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Steven D. Gammon University of Idaho Moscow, ID 83844
A Simplified Mathematical Model for Calculating the Lattice Energies of Binary Group I and II Ionic Compounds: Using Solver to Estimate the Empirical Constants
W
Alan Carter Department of Chemistry, Wellington College, Crowthorne, Berkshire, RG45 7QP, UK;
[email protected] The lattice energies U0 of ionic compounds are widely discussed in undergraduate courses and text books (1–5). They are experimentally calculated from Born–Haber cycles, and are used to interpret the solubility of ionic compounds and the thermal stability of compounds with complex anions. Calculated values of lattice energies are used to estimate the enthalpy of formation of unknown compounds (3–5). The modeling work in this paper was stimulated by an article that showed the use of the nonlinear, least-squares, curve-fitting Microsoft Excel Solver program to model experimental chromatographic data (6 ). The aim was to develop a mathematical model for undergraduate students using basic electrostatics with as few empirical constants as possible. A focus on the Madelung constant (M) for different crystal structures can obscure simple electrostatic relationships. The model simplifies the Born–Mayer equation (1–5) (eq 1); the symbols are explained in the sections that follow.
M NA m n e 2 ρ U0 = 1– R R The simplified model
(1)
Assumes no specific crystal structure; Focuses on the proportionality of attractive potential energies to the size of the cation and anion charges, and the inverse relationship to interionic distances; Involves directly the moles of ions in the empirical formula and considers only nearest neighbor attractions; Uses self-consistent, 6-coordinate Shannon (7 ) ionic radii to estimate interionic distances in the compounds— known interionic distances are not assumed; Uses the Born–Mayer exponential repulsion function, but with empirical constants.
The Solver technique (6 ) was used to test the equation developed (eq 10) against the Born–Haber values for the lattice energy (8a, 8b) and to determine the values of the empirical constants. This was done initially for group I halides. The ease of the Solver technique and the success of the simple model led to an extension of the spreadsheet to group I and II halides, oxides, and sulfides (with the exception of beryllium compounds). This too gave surprisingly good agreement with Born–Haber cycle lattice energies. Initially it was unclear to me why this simple model was so successful. But on reading the literature of lattice energy calculations (1–5, 9–12) more deeply, I was found that a result similar to the Kapustinskii equation (10) (eq 12) had been independently obtained. The reasons for the relative success
of this simplified model are discussed and I make suggestions for the incorporation of this and related modeling exercises. The Mathematical Model 1. The lattice energy of an ionic crystal, U0, is the molar internal energy change, at absolute zero and at 1 atmosphere pressure, accompanying the separation of constituent ions so that they are infinitely removed from one another. For a compound X xYy that contains X n+ and Y m ions, X xYy(s) → x X n+(g) + yY m(g)
U0
(2)
The electrostatic energy of attraction between two ions of charge n+ and m separated by a distance R, where e is the charge on an electron and εo is the vacuum permittivity, is:
E =
n m e2 4 π ε0 R
(3)
2. In the example of CaCl2, there are two Cl ions per Ca ion; thus in 1 mol of solid CaCl2 the number of sets of {ClCa2+Cl} ions is equal to Avogadro’s constant (NA). The attraction between two sets of these ions is shown below, where the solid circles indicate ions belonging to the two sets and the broken circles show ions belonging to other sets in an idealized part of the CaCl2 structure.. 2+
1−
2+
1−
2+
2+
1−
2+
1−
3. The electrostatic energy of attraction between the two sets of ions, using eq 3, is
E=
3 1 2 e 2 π ε0 R
(4)
The factor 3 arises because there are three Ca2+:Cl attractions per empirical formula unit of ions. For 1 mole of the ionic solid X xYy(s), with ion charges n+ and m there are (x + y)NA sets of these ion attractions. In a very simplified way one can assume that the total attractive energy EA, ignoring crystal geometry and attractive forces other than nearest neighbors, is
EA ∝
n m e 2 x + y NA 4 π ε0 R
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4. Since e, NA, and ε0 are constants, we can replace them and the proportion sign by a constant A; Anm x + y EA = (6) R 5. Oppositely charged ions do not attract each so strongly that they collapse to a point. Following the argument in the Born–Mayer equation: a. The electron clouds surrounding each ion repel each other so that the equilibrium interionic distance R0 is reached when the attractive and repulsive forces are in balance.
(7)
where B and C are constants.
is E:
Anm x + y – B exp CR R0
(8)
7. At R0 , the observed internuclear distance, E is a minimum, dE/dR = 0, and
B exp CR 0 =
Anm x + y CR 02
(9)
Substituting for B exp(CR 0 ) in eq 8, and since U0 = E, and putting B = 1/C,
U0 =
Amn x + y R0
1– B R0
(10)
Results and Test of Equation 10 An Excel spreadsheet was set up for 54 group I and II halides, oxides, and sulfides to test eq 10. Beryllium compounds were removed from the final spreadsheets because of relatively poor standard errors with their inclusion. It was assumed that R0 for any compound is equal to the sum of the six coordinate cation and anion Shannon radii. These radii and most of the experimental values of lattice energies were taken from the 1997/98 Handbook of Chemistry and Physics (8a); however, the values for group II oxides and sulfides do not appear in this volume and were taken from an earlier volume (8b). The full spreadsheet (spreadsheet 1) is available via JCE Online;W it gives, in addition to Table 1 below, the 6 coordinate ionic radii, the percentage difference between the Born– Haber cycle values for lattice energies and the calculated values, the Kapustinskii equation values, and the difference between them and the cycle values. Table 1 shows part of this spreadsheet for 23 binary compounds. The Solver technique (3) was used to solve for A and B. In Table 1 the target cell G27 is minimized. This is the sum of the squares of differences between the calculated and experimental Born–Haber values of the lattice energies for the 54 compounds. Solver minimizes cell G27 by varying trial values for A and B in cells E28 and G28, respectively. 1082
1254.7 m n x + y R0
1 – 0.420 R0
(11)
The “jackknife” method (3) was used to determine the standard errors for A and B. Values found with their standard errors and coefficients of variation (SE ÷ mean × 100%) are
The attractive term (1254.7nm(x + y)/R0) has a larger contribution to the lattice energy than the repulsion term {[(0.420)(1254.7nm(x + y))]/R02}: on average, the repulsion term is 14.6% of the attractive term (opposite signs).
6. The total simplified electrostatic energy of the crystal
E = EA – ER =
U0 =
A = 1254.7 ± 15.3 kJ mol1 Å (± 1.2%) B = 0.420 ± 0.027 Å (± 6.5 %)
b. Quantum mechanics suggests that for spherical ions the electron density of the outer shell electrons falls off almost exponentially with distance. Thus for group I and II oxides, sulfides, and halides, the repulsive potential energy ER can be put in the form ER = B exp(CR )
The full spreadsheet shows that A = 1254.7 kJ mol1 Å and B = 0.420 Å, and eq 10 becomes:
Discussion Equation 11 gives good agreement with the experimental values of lattice energy. The percentage differences between the two sets of values are shown in column I of spreadsheet 1 available via JCE Online. The calculated values of the lattice energies for half the compounds are within ±2%, and 95% are within ±5% of the experimental values The greatest differences occurred in the group I halides, which averaged 4.2% below experimental values. The adequacy of the model fitted was checked by means of residual plots; that is, the differences between the experimental values and eq 11 values were plotted against eq 11 values. No obvious pattern was observed, demonstrating that the model was a good fit. Equation 11 predicts values smaller than the experimental ones by an average of 1.4%. The good agreement between the experimental lattice energies and calculated values is pleasing. However, considering the simplified nature of the model and the wide range of group I Table 1. Part of the Spreadsheet Used to Test eq 10 and Calculate the Empirical Contants A and B A B C 1 COMPOUND x+y n.m 2 ions/mol 3 Na2O 3 2 3 2 4 Cs2O 3 2 5 Li2S 3 2 6 K2S 7 MgO 2 4 8 CaO 2 4 9 SrS 2 4 10 BaS 2 4 11 NaF 2 1 12 KF 2 1 13 NaCl 2 1 14 CsCl 2 1 15 LiBr 2 1 16 RbBr 2 1 18 CsI 2 1 3 2 19 MgF2 3 2 20 MgCl2 3 2 21 CaBr2 3 2 22 SrCl2 3 2 23 BaCl2 3 2 25 BaI2 26 27 28 Constants
D E Ro LE cycle /Å /kJ mol1 2.42 2478 3.07 2063 2.6 2472 3.22 2052 2.12 3791 2.4 3401 3.02 2848 3.19 2725 2.35 923 2.71 821 2.83 786 3.48 659 2.72 807 3.48 660 3.87 604 2.05 2957 2.53 2526 2.96 2176 2.99 2156 3.16 2056 3.55 1877
A. . 1254.7
Journal of Chemical Education • Vol. 77 No. 8 August 2000 • JChemEd.chem.wisc.edu
F G LE eq 10 (LEcyc-LE10)2 /kJ mol1 2570.6 8570.0 2116.5 2860.8 2427.4 1986.9 2032.8 368.5 3796.1 25.8 3449.9 2395.2 2861.2 173.4 2732.0 49.4 876.9 2129.2 782.4 1492.0 755.0 959.1 634.0 624.6 780.0 727.7 634.0 675.6 578.0 675.6 2919.4 1412.3 2481.3 2000.0 2182.2 38.4 2163.9 62.3 2065.5 90.2 1869.6 55.2 sum of squares Target cell 77904.5 B. . 0.420
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and II compounds with a variety of structure types that were used, this agreement is surprising. The question arises, why is a simple model so successful at estimating the experimental lattice energies? The conventional Madelung constants (Mc) incorporate the highest common factor of the ion charges as discussed by Huheey (4, p 60). Fortunately this confusing practice is not used much in modern literature, and a geometric Madelung constant (Mg) from which the ion charges have been separated is now largely used (3, 5). Equation 11 is simplified in comparison to the Born–Mayer equation, as it uses a constant for all crystal structures in lieu of a crystal-specific Madelung constant and the moles of ions in the formula are explicitly used. Kapustinskii was the first to point out that if the geometric Madelung constant is divided by the number of moles of ions per formula unit, a relatively constant value, the Madelung constant per ion ( Mi ), is obtained for all structures (10). As shown in Table 2, an average Mi is 0.847 ± 0.014 (±1.7%) for the five structure types encountered in group I and II binary compounds: here the standard error in Mi equals – standard deviation/√ n . Thus postulates 3 and 4 in the model work at acceptable levels of accuracy because the constant of proportionality Mi between electrostatic attraction and lattice energy varies only by ±1.7% for each crystal structure. These values of Mi represent the increase in electrostatic energy on forming a threedimensional lattice, compared to the value of 0.5 per ion or 1.0 per pair (3). Table 2. Relationship between Structure, Coordination Number, and Various Types of Madelung Constants Coordination Conventional Number Mc
Structure
Geometric Mg
M i (Mg /mol of ions in formula)
Rock salt
6:6
1.748
1.748
0.874
Fluorite
8:4
5.039
2.519
0.840
Anti-fluorite
4:8
5.039
2.519
0.840
Rutile
8:4
4.816
2.408
0.802
Cesium chloride
8:8
1.763
1.763
0.881
Average
0.847
Kapustinskii also pointed out that both Mi and the ionic radius increase with increasing coordination number, as seen above, and that this can further decrease the differences between Mi/R0 for each crystal structure (10). He proposed that there exists a hypothetical rock salt structure that is energetically equivalent to the actual structure of any ionic substance. Lattice energies can be calculated using Kapustinskii’s equation, eq 12.
U0 =
1214.0 n m x + y R0
1 – 0.345 R0
(12)
The constant 1214.0 includes e 2, 4 πε0, and the geometric Madelung constant for the rock salt structure. A repulsion constant of 0.345 Å was used; this comes from compressibility data for alkali metal halides, although it is a variable for accurate work (2, 9). The sum of 6-coordinate Shannon ionic radii may be used to estimate R0 for lattice energy calculations. The Shannon 6-coordinate ionic radii in the simple model developed in this paper were used to avoid assuming a crystal structure and to achieve a “fair test”. Thus the simplified model achieves a good agreement with experimental values partly for the reasons first developed by Kapustinskii. It was interesting to “rediscover” this type of relationship, albeit with different values for the two constants. The value of 1214.0 kJ mol1 Å was recalculated with values for fundamental constants taken from CRC Handbook of Chemistry and Physics 1997/98; different values appear in the literature, for example 1203.3 (12) and 1210 kJ per mol1 Å (5). Values for the lattice energy calculated using the Kapustinskii equation are shown in column N in the full spreadsheet (spreadsheet 1) available via JCE Online.W The percentage differences between Born–Haber values and these values are shown in column O. It can be seen that eqs 11 and 12 give very similar values. For group I and II binary compounds eq 11 gives values that are 1.4% lower and the Kapustinskii equation gives values that are 2.2% lower on average than experimental values. Thus the simple model developed in this paper gives values slightly closer to the experimental values than does the Kapustinskii equation. Teaching Approach and Other Modeling Exercises
4000
Group II oxides and sulfides Group II halides
Lattice energies / (kJ mol−1)
3500 Group II oxides and sulfides 3000
2500
2000
Group II halides Group I oxides and sulfides
1500
1000
y = 6915.5 x −0.801 R 2 = .9962
Group I oxides and sulfides Group I halides
y = 5352.8 x −0.8227 R 2 = .9813
y = 5275 x −0.827 R 2 = .9805 y = 1924.4 x −0.8584 R 2 = .9975
Group I halides
1. The lattice energies = KR 0 x for similar sets of compounds such as group I halides; x varies between 0.80 and 0.86 for the four sets of compounds. Thus the lattice energy varies roughly as 1/R 0.
500
0 2.6
2.8
3.0
3.2
3.4
I have successfully used this model and Solver in my teaching. As a first step I give students an Excel spreadsheet with the 6-coordinate ionic radii and the experimental values for the lattice energies of group I and II oxides, sulfides, and halides. Students use the spreadsheet to estimate the interionic distances R0, graph R0 versus lattice energies for the four sets of compounds (group I halides, etc.), and insert the trend lines to maximize R 2 to obtain Figure 1. A class discussion of Figure 1 leads to the following conclusions:
3.6
3.8
Interionic radii R 0
Figure 1. The lattice energies of group I and II halides, oxides, and sulfides versus the interionic radii R0
2. K is a function of the product of the cation and anion charges. For example, group I oxides and sulfides have a value for K 3.6 times as large as that for group I halides. The use of Coulomb’s law of attraction of elec-
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from the use of eqs 11 and 14, but the exercise is illustrative of a general method (4–6, 10–12) for estimating thermochemical radii of ions known to be nonspherical.
trostatic charges suggests that they should be 4 times larger, but there is a strong correlation between K and the product of the ion charges. 3. K is also a function of the moles of ions in the empirical formula. For example, group I oxides and group II halides have similar x and K values, but the value of K is on average 2.76 times greater than for group I halides. Coulomb’s law on its own would suggest that they should only be 2 times as large The difference can be explained by the number of ions per empirical formula unit, 3 moles and 2 moles, respectively. This suggests that ideally the lattice energies of group I oxides and sulfides should be 3 times larger, rather than the 2.76 times found empirically.
Conclusions
I then introduce the mathematical model to the students, as detailed in this paper, and explain the Solver technique using computer video projection of the data for the group I halides. I set the students the task of setting up the worksheet for the 54 binary compounds and using Solver to calculate the two constants A and B. It is straightforward to extend eq 10 in a similar way to the extended Born–Mayer equations (reviewed by Waddington [9]). Equation 13 includes a term C/R 06 to model the van der Waals forces and a fourth constant, D, for the zeropoint energy.
A mn x + y U0 = R0
1– B + C +D R0 R 06
(13)
The Solver calculation to find A, B, C, and D in eq 13 is included in spreadsheet 1 and is available via JCE Online.W Not surprisingly, the use of four constants significantly improves the agreement between experimental and calculated values. Seventy-eight percent of calculated values are within ±1%, 85% within ±2%. The average difference falls between 1.4% and 0%. Equation 13 becomes: 1249.3 m n x + y U0 = R0
1 – 0.462 + 5794.6 + 34.1 (14) R0 R 06
However the simplicity of eq 11 is lost. An additional useful modeling exercise involves using eqs 11 and 14 to estimate an “average” thermochemical radius for the highly polarizable hydride in nine group I and II salt like hydrides (i.e., LiH to CsH and MgH2 to BaH2). Spreadsheet 2 is available via JCE Online. The Solver “target cell” now contains a variable cation radius plus the unknown “constant” radius of the H ion. Experimental values for the lattice energy (8a) and 6-coordinate ionic radii for cations (7 ) were used. Solver minimized the differences between calculated (eqs 11 and 14) and experimental lattice energies by varying the H radius. The jackknife procedure (6 ) gave the standard errors. Values found for the ionic radii of the hydride ion were 1.62 ± 0.04 Å from eq 11, 1.63 ± 0.04 Å from eq 14, and 1.62 Å from the Kapustinskii equation. These calculated values are physically reasonable, as the radius of the free ion has been calculated to be 2.08 Å, and the radius of the H ion varies from 1.30 in LiH to 1.54 Å in RbH (1). These experimental values are smaller than the value found 1084
The simplified model developed in this paper gives good estimates for lattice energies without knowledge of the crystal structure or actual lattice parameters. It allows the student to focus on basic electrostatics—ion charge, the moles of ions per empirical formula, and self-consistent ionic radii. Clearly, eq 11 and the Kapustinskii equations cannot compete with the extended Born–Mayer equations (1–5, 10) and the more recent work of Holbrook et al. (13), who used novel attraction and repulsion functions to calculate the lattice enthalpy for most ionic compounds within 1%; but these methods require knowledge of the crystal structure and actual interionic distances. Equation 14, however, gives excellent calculated values, similar in accuracy to those in the literature (1–5, 13, 19). Equation 11 gives more reliable values for lattice energies than the Kapustiskii equation, and eq 11 or 14 could be used instead of it for determining self-consistent thermochemical radii, electron and proton affinities, and the calculation of the enthalpy of formation of hypothetical compounds (4, 5, 9–11). Acknowledgments I am very grateful to P. G. Barratt, Department of Physics, Wellington College, Crowthorne, Berkshire, UK, for his critical and constructive reading of this paper, and to P. A. Bowtell, Department of Department of Applied Statistics, University of Reading, Berkshire, RG6 6FN, UK, for his useful comments and advice on the statistical analysis in the paper. W
Supplemental Material
Two spreadsheets are available as supplemental material for this article in this issue of JCE Online. Literature Cited 1. Greenwood, N. N.; Earnshaw, A. Chemistry of the Elements; Pergamon: Elmsford, NY, 1984; p 95. 2. Atkins, P. W. Physical Chemistry, 3rd ed.; Oxford University Press: New York, 1986; p 596. 3. Alcock, N. W. Bonding and Structure; Ellis Horwood: London, 1990; p 85. 4. Huheey, J. E. Inorganic Chemistry, 3rd ed.; Harper International: Cambridge, UK, 1983; p 59. 5. Shriver, D. F.; Atkins, P. W.; Langford, C. H. Inorganic Chemistry, 2nd ed.; Oxford University Press: Oxford, 1994; pp 166–179. 6. Harris, D. C. J. Chem. Educ. 1998, 75, 119. 7. Shannon, R. D. Acta Crystallogr., Sect. A 1976, 32, 751–767. 8. (a) Jenkins, H. D. B. In CRC Handbook of Chemistry and Physics, 78th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1997/98; pp 12-14, 12-22. (b) Jenkins, H. D. B. In CRC Handbook of Chemistry and Physics, 63rd ed.; Weast, R. C.; Astle, M. J., Eds.; CRC Press, Boca Raton, FL, 1982. 9. Waddington, T. C. Adv. Inorg. Radiochem. 1959, 1, 157. 10. Kapustinkii, A. F. Q. Rev. Chem. Soc. 1956, 20, 203. 11. Moody, G. J.; Thomas, J. D. R. J. Chem. Educ. 1965, 42, 204. 12. Holbrook, J. B.; Sabry-Grant, R.; Smith, B. C.; Tandel, T. V. J. Chem. Educ. 1990, 67, 304.
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