Ind. Eng. Chem. Res. 1992,31,2362-2369
2362
Laroche, L.; Bekiaris, N.; Andersen, H. W.; Morari, M. The Curious Behavior of Homogeneous Azeotropic Distillation-Implications for Entrainer Selection. AIChE J. 1990,in press. Laroche, L.; Bekiaris, N.; Andersen, H. W.; Morairi, M. Homogeneous Azeotropic Distillation: Separability and Flowsheet Synthesis. Znd. Eng. Chem. Res. 1992,31, 2190-2209. Levy,S.G.; Van Dongen, D. B.; Doherty, M. F. Design and Synthesis of Homogeneous Azeotropic Distillations. 2. Minimum Reflux Calculations for Nonideal and Azeotropic Columns. Znd. Eng. Chem. Fundam. 1985,24,463-473. Malesineki, W. Azeotropy and Other Theoretical Problems of Vapour-liquid Equilibrium; Interscience: New York, 1965. Nikolaev, N. 5.;Kiva, V. N.; Mozzhukhin, A. S.; Serafiiov, L. A,; Goloborodkin, S. I. Utilization of Functional Operators for Determining the Regions of Continuous Rectification. Theor. Found. Chem. Eng. 1979,13,418-423. Petlyuk, F. B. Rectification of Zeotropic, Azeotropic, and Continuous Mixtures in Simple and Complex Infinite Columns with Finite Reflux. Theor. Found. Chem. Eng. 1978,12,671-678. Petlyuk, F. B.; Platonov, V. M.; Slavinsii, D. M. Thermodynamically Optimal Method for Separating Multicomponent Mixtures. Znt. Chem. Eng. 1966,5(2),309-317. Poellmann, P. Untersuchung dea Trennverlaufs bei der Rektiihtion realer Mehrstoffgemieche. Diploma Thesis, Technical University of Munich, Germany, 1989. Reid, R. C.; Prausnitz, J. M.; Poling, B.E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. Schreinemakers, F. A. H. 2.Phys. Chem. 1901,36,257. Stichlmair, J. Personal communication, 1991. Stichlmair, J.; Fair, J. R.; Bravo, J. L. Separation of Azeotropic Mixtures via Enhanced Distillation. Chem. Eng. h o g . 1989,85 (l),63-69. Van Dongen, D. B. Distillation of Azeotropic Mixtures. The Application of SimplaDistillation Theory to the Design of Continuous Processes. Ph.D. Dissertation, University of Massachusetts, Amherst, MA, 1983. Van Dongen, D. B., Doherty, M. F. On the Dynamics of Distillation
Processes-V (The Topology of the Boiling Temperature Surface and its Relation to Azeotropic Distillation). Chem. Eng. Sci. 1984, 39,883-892. Van Dongen, D. B.; Doherty, M. F. Design and Synthesis of Homogeneous Azeotropic Distillations. 1. Problem Formulation for a Single Column. Znd. Eng. Chem. Fundam. 1986,24, 454-463. Vogelpohl, A. Rektifikation von Dreietoffgemischen. Teil 1: Rektifikation als Stoffaustauschvorgang und Rektifikationslinien idealer Gemische. Chem.-Zng.-Tech. 1964,36 (lo), 1033-1045. Wahnschafft, 0. M. "Syntheaisof Separation System for Azeotropic Mixtures with an Emphasis on Distillation-Based Methods; Research Report, Engineering Design Research Center, Carnegie Mellon University: Pittsburgh, P A 15213,1992a. Wahnschafft, 0.M. Synthese von Trennprozessen zur Zerlegung azeotroper Vielstoffgemhhe unter beaonderer Berueckeichtigung der Rektifikation. Ph.D. Dissertation, Department of Chemical Engineering, Technical University of Munich, Germany, 1992b. Wahnschafft, 0. M.; Westerberg, A. W. Improving the Economics of Azeotropic Distillation Processes through Intermediate Heat Exchange. Document prepared for US. patent application, Carnegie Mellon University, Pittsburgh, 1991. Wahnschafft, 0. M.; Westerberg, A. W. The Product Composition Regions of Azeotropic Distillation Columns. 11. Separability in Multi-Feed Columns and Entrainer Selection. Submitted for publication in Znd. Eng. Chem. Res. 1992. Wahnschafft, 0. M.; Jurain, T. P.; Westerberg, A. W. SPLIT a Separation Process Designer. Comput. Chem. Eng. 1991,15 (8), 565-581. Wahnechafft, 0.M.; LeRudulier, J. P.; Westerberg,A. W. A Problem DecompositionApproach for the Syntheais of Complex Separation Processes. Submitted for publication in Znd. Eng. Chem. Res. 1992.
Received for review August 9,1991 Revised manuscript received November 1,1991 Accepted July 16, 1992
GENERALRESEARCH Prediction of Enthalpies of Formation for Ionic Compounds C. Dianne Ratkey and B. Keith Harrison* Chemical Engineering Department, University of South Alabama, Mobile, Alabama 36688
An improved generalized correlation for the prediction of enthalpies of formation of ionic compounds was developed. Four correlations were examined: (1)the Wilcox/Bromley/Brandenbug method (WBB), (2)an empirical model based on Hisham and Benson's method, (3) the Kapustinskii-based method (KBM), and (4) a modified lattice energy method (MLE). The WBB method was not modified by the authors; however, the other three represent either generalizations of specialized methods or, in the last case, a modified lattice energy method. The MLE method exhibited the highest degree of accuracy of the four methods tested with an average absolute error of 23.8 kJ/mol for a database of 806 compounds. This represents about a 50% improvement to the WBB method while using fewer parameters. Additional inorganic compounds were examined to test the predictive abilities of the modified lattice energy method. The enthalpies of formation for these compounds were predicted with an average absolute error of 31.4 kJ/mol. Introduction Chemical engineers have become accustomed to routinely and somewh& confidently predicting themochemical properties for organic chemicals when experimental
* Author to whom correspondence should be addressed.
data are not available. In contrast to the situation for organic c h d & COm~ativelY work been done in develop@ generalized predictive methods for them* chemicalproprties for inorganic molecdea. For instancey current correlations for ionic heats of formation suffer from a lack of generality (being restricted to a small number of compounds) or from a lack of accuracy. When the Am-
08sS-5~5/92/2631-2362$03.00/0 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 2363 erican Society of Testing and Materials (ASTM) Committee E27-07 attempted to extend their program CHETAH (Frurip et al., 1989) from organics to include inorganics using an electronegativity-based method (Brandenburg, 1978),they found unacceptable accuracy in the results. The program CHETAH, which predicts potential explosive properties of chemicals, would too frequently provide a bad prediction because of large errors in predicted values of ionic heats of formation. Certainly other applications that need generalized routine predictve methods for inorganic thermochemical properties have been similarly troubled. This study attempts to address the problem of formulating a better general method for the prediction of ionic heats of formation. The focus is only on the single method found to be applicable to a wide range of ionic compounds (Brandenburg, 1978) and those methods that could be generalized to cover a wide range. An attempt was made to generalize two existing methods: an empirical method of Hisham and Benson (1988) and a method based on lattice energies (Waddington, 1959). In addition a new general method for prediction of ionic heats of formation is proposed here based on an extension of earlier lattice energy models. It was found in comparing the four models studied that the new model was the most accurate with an average percent error of 2.6% for the database compounds. When tested as a predictive model for compounds not in the database, the average error was 5.0%. Significantly, when compared to the other models, the number of predictions greatly in error was reduced. The new model should be accurate enough to allow acceptable thermochemical a p plication computer programs to be developed.
Background and Model Development The standard enthalpy of formation of a substance is the amount of heat evolved or absorbed in the formation of the pure compound from its elements at 25 OC and 101.33 kPa pressure, all substances being in their standard states. Three different schemes are presented for structuring a general correlation for the enthalpy of formation for ionic substances. They can be termed (1)an electronegativity approach, (2) an empirical approach, and (3) a lattice energy approach. Electronegativity-Based Model. Pauling (1932) introduced the concept of electronegativityto describe the ionic resonance energy, D(C,A),of a bond between unlike atoms C and A, defined as shown in eq 1,where E(C,C), D(C,A) = E(C,A) - O.B[E(C,C) + E(A,A)] (1) E(A,A),and E(C,A) are the energies of the bonds between atoms C and C, A and A, and C and A, respectively. This resonance energy, D(C,A),is expremed in electronvolts and is related to the electronegativity difference of the atoms A and C as follows: D(C,A)
[K(C) - K(A)]'
(2)
where K(C) and K(A) are the electronegativity values for C and A. In this manner, Pauling was able to set up a scale of electronegativities for various substances. In some cases, the individual bond energies of eq 1are not available, 80 Pauling then expanded the scale to cover these cases by indicating that D(C,A) can be approximated by A&P,the enthalpy of formation of the crystalline state at 298 K, as shown in eq 3, where 96.2 is a factor which -A@O(C,A) 96.2[K(C) - K(A)]' (3) converts electronvolts to kilojoules per mole. Pauling then
extended eq 3 empirically to predict the enthalpies of formation of compounds containing only carbon, hydrogen, oxygen, and/or nitrogen: -AfHo = 96.2C[K(C) - K(A)I2- 231.8nN- 108.8no (4) where nN = number of single-bondednitrogen atoms and no = number of single-bonded oxygen atoms. Contributions are summed over all single bonds in. the molecule. Anderson and Bromley (1959) augmented eq 4 to correlate enthalpies of formation of metal halides; they obtained -AfHo = n[X(C) - X(A)]' + cY(C) + aY(A) ( 5 ) where C = cation, A = anion, X(C) and Y(C) = empirical cation parameters, X(A) and Y(A) = empirical anion parameters, n = number of apparent single bonds, c = number of cations, and a = number of anions. This equation works well for metal halides but fails when used for carbonates, nitrates, oxides, and sulfides. Wilcox and Bromley (1963) then extended this equation by adding two additional empirical parameters (W(A), W(C)) to obtain -AfHo = n[X(C) - X(A)]' + cY(C) + aY(A) + nW(A)/W(C) (6) This equation correlates the enthalpies of formation of inorganic compounds composed of many different anions and cations. Brandenburg (1978) modified the equation to avoid possible division by zero and obtained the mathematically equivalent equation -AfHo = n[X(C) - X(A)]' + CY@)+ aY(A) + nW(A)W(C) (7) where "(A) and W(C) are again empirical parameters, but different in value from those in eq 6. AE Wilcox had done before him,Brandenburg evaluated a single set of parameters for each cation and two sets of parameters for each anion, one set to be used when the anion is paired with cations of valence +1 or +2 and another for use when paired with cations of valence +3 or +4. This modification of Brandenburg (eq 7) we will label as the Wilcox, Bromley, Brandenburg method (WBB). Empirical-Based Model. Hisham and Benson (1988) have reported an empirical equation which relates the enthalpy of formation of an ionic compound ( C A ) to that for the corresponding compounds of the cation coupled with the chloride (CcC1,) and oxide (C,Oa): (1/a)AfHO(CA,) = mAfH"(CcCL) + z+[(1- m)/2lAfHO(CcOa)+ 8 (8) where all compounds are stoichiometrically balanced, Z+ is the valence the cation, m and Q are anion parameters, and AfHo(CcC1,)and A d o (CcO,) are enthalpies of formation. To use this correlation, it is necessary to obtain the enthalpy of formation of the chloride compound and oxide compound for each cation. In addition, for the anions two fitted parameters, m and Q, are required. To obtain good fits, these two anion parameters are revised each time they are paired with a different cation valence. In theory, knowledge of high-quality thermodynamic data for 300 compounds would allow accurate prediction of enthalpy of formation values for 4032 compounds (Hisham and Benson, 1989). Unfortunately, much of the needed data are not available. In this study, it was first attempted to generalize the Hisham and Benson (1988) method to a four-parameter correlation, having two parameters for each anion and
2364 Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992
Figure 1. Born-Haber cycle.
cation. For the anion the parameters rn and Q were fit. Unlike the original, rn and Q were not varied with valence state to restrict the number of degrees of freedom in the model, For the cation the needed parameters are the heats of formation for the cation oxide and chloride. If experimental values were available for these heats of formation, they were used. Otherwise, these two cation parameters were also treated as fitted parametars. The results of this effort were discouraging and further efforts to generalize the Hisham and Benson method were abandoned. Lattice-Energy-Based Models. Born and Haber (Huheey, 1972) applied Hew' law to the evaluation of the enthalpy of formation of an ionic solid. Figure 1 illustrates the Born-Haber cycle for the formation of a typical ionic crystal from its elementa. If energy is to be conserved, then = AH, + AHA, + AHm + AHW + Uo (9)
Aw
where AJP = enthalpy of formation of an ionic compound, AHb = enthalpy of sublimation of an atom of C, A H A A = enthalpy of dissociation into an atom of A, AH, = ionization energy of an atom of c, AHW = electron affinity of the anion radical, and Uo= lattice energy. Thus it is possible to use the Born-Haber cycle and data available in the literature to predict enthalpies of formation for many ionic compounds (Sharp, 1986). However, the lack of readily available values for all the needed energy terms has prevented its use as a generalized predictive method for enthalpies of formation. Values for the lattice energy term are especially difficult to obtain. The lattice energy of an ionic crystal is defined here as the negative of the energy neceesary to separate the static crystalinta noninteradng ions. The theoretical treatment of the ionic lattice energy was initiated by Born and Lande. They used a simple electrostatic model to develop an equation which predicts this lattice energy [as reported in Huheey (1972)l. If one considers the energy of a hypothetical ion pair, C+z, A-z, which are separated by a distance r, then the electrostatic energy of attraction (Ec) between the ions is Ec = Z+Z-e2/r (10) where Z+ = valence of cation and 2- = valence of anion, and the ionic charges are expressed as multiples of the electronic charge, e = 4.8 X 10-lo esu. In a crystal lattice there are many more interactions to be evaluated due to forces exerted by geometrical neighbors in the lattice. The summation of all of these geometrical interactions is known as the Madelung constant (M). The energy of an ion pair in a lattice can then be written as
Ec = MZ+Z-e2/r
(11)
The value of the Madelung constant is dependent only upon the geometry of the lattice.
Quation 11 expresses the attractive Coulombic energy between the ions, but unless there is also a repulsive energy term, no stable lattice could exist. Born expressed this repulsive energy term as ER = B / d (12) where B is a constant and the value of j is determined empirically from the compressibility data. The total energy for a mole of the crystal (U,)can then be expressed as Uo Ec + ER = MNAZ+Z-e2/ro+ NAB/ri (13) where NAis Avogadro's number. The constant B can then be evaluated since at the equilibrium separation (ro)in the lattice dUo/dr is zero, so then dU/dr = 0 = -MNAZ+Z-e2/ro2- jNAB/rOj+l (14) and B = -MZ+Z-e2r&'/j (15) Hence the lattice energy is expressed by the Born-Lande equation (Huheey, 1972) as uo = (MNAz+z-e2/ro)(l- l/j) (16) Born and Mayer then attempted to evaluate the constant j in the above equation and found that l/j could be approximated b p / r o ,where p was found to be a constant equal to 0.345 , to give the Born-Mayer equation: Uo= (MNAZ+Z-e2/ro)(l- p / r o ) (17) In principle, the evaluation of the Madelung constant for any crystal lattice is straightforward, but in actuality, the computations become very difficult. Also, the Madelung constant limits the predictive powers of the lattice energy equation in that it requires the exact crystal structure be known. In order to overcome these shortcomings, Kapustinskii proposed a legs complex form of the Born-Mayer equation (Waddington, 1959). Using eq 17 as a starting point, one can multiply the first term by u/2, where u is the number of ions in the molecule, to obtain uo = (N~U/2)&z+z-e~/F,)(l- p / r o ) (18) where g = M/(u/2). The constant, g, is not the same for every lattice type; however, Kapustinskii found that the change in the value for one lattice type compared to another was proportional to the change in the interatomic distance, r,. He proposed that every crystal can be assumed to transform into a rock-salt lattice without changing ita lattice energy; thus the value of the constant, g, for the rock-salt lattice (g = 1.745 A) can be substituted into eq 18. If one estimates the value of ro for the rock-salt coordination number of 6 to be given by ro = (rc + rA) and realizing that NAe2is equal to 1379.5 kJ/A, one obtains the Kapustinskii equation: uo = [1201.6uZ+Z-/(rc + r ~ ) ] [ -l 0.345/(rc + rA)] (19) where rc = radius of the cation and r A = radius of the anion. Kupustimkii-Based Method. This model for lattice energy when coupled with the Born-Haber cycle can furnish a correlation for ionic heats of formation. This method, which will be termed the Kapustinskii-based method (KBM), is summarized in the equation A&'= [1201.6uZ+Z-/(rc + r ~ ) ] [ 1- 0.345/(rc +PA)]+ cAHIE+ aAHM (20) The method requires two parameters for each anion and cation. Anion parameters are the radius of each anion and
1
Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 2365
the electron affinity. The cation parameters are the radius of the cation and the ionization energy. It is assumed that the enthalpy of sublimation of the cation is relatively small and can be included in the ionization energy term. It is further assumed that the enthalpy of dissociation of the anion ia relatively small and can be included in the electron affinity term. Available literature data were used as initial values for all parameters, but these were allowed to adjust to best fit values. Modified Lattice Energy Method. Royer (1968) has reported several possible schemes for increasing the accuracy of the Kapustinskii lattice energy equation. These schemes all include terms to account for van der Waals and other miscellaneous forces acting at extremely short ranges. In fact, Hisham and Benson (1988) have pointed out that bonding in salts appears in general to be short range and that next-neighbor interactions are of minor importance. This background led to the inclusion in this study of a third parameter for each ion which was introduced in the form of an inverse sixth power of the ion radius. The inverse sixth power relation ensures a short-range treatment for the added parameter. The equation utilized is
+
+
[1201.6~Z+Z-/(rc rA)][1 - 0.345/(rc rA)] + + a m m + [(Tc + TA)(aC)/(rc rAI6I (21)
Cmm
where Tc and TA are fitted parameters representing the contribution to the overall van der Waals forces by the cation and anion, respectively. The modified lattice energy method requires three parameters for each ion: for the anion, radius (rA), electron affinity (AHm), and van der Waals forces (TA);for the and van der cation, radius (rc), ionization energy (AHCE), Waals forces (Tc). As for the basic Kapustinskii-based method, initial values for radii, ionization energy, and electron affiiities were furnished by reported values where available. Tuning of these values and fitting of the third parameters were accomplished by data regression.
Methodology A large database of ionic enthalpies of formation was needed. The database developed by ASTM Committee E27-07 from a variety of respected thermochemical data sources was utilized (Ratkey, 1990). The database contains information for 1545 inorganic compounds. Investigators assembling the database included quality ratings for the information for each compound. This rating was based primarily on published error analyses. If no published error analysis for an individual item was available, that item received a low quality rating. The authors here choose to use only those data having a high quality rating. The r e d u d , higher quality database contained a total of 806 enthalpies of formation. The compounds which comprise the database represent 117 ions, 75 of which are cations and 42 are anions. Each of the 117 ions is represented in no fewer than 3 and a average number of 14 compounds. The experimental enthalpies of formation contained within the database range in absolute value from 0.84 kJ/mol for FeHz to 13 744 kJ/mol for Calq(P04)6Fz. The reduced database utilized comes almost entwely from the following sources: Chase et al. (19851, CODATA (19891, Glyshko and Mededeva (19661, Glyshko and Gurvich (19791, Parker et al. (1976),and Robie et al. (1979). The sequential simplex minimization (Beveridge and Schechter, 1970) technique was used as the fitting routine for determining parameter values. The s u m of the squared errors between the predicted and experimental enthalpies of formation was used as the objective function to be
Table I. Three Correlations Investigated method no. of cation params no. of anion params WBB 3 6 2 KBM 2 MLE 3 3 Table 11. Error Associated with Each Correlation method av absolute error (kJ/mol) atd dev (kJ/mol) 44.4 WBB 68.6 54.8 38.1 KBM 21.8 MLE 33.6
.. .
rrrrmrmzed. Percentage deviations were originauy used but were abandoned due to undesired extreme weighting of the low-magnitude data. Since it was desired for any new correlation to exceed the performance of the WBB method, parameters for the WBB method were refit to the database used here to ensure a fair comparison. Parameters for each of the other three methods were fit in turn using the same database.
Results and Discussion In all, three correlations were examined in detail: (1) the Wilcox/Bromley/Brandenburg Method (WBB), (2) the Kapustinskii-based method (KBM),and (3) the modified lattice energy method (MLE). In each case the correlation was structured to provide values for compounds composed of 75 different cations and 42 different anions. This makes it possible to calculate 4914 enthalpy of formation values for compounds composed of a single cation species and a single anion species. The three correlations along with the number of parameters required to be fitted with each method are listed in Table I. Models with fewer parameters are to be desired provided acceptable accuracy is achieved. A weakness of the WBB method is ita requirement for a relatively large number of parameters. For the 117 ions available in the correlation a total of 477 parameter values are derived from a database of 806 compounds. Although such a model may be able to fit the database values well, its predictive abilities for other compounds are in question. Accuracy statistics are shown for the three correlations in Table 11. The WBB method demonstrated an average absolute error of 44.4 kJ/mol, but is a nine-parameter model. The KBM method gave resulta comparable to the WBB method but has the advantage of being only a four-parameter model. The modified lattice energy method, which has six parameters, showed the highest accuracy with a 21.8 kJ/mol average absolute error. For the MLE method, the magnitude of the average error represents 2.6% of the average enthalpy of formation value. However, percentage errors can reach high values for compounds having small enthalpies of formation. In the study the emphasis was placed on reducing the magnitude of the enthalpy of formation error, rather than reducing percentage error. More detailed performance statistics are available for the modified lattice energy method. Table I11 shows the average error for ions grouped by valence. The correlation shows no signifcant trend, although enthalpy of formation values for ions having valences of -3 and -4 are slightly less accurate. Table IV shows the distribution of errors categorized by each of the 117 ions. These range in value from a low of 0.2 kJ/mol for Mn3+to a high of 81.5 kJ/mol for Te4+. It is especially important that an enthalpy of formation correlation avoid as much as practical the extreme outlier. Large errors, even if relatively rare, have a dramatic impact when such a correlation is used for reactive hazard pre-
2366 Ind. Eng. Chem. Res., Vol. 31, No. 10,1992
Table 111. Error by Valenoe G~OUD Using MLE' valence +1 +2 +3 +4 -1 -2 -3 -4
no. of ions 10 24 30 11 25 14 2 1
no. of compds 176 356 225 49 462 299 33 12
av absolute error (kJ/mol) 19.8 22.2 22.6 22.5 22.6 19.8 25.2 30.4
'Total of 117 ions.
dictions for instance. To this end, the distribution of errors for the existing generalized correlation (WBB) is shown
in Figure 2. Although this correlation is usually quite good, the figure demonstrates a propensity for occasional large errors. Approximately 4% of the values have errors larger than 150 kJ/mol. The error distribution for the modified lattice energy method is shown in Figure 3. Here only 0.6% of the values have errors larger than 150 kJ/mol, a significant improvement. Table V presents all compounds with errors greater than 100 kJ/mol. To further test the modified lattice energy method, the enthalpies of formation for an additional 50 compounds were predicted. These compounds were not used in the parameter fitting procedure and represent data from Wagman et al. (1982). The results are summarized in Table VI. Predicted and experimental values for the
Table IV. Error for Each Ion Using MLE ion AI3 A1 Au Au Ba Be Bi Ca Cd Ce Ce co Cr Cr cs cu cu
DY
Er Fe Fe Ga Gd Hf HI32 HI3 Ho In
K
La Li Lu MI3 Mn Mn Mo Mo Mo Na Nd Ni 4"
NP
NP
Pd Pd Pr Pt Pt Pu Ra
Rb Re Sb
sc
Sm Sn SI Te
valence 1 3 1 3 2 2 3 2 2 3 4 2 2 3 1 1 2 3 3 2 3 3 3 4 2 2 3 3 1 3 1 3 2 2 3 2 3 4 1 3 2 1 3 4 2 2 3 2 4 3 2 1 4 3 3 3 2 2 4
no. of ComDdE 31 13 3 5 31 18 9 33 23 13 3 17 5 7 11 12 18 7 5 13 7 12 4 4 11 15 6 7 27 17 20 3 25 11 3 3 4 4 30 15 20 6 6 4 21 6 11 5 3 5 8 9 5 7 8 14 14 25 6
av error (kJ /mol) 27.1 31.2 8.6 1.1 13.7 53.1 45.3 28.1 30.0 27.2 1.1 11.9 11.3 27.3 26.5 45.8 14.6 16.4 26.2 22.8 31.9 19.7 12.9 18.0 17.9 20.8 17.2 24.0 10.4 33.8 25.8 1.3 33.4 10.6 0.2 4.2 14.7 2.3 10.7 20.7 15.6 6.2 16.6 5.5 25.4 28.2 24.6 22.7 2.0 4.6 15.4 10.4 63.3 27.3 14.7 21.0 20.0 20.5 81.5
ion Th Ti Ti T1 T1 Tm U U
v
W W Y Zn Zr Zr Zr BH4 Br CHOZ c1 ClOZ C103 C104 CN CNO CNS
co3
Cr04 Cr207 C2H302 0
F H
4
HC03 HP04 HS HS04 I 103 Mn04 Moo4 N N3 "2
NO2 NO3 0 2
0 OH PO4 Reo4 S
so3 so4
Se04 SiOs SiOl
wo4
valence 4 2 3 1 3 3 3 4 3 2 4 3 2 2 3 4 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -2 -2 -2 -1 -2 -1 -1 -1 -2 -1 -1 -1 -1 -1 -2 -3 -1 -1 -1 -1
-2 -2 -1
-3 -1
-2 -2 -2 -2 -2 -4 -2
no. of comDds 7 6 6 27 3 4 4 3 3 3 4 13 21 4 4 6 4 57 15 69 4 6 11 14 10 9 25 19 3 18 11 57 16 3 4 3 3 58 18 3 25 18 12 3 7 22 6 55 33 15 7 44 12 39 15 12 12 29
av error (kJ /mol)
16.9 11.7 19.4 1.0 20.6 10.5 0.6 1.1 5.0 6.6 34.8 16.3 26.0 24.8 17.8 2.0 26.9 13.8 21.0 10.4 9.3 13.8 23.6 32.4 9.1 11.9 17.6 2.4 15.4 6.5 26.0 44.3 1.5 1.3 0.4 0.3 30.2 35.5 1.7 20.7 13.8 15.7 1.3 9.0 13.7 12.6 19.0 22.2 38.8 13.7 23.3 18.7 26.8 18.6 17.4 30.4 27.4
Ind. Eng. Chem. Res., Vol. 31,No. 10,1992 2367
w
-11
1-11
-1
w
M
PUD
QUI,
Awry(. A h l u t e Error (kJ/mol) Figure 2. Error analysis for the WBB method. 1I
1
M
DID
lo11
10010
-1
zm.tw
lloIDQ
OOIO
4oow
Average Absolute Error (kJ/mol) Figure 3. Error analysis for the MLE method. Table V. Species with Calculated Errors Greater Than 100 kJ/mol &H' (kJ/mol) comud calcd eXDt1 error (kJ/mol) &,PO4 -1106.3 -990.4 -115.9 A d 86.6 -32.8 119.2 -166.4 -188.7 Be& -355.1 -353.5 -106.7 BeBr, -460.2 -140.5 -233.5 BeS -374.0 125.5 -2149.3 BezSi04 -2023.8 -110.0 -910.4 BiF3 -1020.6 -239.5 -710.0 Ca(I03h -949.5 -120.25 -1157.9 CrF, -1278.2 -163.4 110.0 CuCNO -53.3 -110.1 -75.7 MgHz -185.8 -158.4 -1916.3 Re(S04h -2074.6 -201.6 -1828.4 Te(S04)z -2030.0
enthalpy of formation and the absolute error for each compound are given. For these 50 compounds the average absolute error (with no additional fitting of parameters) was 43.7 kJ/mol. The magnitude of this error represents 5% of the average enthalpy of formation value in this list of 50 compounds. The three fitted parameters utilized by the modified lattice energy method for each ion are given in Table VI1 for the cations and in Table VIII for the anions. The fitted values obtained for the ion radii are in reasonably close agreement with the initial literature values. The other parameters exhibit larger differences when compared to the literature values; however, for our correlation these parameters are compite valuea including subhation and dissociation energies. It is also interesting to note that Royer (1968)has reported that Born-Madelung type expressions (on which the new model shown here is based) often work well for nonionic compounds. For example, consider CUI; the difference in the electronegativity between copper and
Table VI. ArHoPredicted by the MLE Method" A&Zo (kJ/mol) compd calcd exptl error (kJ/mol) -4108 -3960 -148 20 -500 -520 102 -125 226 -965 -956 -10 -474 -57 -417 -303 -342 39 -21 -284 -263 -1194 -47 -1146 13 -430 -443 7 -1418 -1425 -4 -745 -741 -2 1 -1168 -1146 -744 -14 -730 -10 -380 -370 -1090 27 -1117 -1176 -18 -1158 -340 48 -387 -1025 56 -1080 -1152 -10 -1142 -228 41 -269 -3044 72 -3117 -3 -1067 -1064 -1301 20 -1321 -412 -25 -387 3 -72 -75 -329 -1746 -2075 1 -57 -57 -17 -270 -287 53 -852 -799 48 -313 -265 -5 -618 -624 13 -1111 -1124 7 -467 -460 -1 -265 -264 -365 -10 -375 -241 -1815 -1574 -21 -908 -887 0 -1179 -1176 -21 -1045 -1024 10 -281 -290 5 -84 -78 -1012 -56 -956 28 -301 -328 -45 -458 -414 29 -319 -348 2 -261 -259 -14 -1159 -1141 -28 -1805 -1832 -61 -1825 -1764 178 -1542 -1720 "Total number of compounds = 50; average absolute error (kJ/mol) = 43.7;standard deviation (kJ/mol) = 79.9.
iodine is only about 0.5 unit which indicates that the CUI bond exhibits approximately 10%ionic character and 90% covalent character. Yet the evaluation of the lattice energy for CUI using the Born-Madelung expression results in a calculated value that is less than 7 % below the experimental value. This behavior is typical of a large number of compounds. Royer's explanation for this unexpected behavior is that the effective charges on the ions, thus the electrostatic interactions, decrease as the covalent interactions increase. The changes in these two types of interactions appear to almost exactly cancel each other over a wide range of values. Thus a Born-Madelung type expression can be applied to almost any crystal except one that is completely covalent in character. Conclusions The new modified lattice energy method appears to have satisfied the goal of an improved generalized computer method to predict the enthalpy of formation for inorganic
2368 Ind. Eng. Chem. Res., Vol. 31,No. 10, 1992
Table VII. ion Ag Al Au Au Ba Be Bi Ca Cd Ce Ce co Cr Cr CS cu cu
DY
Er Fe Fe Ga Gd Hf Hg2 Hg Ho In K La Li Lu Mg Mn Mn Mo Mo Mo Na Nd Ni 4"
NP NP Pb Pd Pr Pt Pt Pu
Ra Rb
Re Sb
sc
Sm Sn Sr Te Th Ti Ti
T1
T1 Tm U U V W W
Y
Zn Zr Zr Zr
MLE Parameters for Each Cation z+ rc (A) A h (kJ/mol) 1 3 1 3 2 2 3 2 2 3 4 2 2 3 1 1 2 3 3 2 3 3 3 4 2 2 3 3 1 3 1 3 2 2 3 2 3 4 1 3 2 1 3 4 2 2 3 2 4 3 2 1 4 3 3 3 2 2 4 4 2 3 1 3 3 3 4 3 2 4 3 2 2 3 4
0.921 0.851 0.301 0.319 1.092 0.595 0.914 0.892 0.730 1.005 0.830 0.686 0.665 1.020 2.374 0.836 0.601 0.958 1.123 0.659 0,869 0.834 0.960 0.857 1.452 0.615 0.948 0.871 1.801 0.980 1.316 1.087 0.772 0.658 0.488 0.594 0.722 0.712 1.379 1.009 0.658 1.325 1.046 0.852 0.758 0.595 0.997 0.462 1.233 1.300 1.189 1.876 0.896 0.843 0.944 0.977 0.673 1.025 1.098 0.960 0.684 0.836 1.004 0.260 1.160 1.033 0.889 0.742 4.275 0.724 0.957 0.658 1.001
0.850 0.885
1556 6407 1695 7331 2999 3597 6470 3148 3735 5861 9655 3794 3713 6343 1095 1462 3909 5925 5894 3783 6644 6478 5970 9666 3359 3819 5931 6438 1127 5884 1140 5909 3360 3677 6487 3829 6863 10204 1183 5886 3814 1271 5951 9620 3727 3865 5799 3977 8281 5681 2919 1149 10367 6605 6043 5907 3733 3034 9990 9355 3640 6329 1430 7194 5916 6078 9586 6530 2649 10296 5902 3698 3533 6256 9578
TC -33269 -16748 433 18245 -15057 -4919 5828 -5103 -7765 -12827 28733 -3520 -4087 -27553 -1077084 -8531 2654 -9801 -52875 -4008 -6814 -26 -19987 16432 217 15812 -8780 2339 -316686 -10299 -76516 -41425 -5704 -5932 24423 589 -14695 48243 -109244 -14453 -2889 -84021 -18083 27577 -3688 9690 -5237 11337 537247 -49805 -26870 -419333 3687 3600 -17879 -10408 4222 -10886 -10860 2950 -19202 -4376 -36801 17402 -69161 -53475 18747 5717 -72358 40045 -9115 -320 -57271 431 19170
Table VJII. MLE Parameters for Each Anion ion Zra (A) AHwA(kJ/mol) -341 -1 1.332 BH4 -1 Br -917 2.282 -1 -1117 1.938 CHOZ -915 2.109 -1 c1 -1 ClOZ -730 1.903 -1 C103 -850 1.963 -1 -812 1.837 C104 -1 1.892 -490 CN 1.413 -579 CNO -1 1.883 -597 -1 CNS 1.599 -1150 -2 co3 1.543 -2 -1321 Cr04 2.510 -2 -2773 Crz01 1.735 -1 -1095 CZH302 -2 -1197 1.498 c204 1.641 F -1 -891 1.796 H -1 -428 -1 LOO0 HCOB -695 1.094 -2 HP04 -1093 1.744 -1 HS -625 -1722 2.318 -1 HS04 2.243 I -1 -826 1.570 -1 -768 103 2.423 -1 MnOl -1384 1.611 -2 Moo4 -1509 1.187 -3 N 2198 1.858 -1 -403 N3 2.071 -1 -515 2" 1.902 -1 -796 NO2 1.997 -1 -940 NO3 1.885 -2 -736 0 2 1.537 0 -2 -308 1.588 -735 -1 OH 2.055 -1553 -3 PO4 2.712 -1599 Reo4 -1 2.001 -2 -661 S 2.030 -2 -1453 so3 1.544 -1345 -2 so4 1.467 -915 -2 Se04 1.704 -1663 -2 SiO3 2.654 -4 -1276 Si04 1.540 -1495 -2 wo4
T. 6192 -71 -4299 8500 -17261 34497 21566 -27291 4455 -7334 34355 35975 304833 290 30084 104 -31508 5789 13156 10501 92134 -12834 130 724 37006 7787 277 -113468 -394 5941 -4905 -4202 -8848 47648 -180647 -25534 34867 38083 28481 33797 82 28493
compounds. For the database compounds the new correlation exhibits approximately a 50% reduction in the average absolute error while utilizing fewer parameters as compared to the WBB correlation. The new correlation shows somewhat higher errors associated with the -3 and -4 valence states than it does for other valence states. The compound classes for which the correlation should perhaps be applied with special caution are the compounds containing Be2+,b4+, and Te4+. These cations exhibited the highest average absolute errors. The predictive powers of the modified lattice energy method were tested by the examination of 50 additional inorganic compounds. The enthalpies of formation for these compounds were predicted with a reasonably high degree of accuracy. However, the predictive powers of the correlation are of lower accuracy than that implied by the fit to the original database. The MLE correlation, although a significant improvement to the WBB method, is of much lower accuracy than estimation methods used for organic chemicals. A generalized correlation is valuable for incorporating into computer programs for quick routine predictions of moderate accuracy. However, in most cases, exhaustive hand calculations based on detailed thermochemical knowledge of particular compounds would be expected to yield more accurate predicted values for enthalpies of formation. The correlation developed here is of sufficient
Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 2369 accuracy to be useful in many computer-based routine estimations.
Acknowledgment We acknowledge the cooperation of members of ASTM Committee E27-07for the use of their inorganic thermochemical database and their helpful discussions.
Nomenclature A = anion a = number of anions B = constant C = cation c = number of cations D = ionic resonance energy between unlike atoms, eV e = charge of an electron, esu E = bond energies between two atoms, kJ/mol Ec = electrostatic energy of attraction, kJ/mol ER = electrostatic energy of repulsion, kJ/mol g = constant A",, = enthalpy of dissociation into an atom of A, kJ/mol AHk = enthalpy of sublimation of an atom of C, kJ/mol AHEA= electron affinity of the anion radical, kJ/mol AHH, = ionization energy of an atom of C,kJ/mol 4Zfo = enthalpy of formation of the pure stable state at 298 K and 101.33 kPa, kJ/mol j = constant K = electronegativity of an atom M = Madelung constant m = empirical anion parameter N A = Avogadro's number, molecules/mol n = number of apparent single bonds nN = number of single-bonded nitrogen atoms no = number of single-bonded oxygen atoms p = constant Q = empirical anion parameter r = interionic distance, A rc = radius of the cation, A rA = radius of the anion, A ro = equilibrium interionic separation in a lattice, A Tc = van der Waals force contribution of cation TA = van der Waals force contribution of anion Vo= lattice energy, kJ/mol u = number of ions in a molecule W = empirical parameter X = empirical parameter Y = empirical parameter Z+ = valence of cation 2- = valence of anion Literature Cited Anderson, H. W.; Bromley, L. A. A Method for Estimating the Heat of Formation of the Halides. J. Phys. Chem. 1959, 63, 1115.
Beveridge, G. S.; Schechter, R. S. Multivariable Search-Analytical Methods. In Optimization: Theory and Practice; McGraw-Hilk New York, 1970;pp 382-383. Brandenburg, N. P. Methods for Estimating the Enthalpy of Formation of Inorganic Compounds; Thermochemical and Crystallographic Investigations on Uranyl Salts of Group VI Elements. Doctor of Natural Sciences Thesis, University of Amsterdam, Amsterdam, Holland, 1978. Chase, M. W., Jr.; Davies, C. A.; Downey, J. R., Jr.; Fnuip, D. J.; McDonald, R. A.; syverud, A. N. JANAF Thermochemical Tables, Third Ed. J. Phys. Chem. Ref. Data 1985, 14 (Suppl. No. 1). CODATA Key Values for Thermodynamics. Cox, J. D.; Wagman, D. D., Mededeva, V. A.; Eds.; Hemisphere: New York, 1989. Frurip, D. J.; Freedman, E.; Hertel, G. R. A New Release of the ASTM CHETAH Program for Hazard Evaluation: Versions for Mainframe and Personal Computer. Plantloper. Prog. 1989,8 100. Glyshko, V. P.;Mededeva, V. A. Thermal Conutants of Substances; Viniti: Moacow, 1966; Vol. 1-10. Glyshko, V. P.; Gurvich, L. V. Thermodynamic Properties of Indi1979; Vol. 2. vidual Substances; Science: MOSCOW, Hisham, M. W. M.; Benson, S. W. Thermochemistry of Inorganic Solids. 8. Empirical Relations among the Enthalpies of Formation of Different Anionic Compounds. J. Phys. Chem. 1988,92, 6107-6112. Hisham, M. W. M.; Benson, S. W. Estimation of the Enthalpieg of Formation of Solid Salts. In From Atoms t o Polymers: Isoelectronic Analogies; Liebman, J. F., Greenberg, A., Eds.; VHC P u b lishers: New York, 1989; Chapter 10. Huheey, James E. Ionic Bonding. In Inorganic Chemistry; Harper and Row: New York, 1972;pp 56-91. Parker, V. B.; Wagman, D. D.; Garvin, D. 'Selected Thermochemical Data Compatible with the CODATA Recommendations". National Bureau of Standards Internal Report 75-968(PB 250845), 1976. Pauling, L. Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms. J. Am. Chem. SOC.1932,54,3570. Ratkey, C. D. Prediction of Enthalpies of Formation for Ionic Compounds. M.S. Thesis, University of South Alabama, Mobile, AL, 1990. Robie, R. A.; Hemingway, B. S.; Fisher, J. R. Thermodynamic Properties of Minerale and Related Substances at 298.16K and 1 Bar Premure and at Higher Temperatures. US.Geol. Sum. Bull. 1979,No. 1452. Royer, Donald J. The Solid State. In Bonding Theory; McGraw-Hik New York, 1968; pp 211-247. Sharp,A. G. The Structures and Energetics of Inorganic Solids. In Inorganic Chemistry;Longman Group: Eseex, 1986,pp 160-166. Waddington, T. C. Lattice Energies and Their Significance in Inorganic Chemistry. In Advances in Inorganic Chemistry and Radiochemistry; Emeleus, H. J., Sharp, A. G., Eds.; Academic Preas: New York, 1959; Vol. 1, Chapter 4. Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R H.; Halow, I.; Bailey, S. M.; Churney, K.L.; Nuttall, R.L. The NBS Tables of Chemical Thermodynamic Properties. J. Phys. Chem. Ref. Data 1982,11 (Suppl. No. 2). Wilcox, D. E.; Bromley, L. A. Method for Estimating the Heat of Formation and Free Energy of Formation of Inorganic Compounds. Ind. Eng. Chem. 1963,55,32.
(a,
Receiued for review December 17,1991 Revised manuscript receiued June 18, 1992 Accepted July 17,1992
