components and mixtures by means of the Redlich-Kwong equation of state. First, an alternative way is suggested to represent the temperature dependence of the parameter a in the RK equation. Second, a correlation is given for some homologous series to estimate the RK parameters of pure components solely from a knowledge of their boiling points. The method does not require critical data and is therefore applicable to systems containing components whose critical points are not available or cannot be measured easily.
stants, application of the equation to mixtures raises the problem of reliable mixing rules. A remarkable achievement was made by Redlich and Kwong’ in extending the van der Waals equation of state in the following way: RT ‘=v_b-@V(V+b)
a
(1)
where R is the gas constant, p is the pressure, V the molar volume and T the absolute temperature. a and b are two adjustable parameters which are found from critical data according to the expressions
a=
0.4278R2 T$’
b = O.O867RT,
PC The proposed mixing rules are
INTRODUCITON
Various techniques have been suggested in the literature to calculate vapour-liquid equilibria by means of an equation of state. The equation of state for this purpose must represent reasonably well not only the pressure-volume-temperature behaviour including the liquid region but also the vapour pressure of a pure substance over a wide range of temperature. In addition, reliable mixing rules for the parameters occurring in such an equation must be available. Van Laar made use of the well-known van der Waals equation of state as a basis for calculations of this type. The results obtained by this equation when used in its original form are generally inadequate. An improved equation of state containing eight constants was developed by Benedict, Webb and Rubin. For light hydrocarbons and in some other cases this equation yields good agreement between theory and experiment. However, to determine the eight constants, extensive experimental work is required and, in view of the large number of con-
a=5
+t
?X$+lij j=1
PC
aij = (a$ljj)3 (3)
b = 5 xtbi i=l
An advantage of Redlich and Kwong’s equation of state compared with that of Benedict, Webb and Rubin is the smaller number of adjustable parameters. Moreover, Shah and Thodos* found that the RK equation represents the p VT behaviour of hydrocarbons with about the same accuracy as is achieved with the BWR equation of state. Erbar and West3 came to similar conclusions. Wilson4 demonstrated that the RK equation of state yields a suitable basis for the calculation of thermodynamic properties such as vapour pressure, enthalpy and equilibrium concentrations between phases. The temperature dependence was accounted for by introducing the dimensionless expression aT-‘.‘b -‘R -’ as a function 183
184
H. HEDERER,
of temperature and of Pitzer’s’ acentric factor w. Parameter b was taken to be independent of temperature. Wilson’s suggestions were further developed by Joffe and Zudkevitch6. In a similar fashion Soave’ introduced the temperature dependence of a by multiplying the corresponding parameter found from Redlich’s original expression with a suitable factor which he was able to express as a function of the reduced temperature and Pitzer’s factor o. These suggestions to improve the RK equation by a temperature-dependent parameter a require the critical data for the substances in question. As the accuracy with which critical data are known is generally restricted and, for many substances of technical interest, these data cannot be measured because of thermal decomposition, the object was to develop a technique to introduce temperaturedependent RK parameters which would not require critical data. No attempt was made here to modify the mixing rules or to consider the parameter h as a function of temperature. AN ALTERNATIVE DEPENDENCE
APPROACH
OF PARAMETER
U+=UY
RT ~_ p= VP b
(41
3.0 I I ---i_
Fig. 1. Parameter
a+ as a function
of temperature.
__-a’lQ V(Vtb)
With b = 0 in the second term and cr = 0, this equation reduces to the van der Waals equation of state, whereas b = 0 and cr = - 1 yields Berthelot’s equation of state. (Y= -0.5 yields the original RK equation. For butane the value of 01taken from the slope of the relevant line in Fig. 1 is -0.5 1. This is very nearly the exponent assumed by Redlich and Kwong. However, for other substances 01deviates from this value depending on the molecular weight and the structure of the compound, as will be discussed later. For instance, for decane CY = -0.72. To calculate vapour-liquid equilibria adequately, the parameters of the modified RK equation must be fitted to the vapour pressures and liquid density of the pure substances. To increase the accuracy.
a
InT -
(51
where a and CYare constants for a specific substance. Therefore the RK equation may be modified to
FOR TIIE TEMPERATURE
fz* - V(I/
H. WENZEL
Parameters b and a+ were determined from vapour pressure and liquid density data at the same temperature. This procedure was carried out at a sequence of temperatures for hexane. The parameter b was found to be approximately independent of temperature, whereas the parameter a+ depends on temperature as shown in Fig. 1. In this plot the results obtained for some other substances are also given. These results suggest the following relationship for the temperature dependence of the parameter a+:
The RK equation may be written in the following form which does not specify the way in which the parameter a+ varies with temperature: RT p = 7-b
S. PETER,
6.0
CALCULATION
OF VAPOUR-LIQUID
EQUILIBRIA
185
these data should be selected from a temperature range in which vapour-liquid equilibria are to be calculated (generally between 0 “C and 200 “C). In particular, the parameters must be determined so that the following conditions are satisfied. (a) The liquid volume is given by eqn. (6). (b) Application of Maxwell’s integral relationship’ to eqn. (6) yields the vapour pressure at a given temperature. The resulting expression is
h(s)- b(v%-&)+&x
n-hexane n-butanol
x(ln($$$,)+b(&-&)) =R (c) The enthalpy the expression A&
= jpdV+
jj$$
V’
=p(V-
of vaporization
a = 986.12 12= 10876
b = 0.10851 b = 0.081419
a! = -0.5877 OL= -0.99209
These values were used to calculate thermodynamic properties such as vapour pressure, enthalpy of vaporization and density of both the vapour and liquid phases. These are listed in Tables 1 and 2 and compared with experimental data.
is described by
-p)dV=
V’
V)+(l
Thus the parameters for a particular substance are obtained by solving eqns. (6)-(g) for a, b and a simultaneously using ra method proposed by Powell” Suitable starting values for the trial-and-error calculation may be found either from critical data or from the general correlations given below in eqns. (9)-(20). Starting values found by the latter method usually yield more.reliable parameters and faster convergence. The parameters for n-hexane and n-butanol obtained by a fit to liquid density, vapour pressure and enthalpy of vaporization are
CORRELATION
-a)bln aF
~~7 V + b P” i
FOR RK PARAMETERS
FOR A
HOMOLOGOUS SERIES
(8) 1
The method discussed so far allows the evaluation of the RK parameters of a component from a knowledge of at least two vapour pressure points and the liquid
with V” and v’ being the vapour and liquid volumes respectively.
TABLE 1 n-Hexane Pressure
Boiling temperature (K)
Gas volume (1 mol-‘)
Liquid volume (1 mol-I)
__ 0.0616
0.1343 0.5294 1.1124 2.109 2.803 4.706 5.917 9.230 11.270 13.610 16.910 19.510 23.020 29.6
Enthalpy of vaporization (kcal mol-*)
(atm) expt 213.11
288.76 322.06 344.26 366.46 377.56 400.16 411.16 433.16 444.16 455.16 466.16 477.16 489.16 507.96
talc
% diff:
expt
talc
% diff:
expt
talc
%diff:
expt
talc
% diff:
273.91 289.03 321.93 344.30 367.09 378.49 401.69 413.54 437.51 449.61 461.75 476.67 487.09 499.77 -
-0.27
374.193 180.505 50.059 24.831
374.49 180.94 50.45 25.22 13.80 10.53 6.38 5.03 3.21 2.58 2.09 1.57 1.31 1.02 -
-0.08 -0.24 -0.78 -1.57 -2.29 -2.58 -3.55 -4.10 -5.52 -5.99 -6.66 -0.33 -6.49 -6.31 -
0.12726 0.12993 0.13622 0.14079 0.14639 0.14923 0.15612 0.16017 0.16948 0.17516 0.18163 0.19076 0.20161 0.21798 0.368
0.128 0.130 0.136 0.141 0.146 0.149 0.156 0.160 0.170 0.176 0.182 0.189 0.198 0.208 -
-0.30 -0.06 +0.25 +0.16 +0.24 +0.05 -0.19 -0.21 -0.36 -0.33 -0.28 +o.a7 +1.99 WI.43 -
7.894 7.6716 7.2022 6.8454 6.4619 6.2600 5.7916 5.5259 4.9161 4.5845 4.2105 3.9433 3.3071 2.7129
8.229 7.905 7.262 6.850 6.437 6.227 5.781 5.549 5.045 4.767 4.466 4.082 3.736 3.258 -
-4.24 -3.05 -0.83 -0.07 +0.38 +0.53 +0.18 -0.42 -2.63 -3.99 -6.07 -3.49 -12.96 -20.09 -
-0.09 +0.04 -0.01 -0.17 -0.25 -0.38 -0.58 -1.0 -1.23 -1.45 -2.255 -2.08 -2.17 _
13.493 10.262 6.160 4.834 3.041 2.438 1.956 1.568 1.232 0.956 0.368
Experimental data according to Landolt-BGrnstein.
H. HEDERER, S. PETER, H. WENZEL
186 TABLE 2 Butanol -~ Pressure (atm)
0.0014 0.0057 0.0247 0.0824 0.2256 0.5286 1.0331 1.1364 2.1489 3.8846 6.3537 9.8666 21.2830 29.3410 42.462 50.1
Boiling temperature (K)
expt
talc
273.17 293.16 313.16 333.16 353.16 373.16 390.86 393.16 413.16 433.16 453.16 473.16 513.16 533.16 553.16 560.26 ___
280.16 295.84 315.06 334.20 353.30 372.39 389.86 392.55 412.10 433.12 453.15 473.45 515.67 536.41 444.07 ..-
% diff:
-
Liquid volume (1 mol-‘)
Gas volume (1 mot-‘) expt
-2.56 17073.3 -0.92 4359.2 -0.61 1223.4 -0.31 342.48 ~-0.04 132.31 0.21 59.547 0.26 31.755 0.15 29.952 0.26 15.777 0.01 8.562 0.002 5.211 -0.06 3.4756 -0.49 1.5048 -0.61 0.98593 19.72 0.52758 0.2748 -~ -____
talc 17002.2 4353.5 1075.3 341.80 131.73 58.955 31.258 28.523 15.565 8.776 5.418 3.4656 1.4999 1.00279 0.17303 _
-
% diff. 0.42 0.13 12.11 0.20 0.44 0.99 1.56 4.77 1.35 -2.50 -3.97 -0.29 0.33 -1.71 67.20 _
Enthalpy of vaporization (kcal mol-4)
expt
talc
% diff:
expt
talc
% diff:
0.0899 0.0918 0.0931 0.0896 0.0967 0.0987 0.1006 0.1010 0.1037 0.1063 0.1126 0.1200 0.1401 0.1527 0.1830 0.2748
0.0900 0.0915 0.0933 0.0953 0.0976 0.1002 0.1029 0.1033 0.1089 0.1111 0.1162 0.1225 0.1407 0.1549 0.1730 ~
--0.04 0.26 --0.21 -6.41 -0.90 -1.57 -2.29 -2.29 -3.11 -4.51 -3.19 -2.04 -0.45 ml.49 5.46
15.28 14.62 12.05 11.78 11.29 11.04 10.47 10.36 9.84 9.27 8.67 7.41 6.68 5.71 3.33 _
15.91 14.72 13.68 12.74 11.89 11.10 10.45 10.36 9.65 8.94 8.23 7.50 5.82 4.81
-4.13 -0.75 -6.42 -8.19 -5.28 -0.61 0.20 -0.02 1.95 3.54 5.04 ~1.14 12.75 15.84
_
Experimental data according to Landolt-Barnstein.
volume. This procedure is recommended here when such data are available. However, in some cases the only information about a substance may be a vapour pressure datum point, e.g. the normal boiling point. In this case, the relationships described below will yield a convenient method for estimating the RK parameters. It was found that a plot of the parameters II, b and a! versus normal boiling point for a particular group of chemical substances results in a set of smooth curves. For instance, a nearly linear relationship was found for the alkanes when the logarithm of the parameters was plotted against the normal boiling point (Figs. 2 and 3). The series of cycloalkanes and alkylcycloalkanes yields a curve which is displaced slightly downwards from that of the alkanes. The homologous series of alkynes, alkenes, esters, ethers and ketones show a similar behaviour. The scattering of the parameters is over about the same range as that of deviations for the corresponding experimental values of latent heat, vapour pressure and pVT data. Alkunes
In a = 0.928259 + 0.17391186 x lo4 T, ln b = -3 645972 + 0 4210417 x l(T’? o= -0:04236749 lo.15757636 X l+
(9) (10) TS (11)
The average deviation is 2.22% for a, 1.45% for b and 0.87% for (Y. Alkenes
Ina=1.118322t0.16624554x10-‘T, In b = -3.744294 t 0.44205376 x 10d2 T, cy= -0.0897079 - 0.14212048 x 1O-2 T,
(12) (13) (14)
The average deviation is 0.46% for a, 0.68% for b and 0.26% for (IL. Equations (9)-(14) are restricted to substances with molecular weight greater than 60. Alkynes Ina= 1.1816141 + 0.1695205 x IO-‘T, In b = -4.7860861 + 0.71849595 X 10°2TS ar= -0.36881162 - 0.6929789 X 10 -3TS
05)
(16) (17)
The average deviation is 11% for a, 3% for b and 3.2% for o. The procedure is also valid when the parameters are plotted against the boiling points at pressures lower than 1 atm. This is shown in Fig. 4 for the parameters of alkanes and alkenes at 20 mmHg. The method is valuable in cases where the volatility of the substances is very low and the normal boiling point is not available.
* Cycloalkones
-_1 :
187
-r-
d, lsoalkanes 2~
q
Alkenes
, Alkmeb
300
Fig. 2. Parameters
LOO
a and b vs. boiling
T
‘S 1760mmt+$
point.
- 7.2
- 0.8 ci
t
- 0,6
0 100 Fig.
3. Parameter
200 (L w. boiling
300 point.
LOO TV (760mnHg)
500
600
K
188
H. HEDERER,
S. PETER,
H. WENZEL
lo5 I5 D
Alkenes
v
Alklnes
24
,o"l_
I
5, cl, t 2L
I
103,
O0
b
51~
I 2 :-
IO2
5
600
K
Ts(20mmHgl
Fig. 4. Parameters
a and b of alkanes
and alkenes
vs. boiling
temperature
A comparison of calculated and experimental values for n-hexane with RK parameters evaluated from Fig. 4 is given in Table 3. The interpolated parameters differ from the parameters obtained from experimental data by t3 04% for a, - 1.19% for b and +1.3% for CY.The reproduction of experimental data obtained with these correlated parameters is nearly as good as that achieved when the parameters determined from experimental data are used. As an example, for isoalkanes the expressions for the parameters at 20 mmHg are In a = 1.9474828 + 0.18822162 X 10-l T, (18) In b = -3.7421169 + 0.60577347 X lo-‘T, (19) OL= -0.28607734 - 0.10805363 x 1O-2 Ts (20)
at 20 mmHg.
The average deviation of the correlated parameters from parameters obtained from experimental data is 5.4% for a, 2.2% for b and 1.9% for 01.
APPLICATION
TO PHASE EQUILIBRIUM
CALCULATIONS
IN MIXTURES
As mentioned previously, methods to calculate phase equilibria using the RK equation of state have been described by several authors’? &*. These calculations involve the use of mixing rules. The modified RK equation of this work requires no alteration to the mixing rules commonly used (eqns. (4) and (5)).
CALCULATION
OF VAPOUR-LIQUID
189
EQUILIBRIA
TABLE 3
n-Hexane Pressure
Boiling temperature (K)
Gas volume (1 mol-‘)
expt
talc
% diff:
expt
talc
273.61
274.17
-0.36
374.193
314.47
322.06 366.46 400.16 433.16 455.16 477.16 489.16
322.53 368.06 402.96 439.12 463.60 489.18 501.99
-0.15 -0.44 -0.70 -1.37 -1.85 -2.52 -2.62
50.06 13.49 6.16 3.04 1.956 1.23 0.96
Liquid volume (I mol-‘)
Enthalpy o{vaprization (kcalmol-
(atm)
0.0616 0.5294 2.109 4.706 9.23 13.61 19.51 23.02
so.43 13.79 6.36 3.19 2.067 1.28 0.98
)
% diff:
expt
talc
%diffi
expt
talc
% diff:
-0.08
0.12726
-0.75 -2.18 -3.31 -5.0 -5.73 -4.31 -2.16
0.13622 0.14639 0.15612 0.16948 0.18163 0.20161 0.21798
0.12916 0.13743 0.14762 0.15612 0.17155 0.18342 0.19850 0.20881
-1.49 -0.89 -0.84 -1.19 -1.22 -0.99 +I.54 +4.21
7.894 7.202 6.462 5.792 4.916 4.210 3.307 2.713
8.193 7.241 6.426 5.776 5.043 4.464 3.724 3.219
-3.80 -0.54 +0.55 +0.27 -2.60 -6.02 -12.59 -18.67
Experimental data according to Landolt-Bknstein. Calculated data with the parameters from Fig. 4: a = 956.12, b = 0.1098, CI= -0.580.
60
In the present case b = 2 Xtbt i=l a+ = 5 2 i=l
L50
(21)
XiXiaG
j=l
a; = (1 - e)(azai)
X0
+
:30
for i # j, with aA= aiPi. .20
Phase equilibria may be described on the basis of a single set of parameters a, b and LYover a large temperature range when the modified RK equation is used. This is demonstrated in Fig. 5 which shows a temperature-concentration diagram of the toluenephenol system at 1 atm. The dotted line refers to a calculation based on the original RK equation. Phase equilibria of the 20s -squalane system are of interest in chromatography studies using fluids at elevated pressures. Experimental information on squalane is sparse. To determine the RK parameters for this substance, the only thermodynamic quantity available with sufficient accuracy was the boiling point at 24 mmHg. When used in subsequent phase equilibrium calculations, the parameters for squalane found from the correlation expression for isoalkanes (eqns. (18)-(20)) yield the result represented in Fig. 6. The agreement between calculated and experimental values confirms the validity of the method.
:10
LOO
390
.
l..
,
SYSTEM A,, A22=
X.Y -
0.5
0 =
6We.7 2172.9
TOUENE
- PHEML
P.760
B, = 0.09199
PL, = -0.51570
02=omoo3
AL2= -0.67727
TOFU7
Fig. 5. Temperature-concentration diagram of the toluene-phenol system.
190
H. HEDERER, S. PETER, H. WENZEL SC0 0
-
EXP
CALC
Y
0 =0.029 $00
r.___. I SOTHERM
89 7 All= A22= 103727.8
r05
-------
CO2 - SOUALANE
X,Y Tz
100
100
Superscripts I liquid I, gas
1.
373.9
0.02730
Al ,= - 055689
E2=
0.51960
AL*=
- 0.82886
diagram of the carbon
The authors wish to thank Deutsche Forschungsgemeinschaft for financial support.
NOMENCLATURE
NV
: T
REFERENCES
K
ACKNOWLEDGMENT
a a.+ b
Greek symbols modified KK parameter (exponent) ; interaction parameter
--Q
E, =
Fig. 6. Pressure-concentration dioxide-squalane system.
molar volume, 1 mol- ’ molal liquid concentration, mol mol-’ molal gas concentration, mol mol-’
Subscripts C critical number of components i, i n total number of components s boiling temperature
300
P- X-
V X
FCKparameter, atm l2 Ki modified RK parameter, atm I2 Kn RK parameter, 1 mol-’ molal heat of vaporization, kcal mol-r pressure, atm gas constant, 0.084784 1 atm mol-’ K-’ absolute temperature, K
0. Redlich and J.N. S. Kwong, Chem. Rev., 44 (1949) 233. K. K. Shah and G. Thodos,Ind. Eng. Chem., 57 (3) (1965) 30. J. H. Erbar and E. H. West, Proc. 52nd Annu. Conv. of the NGPA, 1973, Natl. GasProc. Assoc., Tulsa, Okla., 1973, pp. 50-61. G. M. Wilson, Adv. Cryogenic Eng., 9 (1964) 168-76; 11 (1966) 392-400. K. S. Pitzer, D. Z. Lippmann, R. F. Curl, C. M. Huggins and D. E. Petersen, J. Am. Chem. Sot., 77 (1955) 3427-33. J. Joffe and D. Zudkevitch, 4.I.Ch.E. J., 16 (1) (1970) 112-119; J. Joffe, G. M. Schroeder and D. Zudkevitch, A.I. Ch.E. J., 16 (3) (1970) 496-98. G. Soave, Chem. Eng. Sci., 27 (6) (1972) 1197-1203. S. Peter and H. Wenzel. Chem. Inn. Techn., 43 (15) (1971) 856-861. J. C. Maxwell, On the dynamical evidence of the molecular constitution of bodies, Nuture (London), I I (1875) 357-77. 10 M. J. D. Powell, Comput. J., 7 (2) (1964) 155; in Ph. Rabinowitz (ed.), Numerical Methods for Nonlinear Algebraic Equations, Gordon and Breach, London, 1970, Chap. 7, pp. 115-161.
Calculation of Thermodynamic Properties from a Modified Redlich-Kwong Equation of State H. HEDERER,
S. PETER and H. WENZEL
Institute fiir Technische Chemie II, Universitiit Erlangen-Niirnberg, Egerlandstrasse 3, 8520 Erlangen (F. R. G.) (Received 30 October 1975)
Abstract A method is presented to improve the calculation of thermodynamic properties and vapour-liquid equilibrium data for pure components and mixtures by means of the Redlich-Kwong equation of state. First, an alternative way is suggested to represent the temperature dependence of the parameter a in the RK equation. Second, a correlation is given for some homologous series to estimate the RK parameters of pure components solely from a knowledge of their boiling points. The method does not require critical data and is therefore applicable to systems containing components whose critical points are not available or cannot be measured easily.
stants, application of the equation to mixtures raises the problem of reliable mixing rules. A remarkable achievement was made by Redlich and Kwong’ in extending the van der Waals equation of state in the following way: RT ‘=v_b-@V(V+b)
a
(1)
where R is the gas constant, p is the pressure, V the molar volume and T the absolute temperature. a and b are two adjustable parameters which are found from critical data according to the expressions
a=
0.4278R2 T$’
b = O.O867RT,
PC The proposed mixing rules are
INTRODUCITON
Various techniques have been suggested in the literature to calculate vapour-liquid equilibria by means of an equation of state. The equation of state for this purpose must represent reasonably well not only the pressure-volume-temperature behaviour including the liquid region but also the vapour pressure of a pure substance over a wide range of temperature. In addition, reliable mixing rules for the parameters occurring in such an equation must be available. Van Laar made use of the well-known van der Waals equation of state as a basis for calculations of this type. The results obtained by this equation when used in its original form are generally inadequate. An improved equation of state containing eight constants was developed by Benedict, Webb and Rubin. For light hydrocarbons and in some other cases this equation yields good agreement between theory and experiment. However, to determine the eight constants, extensive experimental work is required and, in view of the large number of con-
a=5
+t
?X$+lij j=1
PC
aij = (a$ljj)3 (3)
b = 5 xtbi i=l
An advantage of Redlich and Kwong’s equation of state compared with that of Benedict, Webb and Rubin is the smaller number of adjustable parameters. Moreover, Shah and Thodos* found that the RK equation represents the p VT behaviour of hydrocarbons with about the same accuracy as is achieved with the BWR equation of state. Erbar and West3 came to similar conclusions. Wilson4 demonstrated that the RK equation of state yields a suitable basis for the calculation of thermodynamic properties such as vapour pressure, enthalpy and equilibrium concentrations between phases. The temperature dependence was accounted for by introducing the dimensionless expression aT-‘.‘b -‘R -’ as a function 183
184
H. HEDERER,
of temperature and of Pitzer’s’ acentric factor w. Parameter b was taken to be independent of temperature. Wilson’s suggestions were further developed by Joffe and Zudkevitch6. In a similar fashion Soave’ introduced the temperature dependence of a by multiplying the corresponding parameter found from Redlich’s original expression with a suitable factor which he was able to express as a function of the reduced temperature and Pitzer’s factor o. These suggestions to improve the RK equation by a temperature-dependent parameter a require the critical data for the substances in question. As the accuracy with which critical data are known is generally restricted and, for many substances of technical interest, these data cannot be measured because of thermal decomposition, the object was to develop a technique to introduce temperaturedependent RK parameters which would not require critical data. No attempt was made here to modify the mixing rules or to consider the parameter h as a function of temperature. AN ALTERNATIVE DEPENDENCE
APPROACH
OF PARAMETER
U+=UY
RT ~_ p= VP b
(41
3.0 I I ---i_
Fig. 1. Parameter
a+ as a function
of temperature.
__-a’lQ V(Vtb)
With b = 0 in the second term and cr = 0, this equation reduces to the van der Waals equation of state, whereas b = 0 and cr = - 1 yields Berthelot’s equation of state. (Y= -0.5 yields the original RK equation. For butane the value of 01taken from the slope of the relevant line in Fig. 1 is -0.5 1. This is very nearly the exponent assumed by Redlich and Kwong. However, for other substances 01deviates from this value depending on the molecular weight and the structure of the compound, as will be discussed later. For instance, for decane CY = -0.72. To calculate vapour-liquid equilibria adequately, the parameters of the modified RK equation must be fitted to the vapour pressures and liquid density of the pure substances. To increase the accuracy.
a
InT -
(51
where a and CYare constants for a specific substance. Therefore the RK equation may be modified to
FOR TIIE TEMPERATURE
fz* - V(I/
H. WENZEL
Parameters b and a+ were determined from vapour pressure and liquid density data at the same temperature. This procedure was carried out at a sequence of temperatures for hexane. The parameter b was found to be approximately independent of temperature, whereas the parameter a+ depends on temperature as shown in Fig. 1. In this plot the results obtained for some other substances are also given. These results suggest the following relationship for the temperature dependence of the parameter a+:
The RK equation may be written in the following form which does not specify the way in which the parameter a+ varies with temperature: RT p = 7-b
S. PETER,
6.0
CALCULATION
OF VAPOUR-LIQUID
EQUILIBRIA
185
these data should be selected from a temperature range in which vapour-liquid equilibria are to be calculated (generally between 0 “C and 200 “C). In particular, the parameters must be determined so that the following conditions are satisfied. (a) The liquid volume is given by eqn. (6). (b) Application of Maxwell’s integral relationship’ to eqn. (6) yields the vapour pressure at a given temperature. The resulting expression is
h(s)- b(v%-&)+&x
n-hexane n-butanol
x(ln($$$,)+b(&-&)) =R (c) The enthalpy the expression A&
= jpdV+
jj$$
V’
=p(V-
of vaporization
a = 986.12 12= 10876
b = 0.10851 b = 0.081419
a! = -0.5877 OL= -0.99209
These values were used to calculate thermodynamic properties such as vapour pressure, enthalpy of vaporization and density of both the vapour and liquid phases. These are listed in Tables 1 and 2 and compared with experimental data.
is described by
-p)dV=
V’
V)+(l
Thus the parameters for a particular substance are obtained by solving eqns. (6)-(g) for a, b and a simultaneously using ra method proposed by Powell” Suitable starting values for the trial-and-error calculation may be found either from critical data or from the general correlations given below in eqns. (9)-(20). Starting values found by the latter method usually yield more.reliable parameters and faster convergence. The parameters for n-hexane and n-butanol obtained by a fit to liquid density, vapour pressure and enthalpy of vaporization are
CORRELATION
-a)bln aF
~~7 V + b P” i
FOR RK PARAMETERS
FOR A
HOMOLOGOUS SERIES
(8) 1
The method discussed so far allows the evaluation of the RK parameters of a component from a knowledge of at least two vapour pressure points and the liquid
with V” and v’ being the vapour and liquid volumes respectively.
TABLE 1 n-Hexane Pressure
Boiling temperature (K)
Gas volume (1 mol-‘)
Liquid volume (1 mol-I)
__ 0.0616
0.1343 0.5294 1.1124 2.109 2.803 4.706 5.917 9.230 11.270 13.610 16.910 19.510 23.020 29.6
Enthalpy of vaporization (kcal mol-*)
(atm) expt 213.11
288.76 322.06 344.26 366.46 377.56 400.16 411.16 433.16 444.16 455.16 466.16 477.16 489.16 507.96
talc
% diff:
expt
talc
% diff:
expt
talc
%diff:
expt
talc
% diff:
273.91 289.03 321.93 344.30 367.09 378.49 401.69 413.54 437.51 449.61 461.75 476.67 487.09 499.77 -
-0.27
374.193 180.505 50.059 24.831
374.49 180.94 50.45 25.22 13.80 10.53 6.38 5.03 3.21 2.58 2.09 1.57 1.31 1.02 -
-0.08 -0.24 -0.78 -1.57 -2.29 -2.58 -3.55 -4.10 -5.52 -5.99 -6.66 -0.33 -6.49 -6.31 -
0.12726 0.12993 0.13622 0.14079 0.14639 0.14923 0.15612 0.16017 0.16948 0.17516 0.18163 0.19076 0.20161 0.21798 0.368
0.128 0.130 0.136 0.141 0.146 0.149 0.156 0.160 0.170 0.176 0.182 0.189 0.198 0.208 -
-0.30 -0.06 +0.25 +0.16 +0.24 +0.05 -0.19 -0.21 -0.36 -0.33 -0.28 +o.a7 +1.99 WI.43 -
7.894 7.6716 7.2022 6.8454 6.4619 6.2600 5.7916 5.5259 4.9161 4.5845 4.2105 3.9433 3.3071 2.7129
8.229 7.905 7.262 6.850 6.437 6.227 5.781 5.549 5.045 4.767 4.466 4.082 3.736 3.258 -
-4.24 -3.05 -0.83 -0.07 +0.38 +0.53 +0.18 -0.42 -2.63 -3.99 -6.07 -3.49 -12.96 -20.09 -
-0.09 +0.04 -0.01 -0.17 -0.25 -0.38 -0.58 -1.0 -1.23 -1.45 -2.255 -2.08 -2.17 _
13.493 10.262 6.160 4.834 3.041 2.438 1.956 1.568 1.232 0.956 0.368
Experimental data according to Landolt-BGrnstein.
H. HEDERER, S. PETER, H. WENZEL
186 TABLE 2 Butanol -~ Pressure (atm)
0.0014 0.0057 0.0247 0.0824 0.2256 0.5286 1.0331 1.1364 2.1489 3.8846 6.3537 9.8666 21.2830 29.3410 42.462 50.1
Boiling temperature (K)
expt
talc
273.17 293.16 313.16 333.16 353.16 373.16 390.86 393.16 413.16 433.16 453.16 473.16 513.16 533.16 553.16 560.26 ___
280.16 295.84 315.06 334.20 353.30 372.39 389.86 392.55 412.10 433.12 453.15 473.45 515.67 536.41 444.07 ..-
% diff:
-
Liquid volume (1 mol-‘)
Gas volume (1 mot-‘) expt
-2.56 17073.3 -0.92 4359.2 -0.61 1223.4 -0.31 342.48 ~-0.04 132.31 0.21 59.547 0.26 31.755 0.15 29.952 0.26 15.777 0.01 8.562 0.002 5.211 -0.06 3.4756 -0.49 1.5048 -0.61 0.98593 19.72 0.52758 0.2748 -~ -____
talc 17002.2 4353.5 1075.3 341.80 131.73 58.955 31.258 28.523 15.565 8.776 5.418 3.4656 1.4999 1.00279 0.17303 _
-
% diff. 0.42 0.13 12.11 0.20 0.44 0.99 1.56 4.77 1.35 -2.50 -3.97 -0.29 0.33 -1.71 67.20 _
Enthalpy of vaporization (kcal mol-4)
expt
talc
% diff:
expt
talc
% diff:
0.0899 0.0918 0.0931 0.0896 0.0967 0.0987 0.1006 0.1010 0.1037 0.1063 0.1126 0.1200 0.1401 0.1527 0.1830 0.2748
0.0900 0.0915 0.0933 0.0953 0.0976 0.1002 0.1029 0.1033 0.1089 0.1111 0.1162 0.1225 0.1407 0.1549 0.1730 ~
--0.04 0.26 --0.21 -6.41 -0.90 -1.57 -2.29 -2.29 -3.11 -4.51 -3.19 -2.04 -0.45 ml.49 5.46
15.28 14.62 12.05 11.78 11.29 11.04 10.47 10.36 9.84 9.27 8.67 7.41 6.68 5.71 3.33 _
15.91 14.72 13.68 12.74 11.89 11.10 10.45 10.36 9.65 8.94 8.23 7.50 5.82 4.81
-4.13 -0.75 -6.42 -8.19 -5.28 -0.61 0.20 -0.02 1.95 3.54 5.04 ~1.14 12.75 15.84
_
Experimental data according to Landolt-Barnstein.
volume. This procedure is recommended here when such data are available. However, in some cases the only information about a substance may be a vapour pressure datum point, e.g. the normal boiling point. In this case, the relationships described below will yield a convenient method for estimating the RK parameters. It was found that a plot of the parameters II, b and a! versus normal boiling point for a particular group of chemical substances results in a set of smooth curves. For instance, a nearly linear relationship was found for the alkanes when the logarithm of the parameters was plotted against the normal boiling point (Figs. 2 and 3). The series of cycloalkanes and alkylcycloalkanes yields a curve which is displaced slightly downwards from that of the alkanes. The homologous series of alkynes, alkenes, esters, ethers and ketones show a similar behaviour. The scattering of the parameters is over about the same range as that of deviations for the corresponding experimental values of latent heat, vapour pressure and pVT data. Alkunes
In a = 0.928259 + 0.17391186 x lo4 T, ln b = -3 645972 + 0 4210417 x l(T’? o= -0:04236749 lo.15757636 X l+
(9) (10) TS (11)
The average deviation is 2.22% for a, 1.45% for b and 0.87% for (Y. Alkenes
Ina=1.118322t0.16624554x10-‘T, In b = -3.744294 t 0.44205376 x 10d2 T, cy= -0.0897079 - 0.14212048 x 1O-2 T,
(12) (13) (14)
The average deviation is 0.46% for a, 0.68% for b and 0.26% for (IL. Equations (9)-(14) are restricted to substances with molecular weight greater than 60. Alkynes Ina= 1.1816141 + 0.1695205 x IO-‘T, In b = -4.7860861 + 0.71849595 X 10°2TS ar= -0.36881162 - 0.6929789 X 10 -3TS
05)
(16) (17)
The average deviation is 11% for a, 3% for b and 3.2% for o. The procedure is also valid when the parameters are plotted against the boiling points at pressures lower than 1 atm. This is shown in Fig. 4 for the parameters of alkanes and alkenes at 20 mmHg. The method is valuable in cases where the volatility of the substances is very low and the normal boiling point is not available.
* Cycloalkones
-_1 :
187
-r-
d, lsoalkanes 2~
q
Alkenes
, Alkmeb
300
Fig. 2. Parameters
LOO
a and b vs. boiling
T
‘S 1760mmt+$
point.
- 7.2
- 0.8 ci
t
- 0,6
0 100 Fig.
3. Parameter
200 (L w. boiling
300 point.
LOO TV (760mnHg)
500
600
K
188
H. HEDERER,
S. PETER,
H. WENZEL
lo5 I5 D
Alkenes
v
Alklnes
24
,o"l_
I
5, cl, t 2L
I
103,
O0
b
51~
I 2 :-
IO2
5
600
K
Ts(20mmHgl
Fig. 4. Parameters
a and b of alkanes
and alkenes
vs. boiling
temperature
A comparison of calculated and experimental values for n-hexane with RK parameters evaluated from Fig. 4 is given in Table 3. The interpolated parameters differ from the parameters obtained from experimental data by t3 04% for a, - 1.19% for b and +1.3% for CY.The reproduction of experimental data obtained with these correlated parameters is nearly as good as that achieved when the parameters determined from experimental data are used. As an example, for isoalkanes the expressions for the parameters at 20 mmHg are In a = 1.9474828 + 0.18822162 X 10-l T, (18) In b = -3.7421169 + 0.60577347 X lo-‘T, (19) OL= -0.28607734 - 0.10805363 x 1O-2 Ts (20)
at 20 mmHg.
The average deviation of the correlated parameters from parameters obtained from experimental data is 5.4% for a, 2.2% for b and 1.9% for 01.
APPLICATION
TO PHASE EQUILIBRIUM
CALCULATIONS
IN MIXTURES
As mentioned previously, methods to calculate phase equilibria using the RK equation of state have been described by several authors’? &*. These calculations involve the use of mixing rules. The modified RK equation of this work requires no alteration to the mixing rules commonly used (eqns. (4) and (5)).
CALCULATION
OF VAPOUR-LIQUID
189
EQUILIBRIA
TABLE 3
n-Hexane Pressure
Boiling temperature (K)
Gas volume (1 mol-‘)
expt
talc
% diff:
expt
talc
273.61
274.17
-0.36
374.193
314.47
322.06 366.46 400.16 433.16 455.16 477.16 489.16
322.53 368.06 402.96 439.12 463.60 489.18 501.99
-0.15 -0.44 -0.70 -1.37 -1.85 -2.52 -2.62
50.06 13.49 6.16 3.04 1.956 1.23 0.96
Liquid volume (I mol-‘)
Enthalpy o{vaprization (kcalmol-
(atm)
0.0616 0.5294 2.109 4.706 9.23 13.61 19.51 23.02
so.43 13.79 6.36 3.19 2.067 1.28 0.98
)
% diff:
expt
talc
%diffi
expt
talc
% diff:
-0.08
0.12726
-0.75 -2.18 -3.31 -5.0 -5.73 -4.31 -2.16
0.13622 0.14639 0.15612 0.16948 0.18163 0.20161 0.21798
0.12916 0.13743 0.14762 0.15612 0.17155 0.18342 0.19850 0.20881
-1.49 -0.89 -0.84 -1.19 -1.22 -0.99 +I.54 +4.21
7.894 7.202 6.462 5.792 4.916 4.210 3.307 2.713
8.193 7.241 6.426 5.776 5.043 4.464 3.724 3.219
-3.80 -0.54 +0.55 +0.27 -2.60 -6.02 -12.59 -18.67
Experimental data according to Landolt-Bknstein. Calculated data with the parameters from Fig. 4: a = 956.12, b = 0.1098, CI= -0.580.
60
In the present case b = 2 Xtbt i=l a+ = 5 2 i=l
L50
(21)
XiXiaG
j=l
a; = (1 - e)(azai)
X0
+
:30
for i # j, with aA= aiPi. .20
Phase equilibria may be described on the basis of a single set of parameters a, b and LYover a large temperature range when the modified RK equation is used. This is demonstrated in Fig. 5 which shows a temperature-concentration diagram of the toluenephenol system at 1 atm. The dotted line refers to a calculation based on the original RK equation. Phase equilibria of the 20s -squalane system are of interest in chromatography studies using fluids at elevated pressures. Experimental information on squalane is sparse. To determine the RK parameters for this substance, the only thermodynamic quantity available with sufficient accuracy was the boiling point at 24 mmHg. When used in subsequent phase equilibrium calculations, the parameters for squalane found from the correlation expression for isoalkanes (eqns. (18)-(20)) yield the result represented in Fig. 6. The agreement between calculated and experimental values confirms the validity of the method.
:10
LOO
390
.
l..
,
SYSTEM A,, A22=
X.Y -
0.5
0 =
6We.7 2172.9
TOUENE
- PHEML
P.760
B, = 0.09199
PL, = -0.51570
02=omoo3
AL2= -0.67727
TOFU7
Fig. 5. Temperature-concentration diagram of the toluene-phenol system.
190
H. HEDERER, S. PETER, H. WENZEL SC0 0
-
EXP
CALC
Y
0 =0.029 $00
r.___. I SOTHERM
89 7 All= A22= 103727.8
r05
-------
CO2 - SOUALANE
X,Y Tz
100
100
Superscripts I liquid I, gas
1.
373.9
0.02730
Al ,= - 055689
E2=
0.51960
AL*=
- 0.82886
diagram of the carbon
The authors wish to thank Deutsche Forschungsgemeinschaft for financial support.
NOMENCLATURE
NV
: T
REFERENCES
K
ACKNOWLEDGMENT
a a.+ b
Greek symbols modified KK parameter (exponent) ; interaction parameter
--Q
E, =
Fig. 6. Pressure-concentration dioxide-squalane system.
molar volume, 1 mol- ’ molal liquid concentration, mol mol-’ molal gas concentration, mol mol-’
Subscripts C critical number of components i, i n total number of components s boiling temperature
300
P- X-
V X
FCKparameter, atm l2 Ki modified RK parameter, atm I2 Kn RK parameter, 1 mol-’ molal heat of vaporization, kcal mol-r pressure, atm gas constant, 0.084784 1 atm mol-’ K-’ absolute temperature, K
0. Redlich and J.N. S. Kwong, Chem. Rev., 44 (1949) 233. K. K. Shah and G. Thodos,Ind. Eng. Chem., 57 (3) (1965) 30. J. H. Erbar and E. H. West, Proc. 52nd Annu. Conv. of the NGPA, 1973, Natl. GasProc. Assoc., Tulsa, Okla., 1973, pp. 50-61. G. M. Wilson, Adv. Cryogenic Eng., 9 (1964) 168-76; 11 (1966) 392-400. K. S. Pitzer, D. Z. Lippmann, R. F. Curl, C. M. Huggins and D. E. Petersen, J. Am. Chem. Sot., 77 (1955) 3427-33. J. Joffe and D. Zudkevitch, 4.I.Ch.E. J., 16 (1) (1970) 112-119; J. Joffe, G. M. Schroeder and D. Zudkevitch, A.I. Ch.E. J., 16 (3) (1970) 496-98. G. Soave, Chem. Eng. Sci., 27 (6) (1972) 1197-1203. S. Peter and H. Wenzel. Chem. Inn. Techn., 43 (15) (1971) 856-861. J. C. Maxwell, On the dynamical evidence of the molecular constitution of bodies, Nuture (London), I I (1875) 357-77. 10 M. J. D. Powell, Comput. J., 7 (2) (1964) 155; in Ph. Rabinowitz (ed.), Numerical Methods for Nonlinear Algebraic Equations, Gordon and Breach, London, 1970, Chap. 7, pp. 115-161.