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Vapor compositions in binary orqanic systems from p-t-x data coupled with liquid excess properties
0364-5916/89 $3.00 t .OO (c) 1989 Pergamon Press plc
CALPIIAD Vol. 13, No. 4, pp. 401-411, 1989 Printed in the USA.
VAWR ColWoB1T10N8 It0BIlIARYORGAMIC BYBTBx8 BROH P-T-x DATA COUPLED WITH LIQUID EXCBBB PROPERTIES
Paul lC.TALLEY and Christopher 1. BALE CENTRE FOR RESEARCH INTO COMPUTATIONAL TRERMOCHEMISTRY ECOLE POLYTECHNIQUE Box 6079, Station A, Montreal, Quebec, Canada, H3C 3A7
A new reduction method is proposed for obtaining liquid activity coefficients and vapor compositions from PTx data in organic systems. One can also include enthalpies of mixing, Foxy data and activity coefficients at infinite dilution in the treatment, thus providing a complete description of the excess Gibbs energy of the liquid phase over a wide range of conditions. The results for the following five binary systems are presented: ethanol + chloroform: n-propylbenzene + methyl+ cyclohexane; I,l-dichlorobenzene + 1,4-dibromobenzene; methanol N,N-diethylformamide; ethanol + acetone. In principle, the reduction method can be extended to volatile inorganic and metallic systems where precise total pressure data are available.
1.
B
Much of the liquid-vapor and solid-vapor equilibrium data now available record only total pressure, temperature and condensed phase compositions (PTx) because these measurements are the easiest to obtain. For design purposes in chemical engineering, a knowledge of the vapor compositions (y) as a function of these variables is essential. It is possible to obtain this information from PTx data by applying a reduction method provided that the thermodynamic properties of the gas phase are well known. Several reduction methods have been proposed in the literature (Marsh, 1977; Tao, 1961; Christian, 1960; Barker, 1953). In the present article, a new method is proposed which permits the reduction of FTx data at several temperatures and yields a set of descriptive parameters applicable over wide ranges of pressure, temperature and composition. It is possible to include heats of mixing, PTxy data and activity coefficients at infinite dilution in the analysis. As a result, one is then able to generate isobaric phase diagrams and interpolate or extrapolate to temperatures and pressures different from the experimental values. ____-_______________------. Received 5 June 1989
401
402
P.K. TALLEY and C.W. BALE
2.
Theorv
The criteria for equilibrium of a component i in a mixture between a condensed phase and a vapor at temperature T and total pressure p is: YiQP
111
= xi7if;a
where yi and xi are the vapor and condensed phase compositions, 49 is the vapor phase fugacity coefficient, 7i is the activity coefficient of component i in the condensed phase and r;c is the fugacity of pure i in the condensed phase. The PC term is not a directly measurable quantity, but it can be estimated by assuming: f;C
9i(sat)
fl
pi(sat)
where 4i (sat) is the saturation fugacity coefficient and pi (sat) is the saturation (vapor or sublimation) pressure of component i. By combining Eqs. [l-2] and rearranging we obtain: p Jli
=
ziTidi
(sat) pi WW/df
as an expression for the partial pressure of component i. The fugacity coefficients in Eq. [3] can be calculated from an equation of state or an ideal gas may be assumed by setting both coefficients equal to unity. In this case, the virial equation of state is The Tsonopoulos (1974, 1975) truncated at the second coefficient. empirical correlation is used to estimate the second virial coeffiThe saturation pressures can be calculated from standard cients. expressions such as the Antoine equation (as is done here) or by using experimental measurements. In the case of PTx measurements, it follows that there are two values to be calculated: yi and 7i. The present method requires that an expression for the condensed phase activity coefficients be available. This expression should be flexible enough to analytically represent the activity coefficients of highly nonideal solutions over a wide The excess Gibbs range of pressure, temperature and composition. energy of the condensed phase is represented by the polynomial expression: gB= x1x2 E (ai + biT + aiT 1nT) Pi(x2 - x1) where ai, bi and Ci are adjustable parameters polynomial of order i and argument (x2-x1). preferred over Redlich-Kister or simple power (Bale and Pelton, 1974).
['I
and Pi is a Legendre Legendre polynomials are series type expansions
Expressions for the other excess properties can be obtained by the appropriate differentiation of Eq. [4]. hB =
x1x2
C
(ai
-
aiT)
Pi(x2 - x1)
151
403
VAPOR COMPOSITIONS IN BINARY ORGANIC SYSTEMS
2 t
C
-x1x2
(bi + ai(l + In T)) Pi(x2 - x1)
c; = -
x1x*
c
aiPi(x*
-
[61
x,1
c73
The expressions for the excess chemical potentials can also be from Eq. 143. R
91 a~xi
E (ai + biT + aiT 1nT) (Pi(X2-X1)- 2m1Pi(x2-xl))
E g2 = xt C
(ai
+ bill
+ o&T
InT)(Pi(x2-xl)+ 2x2Pi(x2-xl))
derived
Cal c91
where Pj '(x2-x1) is the first derivative with respect to (X2-X1) of the Legendre polynomial of order i. It is important to note that the coefficients in Eqs.(4-71 are the same. This enables one to treat partial properties of both components and integral properties simultaneously.Eqs.fS-91 are used extensively in the treatment of metallurgical systems and have been employed by the authors for representing the thermodynamic properties of mixtures (Bale and Pelton, 1974, 1983a; Pelton and Bale, 1977) including alloys (Bale and Pelton 1983b), molten salts (Pelton et al. 1983) and organic compounds (Talley et al., 1987; Sangster et al., 1988). The total pressure is the sum of partial pressures, which from Eq. 133 for a binary system is: P = xz71pl(sat) )I(sat)/df + X272~2(sat) 92[satl/9:
1101
Upon rearrangement and expansion of Eq. [lo], we obtain: RT [I - "x(72 - ln71) - 92(72 - lm72)l = Qlgl E + a2g:
El11
where Qi = xipi fsat)
di
Eqs. [S-9] are substituted into the give:
tsatl/#$ right-hand-side of
[=I
Eq.
[ll] to
RT [l - 41(71 - ln7,) - a2(r2 - ln7,13 = = alx$? (ai +
biT
+
ciTlnT) [Pi(x2- xi) - 2x& Pl(X2- X%)1
+ a2xtE (ai + biT + clT1nT) [Pi(x2- xpl)+ 2x2 Pl(X2- x1)1
El31
P.K. TALLEY and C.W. BALE
The following iterative procedure is used to determine the adjustable parameters (ai, bi and ci) in Eq. [13]. Initial values for the condensed phase activity coefficients and the vapor compositions are required. If only PTx data are available, the condensed phase is assumed to be ideal. If other excess Gibbs energy data are available, these are treated in an optimization analysis first and the calculated bi and cf parameters are used to estimate the activity coefficient ai I for the PTx data points. In both cases, Eq. [3] is solved iteratively to obtain the vapor phase compositions and the corresponding 47's used in Eq. [12]. Eq. [13] is solved by least squares regression analysis and the revised values of the ai, bi and ci parameters are used to calculate new values for the activity coefficients. The vapor compositions and thus the vapor phase fugacity coefficients, are adjusted giving new values for al and a~. The new values are then substituted into the left-hand-side of Eq. [13] and the procedure is repeated until the difference between the left-hand-side and right-hand-side of the equation reaches a specified maximum. Since Eq. [13] is written in terms of energy/mole, PTxy and HE1 can be treated simultaneously with PTx data in the least squares regression analysis. This reduces the number of iterations required for convergence and insures that the calculated coefficients represent all the data. The convergence criterion is that the root mean square of the percent deviations in pressure (Eq.[12]) no longer changes within a specified limit, where the square of the root mean square of the percent deviations is defined as: (RMS%dev.)2 100
=
(Pcxp - P,,(c) PexP
3.
8 amole caloulatioae
Cl’1
In the following examples, only 5 iterations or less were required to achieve a change of less than 0.001 in Eq.[lQ]. The proposed reduction method was applied to systems where there The are polar - polar and nonpolar - nonpolar molecular interactions. results for five separate organic systems are presented here. 3.1
Ethanol + Chloroform:
PTxy data for the ethanol + chloroform system are available at 35013 (Scatchard et al., 1938). These are shown as experimental points in Figure 1. Only the experimental values for the liquid compositions and total pressures were used in the analysis. (The experimental values composition are plotted for comparison purposes to show of the vapor that the method works). The least squares analysis was performed by using up to 10 ai It was found that 5 coefficients gave satisfactory coefficients. Six or more results with only one extrema in excess Gibbs energy. coefficients had little additional effect or introduced 88wigglest8in the excess Gibbs energy curve.
405
VAPOR COMPOSITIONS IN BINARY ORGANIC SYSTEMS
=R
SERIES
COEFFICIENTS
FOR EO.
131
j
bj
k
_ _ _ -
___ ____B-s ---
_ -
___ _-_ _-_ --_ ---
_ -
-------
SYSTEM
i
Ethanol (1) + Chloroform (2)
1 2 3 4 5
N-propylbenzene (1) + Methylcyclohexane (2)
1 2 3
172.96 -26.7~1 34.868
_ -
--_ -----
1,4-dichlorobenzene + 1,4_dibromobenzene
1 2 3
466.34 -163.02 -68.028
-
-------
-
-------
Methanol (1) + N,N-diethylformamide (2)
1 2 3 4
342.07 157.32 -728.55 420.73
1 2 3 4
-1.9748 0.44499 0.56753 -0.30992
-
---------
Ethanol Acetone
1 2 3
-1265.2 2827.2 -932.90
1 2 3
1 2 3
-7.7826 9.6919 -3.9309
(1) (2)
+
(1) (2)
ai 222.45 1055.3 -660.44 147.68 265.19
50.431 -65.132 25.857
‘=k
406
P.K. TALLEY and C.W. BALE
FIG. 1.
% 250
L’ :g 200 h
Pressure - composition diagram for ethanol + chloroform at 35oC.
Date of Scatcha end Reymond
(1938)
150
Hole fraction of ethanol
PTC_ ___-
3. _-
-3
Pressure - composition diagram for N-propylbenzene + methylcyclohexane at 4OoC.
4
6.0 Solid
2 2 0 4.0 h 2.0
Vapor
o.o~.-.......‘....,~-..I...~...,....I....I..,-~ 0.2
0.0
Hole
Data of Arnanova and Gore1
0.4 fraction
of
0.6 0.8 1.4-dichlorobenzene
(i9EO)
FIG. 3. Pressure - composition diagram for 1,4-dichlorobenzene + 1,4-dibromobenzene at 50~C.
O.Ot.-.‘..-.‘.-LY...--I....I-.-.’-...,-...’.-..I-...1 0.0
0.2
Hole
0.4
fraction
0.6
of
propylbenzene
0.8
i.0
1
1.0
VAPOR COMPOSITIONS IN BINARY ORGANIC SYSTEMS
407
The values of the ai's are given in Table 1 and the calculated bubble and dew lines are shown in Figure 1. We see from this figure that the calculated vapor compositions are in excellent agreement with the independently determined experimental data. The RWS % pressure deviation is 0.18 with a maximum of 0.48 %. The mean deviation in the vapor phase mole fractions is 0.002 with a maximum deviation of 0.006. It is noted that the presence of the azeotrope does not effect the method. 3.2
N:
Only PTx data at 4OaC (Asmanova and Goral 1980) are available for this system. The data were treated as in the previous case and it was found that 3 ai (see Table 1) coefficients yielded an average absolute deviation of 0.053 mm Hg in pressure in a bubble pressure calculation. A phase diagram showing the experimental points and the calculated boundaries is given in Figure 2. Even though the aromatic component is much less volatile than methylcyclohexane, the agreement between the experimental values and the calculated curve near the pure propylbenzene region is still very good. 3.3
1.4_dichlorobenzene
The reduction method can also be applied to PTx data for solidvapor equilibria (Figure 3). Data for the 1,4-dichlorobenzene + 1,4-dibromobenzene (Walsh and Smith, 1961; Callanan and Smith, 1962) 509C were treated in exactly the same way as before. It was found that 3 Qi coefficients reproduced the data with an absolute average deviation of 0.043 mm Hg in the total pressure. This is well within experimental error. The coefficients are listed in Table 1 and the calculated phase diagram is given in Figure 3. 3.4
2 ethan
When PTx data at more than one temperature and heats of mixing are also available, a more detailed analysis may be performed. The expression for the excess Gibbs energy can now have a temperature dependence of the type ai + bi T. This is where the present method differs considerably from the other methods. PTx data have been measured for the methanol + N,N-diethylformamide system at 20 and 4OoC (Quitzsch et al. 1969) (Figure 4). Heats of mixing at 25oC (Quitzsch et al. 1966) (Figure 5) are also available. In this particular case the heats of mixing were treated to obtain the coefficients for Eq. [5]. It was found that 4 coefficients yielded ai an average absolute error of 0.46 % in the heat of mixing. The PTx data were then used to find the bi coefficients. It was found that 4 bi coefficients yielded an average absolute deviation of 0.24 mm Hg in pressure. The calculated coefficients are listed in Table 1. These coefficients can be used to extrapolate the gz expression to other temperatures and pressures assuming that the enthalpy of mixing and excess entropy are temperature independent. It should be pointed out that although this system exhibits negative deviations from ideality, the procedure still provides good results even in regions where this effect is most pronounced. It is possible to treat PTx data and heats of neously, but this rarely gives satisfactory results.
mixing
simulta-
P.K. TALLEY and C.W. BALE
408
FIG.
4.
Heat8 of mixing for methanol + N,N-diethylformamide at 25oC.
Mole fraction of rethsnol 150
oatm of nuitzscll at Sl. FIG.
5.
Pressure - composition diagram for methanol + N,N-diethyl-formamide at 20 and 4tPc.
flsa
--,ioo
# c &o.o
Wole fraction of methanol
260 = p
FIG.
c y
Heat8 of mixing for ethanol + acetone at 25 and 5OeC.
L60
Y 100
0 Cootier end Woruld 20.0
0.0 0.0
6.
(19761
0 Nlcoleides end Eckert IlS7Sl
0.2
0.4 0.6 Mole trection of ethanol
0.6
i.0
VAPOR COMPOSITIONS IN BINARY ORGANIC SYSTEMS
3.5
-01
409
+ Acetone:
When heats of mixing have been measured at several temperatures one can solve for all three sets of coefficients in Eq. [4]. In this system heat of mixing data at 25 and 5O*C (Nicolaidesand Eckert, 1978; Coomber and Wormald, 1976; Paz-Andrade et al. 1973; Findlay, 1961) were treated first to obtain the ai and ci coefficients (eq [53). zh:,3 ti and 3 ci coefficients yield an absolute percent deviation of . The experimental points and calculated curves are given in Figure 6. * PTxy data are available at 1 atm. and 55aC (Vinichenkoand Susarev, 1966; Amer, 1961; Hellwig and Van Winkle, 1953), and lmDIs (Thomas et al. 1982a, b) have been measured for both components at several temperatures. These data were then optimized and the resulting bi coefficients along with the ai and cl coefficients were used to calculate the *seed" values for the point activity coefficients for the reduction method. PTx data at 50oC (Chaudry et al., 1980) and 1 atm. (Amer et al., 1953) were then reduced using the proposed method. It is important to note that the PTxy data and the -lo'swere optimized simultaneously with the PTx data during the least squares regression analysis. This is done to ensure that the resulting coefficients will fit all the data. The coefficients given in Table 1 reproduce the PTx data with an average absolute deviation of 2.09 mm Hg in pressure. The vapor phase compositions of the PTxy data show an average absolute deviation of 0.01. Phase diagrams showing the experimental points and calculated curves are given in Figures 7 and 8.
An algorithm has been developed whereby one can maximize the utility of total pressure data. The method can be applied to systems where only total pressure data are available regardless of the type of condensed phase (solid or liquid), disparity in volatility or the presence of an azeotrope. In addition, one is able to couple the analysis with liquid excess property data from other sources such as enthalpy of mixing data, PTxy data, activity data, etc., in order to obtain a set of descriptive parameters applicable over a wide range of pressure, temperature and composition. The method can be applied to isothermal or isobaric data without any change in either the procedure or the equations. Repetitive differentiation of extensive or intensive properties, numerically or explicitly, has been eliminated. Other methods previously proposed offer one or two of the above advantages, but none allow all of them. We are aware of no other algorithm that offers this degree of flexibility in the treatment of such diverse types of thermodynamic data. 5. A&J&R&&_ The authors wish to thank Dr. F. Ajersch for his helpful comments during the preparation of this paper. Financial support in the form of a strategic grant from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
410
P.K. TALLEY and C.W. BALE
0HollwiSand
Van
Winkle119531
nVlnichenko and Susarw 1iSSSl rAmoret al. li933) ii70.0 o*asr liSS2) z 3 e P 8 SO.0
50.0 0.0
0.2
0.4
0.8
0.8
1.0
Mole fraction of ethanol FIG. 7. Temperature - composition diagram for ethanol + acetone at 1 atmosphere.
I-~-‘.---‘,.--....-.“-‘......, ._...-C
Liquid
Dataof Chwdry at al. MS01
~.......I.......~.......I.........I....I, 0.2 i. 0.4 0.6 0.6 Molefraction of ethanol FIG. 8. Pressure - composition diagram for ethanol + acetone at 5OoC.
VAPOR COMPOSITIONS IN BINARY ORGANIC SYSTEMS
H.H., Thesis Stanford (1961). Amer, H.H., R.R. Paxton and M. Van Winkle, Anal. Chem. 25, 1204 (1953). Asmanova, N. and M. Goral, J. Chem. Eng. Data 25, 159 (1980). Bale, C.W. and A.D. Pelton, Met. Trans., 5, 2323 (1974). Bale, C.W. and A.D. Pelton, Hetall. Trans. 148, 77 (1983a). Bale, C.W. and A.D. Pelton, Bull. Alloy Phase Diagrams 4, 379 (1983b). Barker, J.A., Au&. J. Chem. 6, 207 (1953). Callanan, J.E. and N.O. Smith, J. Chem. Eng. Data 7, 374 (1962). Chaudry, M.M., H.C. Van Ness and M.M. Abbott, J. Chem. Eng. Data 25, 254 (1980). Christian, S.D., J. Phys. Chem. 64, 764 (1960). Coomber, B.A. and C.J. Wormald, J. Chem. Thermo. 8, 793 (1976). Findlay, T.J.V., Au&. J. Chem. 14, 529 (1961). Hellwig, L.R. and M. Van Winkle, Ind. Eng. Chem. 45, 624 (1953). Lunglin, J.J. and H.C. Van Ness, Chem. Eng. Sci. 17, 531 (1962). Marsh, K.N., J. Chem. Thermo. 9, 719 (1977). Nicolaides, G.L. and C.A. Eckert, J. Chem. Eng. Data 23, 152 (1978). Paz-Andrade, M.I., E. Jiminez and D.L. Garcia, An. Quim. 69, 289 (1973). Pelton, A.D. and C.W. Bale, CALPHAD J. 1, 253 (1977). Pelton. A.D.. C.W. Bale and P.L. Lin. Canadian J. Chem. 62, 457 (i983). . Pelton, A.D. and C.W. Bale, Met. Trans. 17A, 1057 (1986). Geiseler, J. Prakt. Quitzsch, K., H.-P. Prinz, K. Suhnel and G. Chem. 311, 420 (1969). Quitzsch, K., H.-P. Hofman, R.P. Pfestorf and G. Geiseler, J. Prakt. Chem. 34, 145 (1966). Sangster, J., P.K. Talley, C.W. Bale and A.D Pelton, Can. J. Chem. Eng. 66, 881 (1988). Scatchard, G,, C-L. Raymond and H.H. Gilman, J. Amer. Chem. Sot. 60, 1278 (1938). Talley, P.K., C.W. Bale and A.D. Pelton, Ind. Eng. Chem. Fundamentals (submitted). Tao, L.C., Ind. Eng. Chem. 53, 307 (1961). Thomas, E.R., B.A. Newman, G.L. Nicglaides and C.A. Eckert, J. Chem. Eng. Data 27, 233 (1982a). Thomas, E-R., B.A. Newman, T.C. Long, D.A. Wood and C.A. Eckert, J. Chem. Eng. Data 27, 399 (198233). Tsonopoulos, C., A.1.Ch.E. J. 20, 263 (1974). Tsonopoulos, C., A.1.Ch.E. J. 21, 827 (1975). Vinichenko, 1-G. and M.P. Susarev, Zh. Prikl. Khim. 39, 1583 (1966). Walsh, P.N. and N.O. Smith, J. Phys. Chem. 65, 718 (1961).
Amer,
411
CALPIIAD Vol. 13, No. 4, pp. 401-411, 1989 Printed in the USA.
VAWR ColWoB1T10N8 It0BIlIARYORGAMIC BYBTBx8 BROH P-T-x DATA COUPLED WITH LIQUID EXCBBB PROPERTIES
Paul lC.TALLEY and Christopher 1. BALE CENTRE FOR RESEARCH INTO COMPUTATIONAL TRERMOCHEMISTRY ECOLE POLYTECHNIQUE Box 6079, Station A, Montreal, Quebec, Canada, H3C 3A7
A new reduction method is proposed for obtaining liquid activity coefficients and vapor compositions from PTx data in organic systems. One can also include enthalpies of mixing, Foxy data and activity coefficients at infinite dilution in the treatment, thus providing a complete description of the excess Gibbs energy of the liquid phase over a wide range of conditions. The results for the following five binary systems are presented: ethanol + chloroform: n-propylbenzene + methyl+ cyclohexane; I,l-dichlorobenzene + 1,4-dibromobenzene; methanol N,N-diethylformamide; ethanol + acetone. In principle, the reduction method can be extended to volatile inorganic and metallic systems where precise total pressure data are available.
1.
B
Much of the liquid-vapor and solid-vapor equilibrium data now available record only total pressure, temperature and condensed phase compositions (PTx) because these measurements are the easiest to obtain. For design purposes in chemical engineering, a knowledge of the vapor compositions (y) as a function of these variables is essential. It is possible to obtain this information from PTx data by applying a reduction method provided that the thermodynamic properties of the gas phase are well known. Several reduction methods have been proposed in the literature (Marsh, 1977; Tao, 1961; Christian, 1960; Barker, 1953). In the present article, a new method is proposed which permits the reduction of FTx data at several temperatures and yields a set of descriptive parameters applicable over wide ranges of pressure, temperature and composition. It is possible to include heats of mixing, PTxy data and activity coefficients at infinite dilution in the analysis. As a result, one is then able to generate isobaric phase diagrams and interpolate or extrapolate to temperatures and pressures different from the experimental values. ____-_______________------. Received 5 June 1989
401
402
P.K. TALLEY and C.W. BALE
2.
Theorv
The criteria for equilibrium of a component i in a mixture between a condensed phase and a vapor at temperature T and total pressure p is: YiQP
111
= xi7if;a
where yi and xi are the vapor and condensed phase compositions, 49 is the vapor phase fugacity coefficient, 7i is the activity coefficient of component i in the condensed phase and r;c is the fugacity of pure i in the condensed phase. The PC term is not a directly measurable quantity, but it can be estimated by assuming: f;C
9i(sat)
fl
pi(sat)
where 4i (sat) is the saturation fugacity coefficient and pi (sat) is the saturation (vapor or sublimation) pressure of component i. By combining Eqs. [l-2] and rearranging we obtain: p Jli
=
ziTidi
(sat) pi WW/df
as an expression for the partial pressure of component i. The fugacity coefficients in Eq. [3] can be calculated from an equation of state or an ideal gas may be assumed by setting both coefficients equal to unity. In this case, the virial equation of state is The Tsonopoulos (1974, 1975) truncated at the second coefficient. empirical correlation is used to estimate the second virial coeffiThe saturation pressures can be calculated from standard cients. expressions such as the Antoine equation (as is done here) or by using experimental measurements. In the case of PTx measurements, it follows that there are two values to be calculated: yi and 7i. The present method requires that an expression for the condensed phase activity coefficients be available. This expression should be flexible enough to analytically represent the activity coefficients of highly nonideal solutions over a wide The excess Gibbs range of pressure, temperature and composition. energy of the condensed phase is represented by the polynomial expression: gB= x1x2 E (ai + biT + aiT 1nT) Pi(x2 - x1) where ai, bi and Ci are adjustable parameters polynomial of order i and argument (x2-x1). preferred over Redlich-Kister or simple power (Bale and Pelton, 1974).
['I
and Pi is a Legendre Legendre polynomials are series type expansions
Expressions for the other excess properties can be obtained by the appropriate differentiation of Eq. [4]. hB =
x1x2
C
(ai
-
aiT)
Pi(x2 - x1)
151
403
VAPOR COMPOSITIONS IN BINARY ORGANIC SYSTEMS
2 t
C
-x1x2
(bi + ai(l + In T)) Pi(x2 - x1)
c; = -
x1x*
c
aiPi(x*
-
[61
x,1
c73
The expressions for the excess chemical potentials can also be from Eq. 143. R
91 a~xi
E (ai + biT + aiT 1nT) (Pi(X2-X1)- 2m1Pi(x2-xl))
E g2 = xt C
(ai
+ bill
+ o&T
InT)(Pi(x2-xl)+ 2x2Pi(x2-xl))
derived
Cal c91
where Pj '(x2-x1) is the first derivative with respect to (X2-X1) of the Legendre polynomial of order i. It is important to note that the coefficients in Eqs.(4-71 are the same. This enables one to treat partial properties of both components and integral properties simultaneously.Eqs.fS-91 are used extensively in the treatment of metallurgical systems and have been employed by the authors for representing the thermodynamic properties of mixtures (Bale and Pelton, 1974, 1983a; Pelton and Bale, 1977) including alloys (Bale and Pelton 1983b), molten salts (Pelton et al. 1983) and organic compounds (Talley et al., 1987; Sangster et al., 1988). The total pressure is the sum of partial pressures, which from Eq. 133 for a binary system is: P = xz71pl(sat) )I(sat)/df + X272~2(sat) 92[satl/9:
1101
Upon rearrangement and expansion of Eq. [lo], we obtain: RT [I - "x(72 - ln71) - 92(72 - lm72)l = Qlgl E + a2g:
El11
where Qi = xipi fsat)
di
Eqs. [S-9] are substituted into the give:
tsatl/#$ right-hand-side of
[=I
Eq.
[ll] to
RT [l - 41(71 - ln7,) - a2(r2 - ln7,13 = = alx$? (ai +
biT
+
ciTlnT) [Pi(x2- xi) - 2x& Pl(X2- X%)1
+ a2xtE (ai + biT + clT1nT) [Pi(x2- xpl)+ 2x2 Pl(X2- x1)1
El31
P.K. TALLEY and C.W. BALE
The following iterative procedure is used to determine the adjustable parameters (ai, bi and ci) in Eq. [13]. Initial values for the condensed phase activity coefficients and the vapor compositions are required. If only PTx data are available, the condensed phase is assumed to be ideal. If other excess Gibbs energy data are available, these are treated in an optimization analysis first and the calculated bi and cf parameters are used to estimate the activity coefficient ai I for the PTx data points. In both cases, Eq. [3] is solved iteratively to obtain the vapor phase compositions and the corresponding 47's used in Eq. [12]. Eq. [13] is solved by least squares regression analysis and the revised values of the ai, bi and ci parameters are used to calculate new values for the activity coefficients. The vapor compositions and thus the vapor phase fugacity coefficients, are adjusted giving new values for al and a~. The new values are then substituted into the left-hand-side of Eq. [13] and the procedure is repeated until the difference between the left-hand-side and right-hand-side of the equation reaches a specified maximum. Since Eq. [13] is written in terms of energy/mole, PTxy and HE1 can be treated simultaneously with PTx data in the least squares regression analysis. This reduces the number of iterations required for convergence and insures that the calculated coefficients represent all the data. The convergence criterion is that the root mean square of the percent deviations in pressure (Eq.[12]) no longer changes within a specified limit, where the square of the root mean square of the percent deviations is defined as: (RMS%dev.)2 100
=
(Pcxp - P,,(c) PexP
3.
8 amole caloulatioae
Cl’1
In the following examples, only 5 iterations or less were required to achieve a change of less than 0.001 in Eq.[lQ]. The proposed reduction method was applied to systems where there The are polar - polar and nonpolar - nonpolar molecular interactions. results for five separate organic systems are presented here. 3.1
Ethanol + Chloroform:
PTxy data for the ethanol + chloroform system are available at 35013 (Scatchard et al., 1938). These are shown as experimental points in Figure 1. Only the experimental values for the liquid compositions and total pressures were used in the analysis. (The experimental values composition are plotted for comparison purposes to show of the vapor that the method works). The least squares analysis was performed by using up to 10 ai It was found that 5 coefficients gave satisfactory coefficients. Six or more results with only one extrema in excess Gibbs energy. coefficients had little additional effect or introduced 88wigglest8in the excess Gibbs energy curve.
405
VAPOR COMPOSITIONS IN BINARY ORGANIC SYSTEMS
=R
SERIES
COEFFICIENTS
FOR EO.
131
j
bj
k
_ _ _ -
___ ____B-s ---
_ -
___ _-_ _-_ --_ ---
_ -
-------
SYSTEM
i
Ethanol (1) + Chloroform (2)
1 2 3 4 5
N-propylbenzene (1) + Methylcyclohexane (2)
1 2 3
172.96 -26.7~1 34.868
_ -
--_ -----
1,4-dichlorobenzene + 1,4_dibromobenzene
1 2 3
466.34 -163.02 -68.028
-
-------
-
-------
Methanol (1) + N,N-diethylformamide (2)
1 2 3 4
342.07 157.32 -728.55 420.73
1 2 3 4
-1.9748 0.44499 0.56753 -0.30992
-
---------
Ethanol Acetone
1 2 3
-1265.2 2827.2 -932.90
1 2 3
1 2 3
-7.7826 9.6919 -3.9309
(1) (2)
+
(1) (2)
ai 222.45 1055.3 -660.44 147.68 265.19
50.431 -65.132 25.857
‘=k
406
P.K. TALLEY and C.W. BALE
FIG. 1.
% 250
L’ :g 200 h
Pressure - composition diagram for ethanol + chloroform at 35oC.
Date of Scatcha end Reymond
(1938)
150
Hole fraction of ethanol
PTC_ ___-
3. _-
-3
Pressure - composition diagram for N-propylbenzene + methylcyclohexane at 4OoC.
4
6.0 Solid
2 2 0 4.0 h 2.0
Vapor
o.o~.-.......‘....,~-..I...~...,....I....I..,-~ 0.2
0.0
Hole
Data of Arnanova and Gore1
0.4 fraction
of
0.6 0.8 1.4-dichlorobenzene
(i9EO)
FIG. 3. Pressure - composition diagram for 1,4-dichlorobenzene + 1,4-dibromobenzene at 50~C.
O.Ot.-.‘..-.‘.-LY...--I....I-.-.’-...,-...’.-..I-...1 0.0
0.2
Hole
0.4
fraction
0.6
of
propylbenzene
0.8
i.0
1
1.0
VAPOR COMPOSITIONS IN BINARY ORGANIC SYSTEMS
407
The values of the ai's are given in Table 1 and the calculated bubble and dew lines are shown in Figure 1. We see from this figure that the calculated vapor compositions are in excellent agreement with the independently determined experimental data. The RWS % pressure deviation is 0.18 with a maximum of 0.48 %. The mean deviation in the vapor phase mole fractions is 0.002 with a maximum deviation of 0.006. It is noted that the presence of the azeotrope does not effect the method. 3.2
N:
Only PTx data at 4OaC (Asmanova and Goral 1980) are available for this system. The data were treated as in the previous case and it was found that 3 ai (see Table 1) coefficients yielded an average absolute deviation of 0.053 mm Hg in pressure in a bubble pressure calculation. A phase diagram showing the experimental points and the calculated boundaries is given in Figure 2. Even though the aromatic component is much less volatile than methylcyclohexane, the agreement between the experimental values and the calculated curve near the pure propylbenzene region is still very good. 3.3
1.4_dichlorobenzene
The reduction method can also be applied to PTx data for solidvapor equilibria (Figure 3). Data for the 1,4-dichlorobenzene + 1,4-dibromobenzene (Walsh and Smith, 1961; Callanan and Smith, 1962) 509C were treated in exactly the same way as before. It was found that 3 Qi coefficients reproduced the data with an absolute average deviation of 0.043 mm Hg in the total pressure. This is well within experimental error. The coefficients are listed in Table 1 and the calculated phase diagram is given in Figure 3. 3.4
2 ethan
When PTx data at more than one temperature and heats of mixing are also available, a more detailed analysis may be performed. The expression for the excess Gibbs energy can now have a temperature dependence of the type ai + bi T. This is where the present method differs considerably from the other methods. PTx data have been measured for the methanol + N,N-diethylformamide system at 20 and 4OoC (Quitzsch et al. 1969) (Figure 4). Heats of mixing at 25oC (Quitzsch et al. 1966) (Figure 5) are also available. In this particular case the heats of mixing were treated to obtain the coefficients for Eq. [5]. It was found that 4 coefficients yielded ai an average absolute error of 0.46 % in the heat of mixing. The PTx data were then used to find the bi coefficients. It was found that 4 bi coefficients yielded an average absolute deviation of 0.24 mm Hg in pressure. The calculated coefficients are listed in Table 1. These coefficients can be used to extrapolate the gz expression to other temperatures and pressures assuming that the enthalpy of mixing and excess entropy are temperature independent. It should be pointed out that although this system exhibits negative deviations from ideality, the procedure still provides good results even in regions where this effect is most pronounced. It is possible to treat PTx data and heats of neously, but this rarely gives satisfactory results.
mixing
simulta-
P.K. TALLEY and C.W. BALE
408
FIG.
4.
Heat8 of mixing for methanol + N,N-diethylformamide at 25oC.
Mole fraction of rethsnol 150
oatm of nuitzscll at Sl. FIG.
5.
Pressure - composition diagram for methanol + N,N-diethyl-formamide at 20 and 4tPc.
flsa
--,ioo
# c &o.o
Wole fraction of methanol
260 = p
FIG.
c y
Heat8 of mixing for ethanol + acetone at 25 and 5OeC.
L60
Y 100
0 Cootier end Woruld 20.0
0.0 0.0
6.
(19761
0 Nlcoleides end Eckert IlS7Sl
0.2
0.4 0.6 Mole trection of ethanol
0.6
i.0
VAPOR COMPOSITIONS IN BINARY ORGANIC SYSTEMS
3.5
-01
409
+ Acetone:
When heats of mixing have been measured at several temperatures one can solve for all three sets of coefficients in Eq. [4]. In this system heat of mixing data at 25 and 5O*C (Nicolaidesand Eckert, 1978; Coomber and Wormald, 1976; Paz-Andrade et al. 1973; Findlay, 1961) were treated first to obtain the ai and ci coefficients (eq [53). zh:,3 ti and 3 ci coefficients yield an absolute percent deviation of . The experimental points and calculated curves are given in Figure 6. * PTxy data are available at 1 atm. and 55aC (Vinichenkoand Susarev, 1966; Amer, 1961; Hellwig and Van Winkle, 1953), and lmDIs (Thomas et al. 1982a, b) have been measured for both components at several temperatures. These data were then optimized and the resulting bi coefficients along with the ai and cl coefficients were used to calculate the *seed" values for the point activity coefficients for the reduction method. PTx data at 50oC (Chaudry et al., 1980) and 1 atm. (Amer et al., 1953) were then reduced using the proposed method. It is important to note that the PTxy data and the -lo'swere optimized simultaneously with the PTx data during the least squares regression analysis. This is done to ensure that the resulting coefficients will fit all the data. The coefficients given in Table 1 reproduce the PTx data with an average absolute deviation of 2.09 mm Hg in pressure. The vapor phase compositions of the PTxy data show an average absolute deviation of 0.01. Phase diagrams showing the experimental points and calculated curves are given in Figures 7 and 8.
An algorithm has been developed whereby one can maximize the utility of total pressure data. The method can be applied to systems where only total pressure data are available regardless of the type of condensed phase (solid or liquid), disparity in volatility or the presence of an azeotrope. In addition, one is able to couple the analysis with liquid excess property data from other sources such as enthalpy of mixing data, PTxy data, activity data, etc., in order to obtain a set of descriptive parameters applicable over a wide range of pressure, temperature and composition. The method can be applied to isothermal or isobaric data without any change in either the procedure or the equations. Repetitive differentiation of extensive or intensive properties, numerically or explicitly, has been eliminated. Other methods previously proposed offer one or two of the above advantages, but none allow all of them. We are aware of no other algorithm that offers this degree of flexibility in the treatment of such diverse types of thermodynamic data. 5. A&J&R&&_ The authors wish to thank Dr. F. Ajersch for his helpful comments during the preparation of this paper. Financial support in the form of a strategic grant from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
410
P.K. TALLEY and C.W. BALE
0HollwiSand
Van
Winkle119531
nVlnichenko and Susarw 1iSSSl rAmoret al. li933) ii70.0 o*asr liSS2) z 3 e P 8 SO.0
50.0 0.0
0.2
0.4
0.8
0.8
1.0
Mole fraction of ethanol FIG. 7. Temperature - composition diagram for ethanol + acetone at 1 atmosphere.
I-~-‘.---‘,.--....-.“-‘......, ._...-C
Liquid
Dataof Chwdry at al. MS01
~.......I.......~.......I.........I....I, 0.2 i. 0.4 0.6 0.6 Molefraction of ethanol FIG. 8. Pressure - composition diagram for ethanol + acetone at 5OoC.
VAPOR COMPOSITIONS IN BINARY ORGANIC SYSTEMS
H.H., Thesis Stanford (1961). Amer, H.H., R.R. Paxton and M. Van Winkle, Anal. Chem. 25, 1204 (1953). Asmanova, N. and M. Goral, J. Chem. Eng. Data 25, 159 (1980). Bale, C.W. and A.D. Pelton, Met. Trans., 5, 2323 (1974). Bale, C.W. and A.D. Pelton, Hetall. Trans. 148, 77 (1983a). Bale, C.W. and A.D. Pelton, Bull. Alloy Phase Diagrams 4, 379 (1983b). Barker, J.A., Au&. J. Chem. 6, 207 (1953). Callanan, J.E. and N.O. Smith, J. Chem. Eng. Data 7, 374 (1962). Chaudry, M.M., H.C. Van Ness and M.M. Abbott, J. Chem. Eng. Data 25, 254 (1980). Christian, S.D., J. Phys. Chem. 64, 764 (1960). Coomber, B.A. and C.J. Wormald, J. Chem. Thermo. 8, 793 (1976). Findlay, T.J.V., Au&. J. Chem. 14, 529 (1961). Hellwig, L.R. and M. Van Winkle, Ind. Eng. Chem. 45, 624 (1953). Lunglin, J.J. and H.C. Van Ness, Chem. Eng. Sci. 17, 531 (1962). Marsh, K.N., J. Chem. Thermo. 9, 719 (1977). Nicolaides, G.L. and C.A. Eckert, J. Chem. Eng. Data 23, 152 (1978). Paz-Andrade, M.I., E. Jiminez and D.L. Garcia, An. Quim. 69, 289 (1973). Pelton, A.D. and C.W. Bale, CALPHAD J. 1, 253 (1977). Pelton. A.D.. C.W. Bale and P.L. Lin. Canadian J. Chem. 62, 457 (i983). . Pelton, A.D. and C.W. Bale, Met. Trans. 17A, 1057 (1986). Geiseler, J. Prakt. Quitzsch, K., H.-P. Prinz, K. Suhnel and G. Chem. 311, 420 (1969). Quitzsch, K., H.-P. Hofman, R.P. Pfestorf and G. Geiseler, J. Prakt. Chem. 34, 145 (1966). Sangster, J., P.K. Talley, C.W. Bale and A.D Pelton, Can. J. Chem. Eng. 66, 881 (1988). Scatchard, G,, C-L. Raymond and H.H. Gilman, J. Amer. Chem. Sot. 60, 1278 (1938). Talley, P.K., C.W. Bale and A.D. Pelton, Ind. Eng. Chem. Fundamentals (submitted). Tao, L.C., Ind. Eng. Chem. 53, 307 (1961). Thomas, E.R., B.A. Newman, G.L. Nicglaides and C.A. Eckert, J. Chem. Eng. Data 27, 233 (1982a). Thomas, E-R., B.A. Newman, T.C. Long, D.A. Wood and C.A. Eckert, J. Chem. Eng. Data 27, 399 (198233). Tsonopoulos, C., A.1.Ch.E. J. 20, 263 (1974). Tsonopoulos, C., A.1.Ch.E. J. 21, 827 (1975). Vinichenko, 1-G. and M.P. Susarev, Zh. Prikl. Khim. 39, 1583 (1966). Walsh, P.N. and N.O. Smith, J. Phys. Chem. 65, 718 (1961).
Amer,
411