Research: Science and Education
Quantifying Molecular Character P. G. Nelson Department of Chemistry, University of Hull, Hull HU6 7RX, UK
One of the basic phenomena of chemistry is that substances vary in character from completely molecular (e.g., hydrogen) to completely nonmolecular (e.g., diamond). Wells classified substances according to their type of structure (1). Jensen has developed this classification, and introduced the concept of the “degree of nonmolecularity” of a substance (2, 3b). In this article I show how Jensen’s ideas can be made quantitative, thereby underpinning their use in the teaching of chemistry. I also show how they can be combined with my classification of substances according to electrical character (4). Structural Classification of Substances Wells and Jensen divide substances into the following types (1, 2, 3b): MOLECULAR SUBSTANCES. Substances that comprise discrete clusters of atoms (molecules) with strong bonding within the clusters and weak bonding between them. CHAIN (1-D NONMOLECULAR) SUBSTANCES. Substances that comprise continuous chains of atoms, with strong bonding within the chains and weak bonding between them. LAYER (2-D NONMOLECULAR) SUBSTANCES. Substances that comprise continuous sheets of atoms, with strong bonding within the sheets and weak bonding between them. FRAMEWORK (3-D NONMOLECULAR) S UBSTANCES . Substances that comprise a continuous framework of atoms with strong bonding throughout. CLATHRATES. Substances that comprise clusters of atoms held within a continuous framework of other atoms, with weak bonding between clusters and framework.
Jensen has shown that all the types except clathrates can be represented on a tetrahedral diagram similar to that in Figure 1 (3b). He illustrated his diagram with examples, but his placement of examples along the edges of the tetrahedron was largely qualitative. Quantification of Structure Type Substances can be assigned to classes by comparing distances (d) between neighboring atoms with sums of van der Waals radii (∑rvdW) and atomic radii (∑rat). If the distance between two atoms lies between ∑rvdW and ∑rat, the bond between them is of intermediate strength. Alcock (5) calls such bonding “secondary”.1 Values of rvdW are given by Bondi (6 ) and Gavezzotti (7 ), and of rat by Slater (8). The latter are independent of the nature of the bonding. As an example, consider crystalline iodine. In this, iodine atoms are bound into diatomic units, the units are bound into layers, and the layers into a continuous framework (1). The bond distance within the units is 2.72 Å, between units within layers 3.50 Å, and between layers 4.27 Å. Comparison of these values with ∑rat ≈ 2.8 Å and ∑rvdW ≈ 4.2 Å (7, 8) shows that the bonds within the layers are of intermediate strength and that solid iodine is a semimolecular–semilayer substance. To specify the extent to which iodine is a semilayer substance, we need to know the strength of the secondary bonds. An index of the relative strength of such a bond is the quantity
n′ =
d vdW – d d vdW – d 1
p′
(1)
INTERMEDIATE SUBSTANCES. Substances in which the bonds holding units together are of intermediate strength, thus giving rise to structures that are intermediate between the above types.
where d1 is the length of a single bond, dvdW is the length of a van der Waals bond (= ∑rvdW ), and p′ is a constant. The value of n′ varies from zero for d = dvdW to one for d = d1. The value of p′ can be fixed by considering symmetrical systems of the type X--Y--X (e.g., HF2᎑, I3᎑), and setting the
Figure 1. Classification of substances after Jensen (3b). Intermediate types: 1, 2, 3, semimolecular; 1, 4, 5, semichain; 2, 4, 6, semilayer; 3, 5, 6, semiframework.
Figure 2. Classification of Figure 1 showing examples. D, diamond; G, graphite; Sψ , fibrous sulfur (1).
JChemEd.chem.wisc.edu • Vol. 77 No. 2 February 2000 • Journal of Chemical Education
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Research: Science and Education Table 3. Degree of Nonmolecularity (D) and Molecular Character () of Different Types of Substances
Table 1. Values of the Parameters in Equation 1 for Some Bonds Bond
d1 /Å
dv d W /Å
F- - H
0.92
2.47
O- - H
p′
b
2.4c
0.96a
2.57b
2.4d
Cl - - H
1.27a
2.94b
2.6e
Cl - - Cl
1.99a
3.54b
2.2f
Br- - Br
2.28a
3.90b
1.6g
I - -I
2.67a
4.20b
0.9h
a
Type
S - -S
2.03a
3.60i
(2.0)j
Se - - Se
2.34a
3.80i
(1.5)j
Te - - Te
2.70
4.12
(1.0)
k
i
j
a Ref
23. b Ref 7. c Calculated from b = 1.4 and d = 1.13 Å (25) for HF ᎑. The value of 2 ᎑ b was calculated from the dissociation energy of HF2 (26) and HF (23). d Calculated from b = 1.3 and d = 1.22 Å (25) for H(O CR) ᎑. The 2 2 ᎑ value of b was calculated from the dissociation energy of H(O2CR)2 (25) and of RCO2H into RCO2 + H (23). e Calculated from b = 1.2 and d = 1.57 Å (25) for HCl ᎑. The value of b 2 ᎑ was calculated from the dissociation energy of HCl2 (27 ) and HCl (23). f Calculated from b = 1.3 and d = 2.27 Å (28 , mean) for Cl ᎑. The 3 ᎑ value of b was calculated from the dissociation energy of Cl3 (29–32) and Cl2 (23). Estimated hydration energies (32, converted to ∆E ) were used to convert the dissociation energy of ref 29 to the gas phase. g Calculated from b = 1.5 and d = 2.54 Å (1) for Br ᎑. The value of 3 ᎑ b was calculated from the dissociation energy of Br3 (31–33) and Br2 (23). Estimated hydration energies (32, converted to ∆E ) were used to convert the dissociation energy of ref 33 to the gas phase. h Calculated from b = 1.7 and d = 2.93 Å (1 ) for I ᎑. The value of b 3 ᎑ was calculated from the dissociation energy of I3 (31, 32, 34, 35) and I2 (23). Estimated hydration energies (32, converted to ∆E ) were used to convert the dissociation energy of ref 34 to the gas phase. i Ref 6. j Estimated from other values in the table. k Ref 24.
Table 2. Bond Numbers and Bond Orders in Triiodide Ions Salt
da b /Å db c /Å
na b
nb c
∑n
n ′a b
n ′b c
∑n′
KI3 ⴢH2 O 2.93
2.93
0.49
0.49
0.98
0.85
0.85
1.70
CsI3
2.84
3.04
0.63
0.37
1.00
0.90
0.78
1.68
NH4 I3
2.79
3.11
0.72
0.30
1.02
0.93
0.74
1.67
I 2… I ᎑
2.67a
4.20a
1.00
0.02
1.02
1.00
0.00
1.00
NOTE: Calculated from eqs 1 and 4. Distances from ref 1. a Table 1.
∆E XYX → XY + X ∆E XY → X + Y
(2)
Some values of p′ obtained in this way are given in Table 1. For the bonds within the layers of iodine (d = 3.50 Å), eq 1 gives n′ = 0.5. This indicates that these bonds have 50% of the strength of a single bond and that iodine should be placed 50% of the way along the line from molecular to layer substances in Figure 1. This is shown in Figure 2, along with other examples discussed below. Equation 1 is more general than other equations that have been proposed. Reed, Curtiss, and Weinhold (9) set p′ = 1. This is a good approximation for I--I, but not for F--H, O--H, Cl--H, or Cl --Cl (Table 1). On the other hand, Pauling (10a) suggested the function d = d1 – c ′log10 n′ (3) 246
µ (%)
Molecular
0
100
Chain
1
67
Layer
2
33
Framework
3
Clathrate
—
0 50
where c ′ is a constant. This function gives n′ > 0 at d = dvdW. The residual is small (< 0.1) for F--H, O--H, Cl--H, and Cl--Cl, but not for Br--Br (0.2) or I--I (0.4).2 Pauling called n′ the “bond order” (10b). This must not be confused with the “bond number”, n, for which he used a similar equation (10a): d = d1 – c log10 n
(4)
Bond numbers are designed to add up to valencies (11). For example, for a system X--Y--X in which the X atoms share a single valency of Y, the bond numbers must add up to one. The bond orders, however, add up to more than one, reflecting the extra bonding in X--Y--X over X–Y + X, which causes X--Y--X to form. This is illustrated for (Ia--Ib--Ic)᎑ ions in Table 2. The bond orders have been calculated from eq 1 with p′ = 0.9 (Table 1); the bond numbers, from eq 4 with c = 0.85 Å (12). Similar values of n and ∑n can be obtained from eq 5 with p = 3.8.
n=
d vdW – d d vdW – d 1
p
(5)
Degree of Molecular Character Jensen (2) defined the “degree of nonmolecularity” or “dimensionality” (D) of a substance as the number of dimensions in which it is not molecular (Table 3). He recognized that this can take intermediate values, but did not quantify them. From the above discussion D can be calculated from the equation D = ν1 + ν2 + ν3
sum of the values of n′ for the two bonds, ∑n′, equal to
b =1+
D
(6)
where ν = 1 for primary bonds and n′ for secondary bonds, ν1 is the value for bonds linking molecules into chains, ν2 for bonds linking chains into layers, and ν3 for bonds linking layers into a framework. A related quantity is the degree of molecular character (µ), defined as the extent to which a substance comprises discrete clusters of atoms. Molecular substances are 100% molecular (Table 3). Chain substances are nonmolecular in one dimension but molecular in two, making them 67% molecular. Layer substances are likewise 33% molecular, and framework substances 0%. Clathrates are 0% molecular with respect to the host and 100% molecular with respect to the guest, averaging 50%. Molecular character can be calculated from the equation µ = 1 – 1⁄3(ν1 + ν2 + ν3)
(7)
µ = 1⁄2 µhost + 1⁄2(1 – n′host-guest)
(8)
For clathrates,
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Figure 3. Classification of substances according to electrical and molecular character.
Figure 4. Classification of Fig. 3 showing examples. Most substances fall within or near the irregular tetrahedral region shown in bold. D, diamond; G, graphite.
Once the basic structure of a substance has been established, the application of these equations is straightforward, as the following examples show. IODINE. The basic structure was derived earlier. For the bonds within the layers, n′ = 0.5. Thus in eqs 6 and 7, ν1 = ν2 = 0.5 and ν3 = 0, whence D = 1.0 and µ = 67%. TELLURIUM. In the ordinary form of this, tellurium atoms are bound into chains and the chains are bound into a continuous framework (1). The bond distance within the chains is 2.84 Å and between chains, 3.50 Å. Comparison of these values with ∑rat ≈ 2.8 Å and ∑rvdW ≈ 4.1 Å (6, 8) indicates that the bonding between the chains is secondary. Equation 1 with p′ ≈ 1.0 (Table 1) gives n′ = 0.4. Therefore in eqs 6 and 7, ν1 = 1 and ν2 = ν3 = 0.4, whence D = 1.8 and µ = 40%. The fact that the Te–Te distance in the chains is 0.14 Å greater than d1 and gives n′ = 0.9 in eq 1 does not affect the calculation because the molecular units are the chains, not the atoms. WATER. Ordinary ice (ice Ih) has a structure in which each oxygen atom is bound to two hydrogen atoms at 1.01 Å and two at 1.74 Å (1, 13).3 Comparison of these values with ∑rat ≈ 0.85 Å (8) or d1 = 0.96 Å (Table 1) and ∑rvdW ≈ 2.57 Å (7 ) indicates that two of the bonds are primary and two are secondary. From eq 1 the order of the secondary bonds is 0.20. The secondary bonds hold the H2O molecules in a framework. Thus in eqs 6 and 7, ν1 = ν2 = ν3 = 0.20. When ice melts a fraction ( f ) of the secondary bonds break, so that ν¯ 1 = ν¯ 2 = ν¯ 3 = 0.20(1 – f ). Thus if f ≈ 10% (14 ), D ≈ 0.5 and µ ≈ 80%. LIQUID HYDROGEN FLUORIDE. Solid hydrogen fluoride comprises HF molecules held together in chains (1). The hydrogen–fluorine distance between molecules is 1.56 Å. This corresponds to n′ = 0.28. On melting, some of the secondary bonds will be broken and the chain structure confined to domains. If 10% of the secondary bonds are broken, the average value of n′ will be 0.25, whence D ≈ 0.3 and µ ≈ 90%. SULFURIC ACID (100%). Solid sulfuric acid comprises H2SO4 molecules held together in layers (1, 15). The hydrogen–oxygen distance between molecules is 1.49 Å. This corresponds to n′ = 0.38. The average value for the liquid will be about 0.34, whence D ≈ 0.7 and µ ≈ 75%. All these examples relate to ordinary conditions. Substances can change character under extreme conditions. All
are molecular (or decompose) at high temperatures, all are nonmolecular at high pressures (16 ). New Classification of Substances Molecular character can be used alongside electrical properties (4 ) to give the classification shown in Figures 3 and 4. The coordinate system is based on a trigonal prism. Electrical properties are plotted along the edges of the triangular base, and µ is plotted vertically. The quantity c is a conductivity index; t is the degree of electrolysis (4b). Most substances lie within or near the tetrahedral region shown in bold in Figure 4, reflecting the fact that nonmolecular substances can be metallic, electrolytic, or insulating, whereas fully molecular substances are only insulating. However, some substances have a higher electrical conductivity than their molecular character might suggest (e.g., HF, H2SO4). The new classification arranges substances in a way similar to tetrahedral classifications based on bond type (17, 18). It is, however, more securely based than the latter, bond type being an imprecise concept (4a, 19). Indeed, published scales of ionicity differ from one another by up to an order of magnitude (20). Jensen (3) has stressed the importance of not confusing different levels of chemical description (Table 4). My classification employs two levels of description (electrical properties belong to level 1, molecular character to level 2), but keeps them distinct.
Table 4. Levels of Chemical Description Level
Name a
Example
1
Bulk
Diamond is a colorless, hard, crystalline substance
2
Atomic
A crystal of diamond comprises a continuous framework of carbon atoms, with each atom bound to four others
3
Electronic
A crystal of diamond comprises a continuous array of C 4+ cores held together by four times as many electrons
aMy terminology (36). Jensen (3) uses “molar”, “molecular”, and “electrical”, respectively.
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Research: Science and Education
Jensen has also argued that chemistry should be taught in a logical order, with levels 1 and 2 before level 3. On this basis, my classification is best presented before bond type. It can then be used to help to introduce the latter. Acknowledgments I am very grateful to William B. Jensen for stimulating correspondence on issues in this paper, and to the reviewers for suggesting improvements to it. Notes 1. This use of “secondary” must not be confused with Werner’s (21). 2. Calculated from values of c′ fixed by the values of b and d in the footnotes to Table 1: 1.4 Å (F--H, O--H, Cl--H), 1.5 Å (Cl--Cl), 2.1 Å (Br--Br), and 3.7 Å (I--I). 3. These values may be means of two sets of values (22).
Literature Cited 1. Wells, A. F. Structural Inorganic Chemistry, 5th ed.; Clarendon: Oxford, 1984. 2. Jensen, W. B. In The Structures of Binary Compounds; de Boer, F. R.; Pettifor, D. G., Eds.; North-Holland: Amsterdam, 1989. 3. (a) Jensen, W. B. J. Chem Educ. 1998, 75, 679. (b) Ibid., p 817. (c) Ibid., p 961. 4. (a) Nelson, P. G. J. Chem. Educ. 1994, 71, 24. (b) Nelson, P. G. J. Chem. Educ. 1997, 74, 1084. 5. Alcock, N. W. Adv. Inorg. Chem. Radiochem. 1972, 15, 1. 6. Bondi, A. J. Phys. Chem. 1964, 68, 441. 7. Gavezzotti, A. J. Am. Chem. Soc. 1983, 105, 5220. 8. Slater, J. C. J. Chem. Phys. 1964, 41, 3199. 9. Reed, A. E.; Curtiss, L. A.; Weinhold, F. Chem. Rev. 1988, 88, 899, eq 25. 10. (a) Pauling, L. The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1960; Sect. 7-10. (b) Ibid., Sect. 7-6. 11. Nelson, P. G. J. Chem. Educ. 1997, 74, 465. 12. Bürgi, H.-B. Angew. Chem., Int. Ed. Engl. 1975, 14, 460. 13. Cotton, F. A.; Wilkinson, G. Advanced Inorganic Chemistry, 5th ed.; Wiley: New York, 1988; Sect. 3-4.
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14. Symons, M. C. R. Acc. Chem. Res. 1981, 14, 179. 15. Moodenbaugh, A. R.; Hartt, J. E.; Hurst, J. J.; Youngblood, R. W.; Cox, D. E.; Frazer, B. C. Phys. Rev. B 1983, 28, 3501. 16. Hall, H. T. Prog. Inorg. Chem. 1966, 7, 1. 17. Grimm, H. G. Z. Elektrochem. 1928, 34, 430; Angew. Chem. 1934, 47, 53. 18. Laing, M. Educ. Chem. 1993, 30, 160. 19. Nelson, P. G. Educ. Chem. 1994, 31, 93. 20. Meister, J.; Schwarz, W. H. E. J. Phys. Chem. 1994, 98, 8245. 21. Werner, A. New Ideas on Inorganic Chemistry; Hedley, E. P., Translator; Longmans: London, 1911. 22. Li, J.; Ross, D. K. Nature 1993, 365, 327. 23. CRC Handbook of Chemistry and Physics, 77th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1996. 24. Kruse, F. H.; Marsh, R. E.; McCullough, J. D. Acta Crystallogr. 1957, 10, 201. Cf. Teller, R. G.; Krause, L. J.; Haushalter, R. C. Inorg. Chem. 1983, 22, 1809. 25. Emsley, J. Chem. Soc. Rev. 1980, 9, 91. 26. Clark, J. H.; Emsley, J.; Jones, D. J.; Overill, R. E. J. Chem. Soc., Dalton Trans. 1981, 1219. 27. Yamdagni, R.; Kebarle, P. J. Am. Chem. Soc. 1971, 93, 7139. Thomson, C.; Clark, D. T.; Waddington, T. C.; Jenkins, H. D. B. J. Chem. Soc., Faraday Trans. 2 1975, 71, 1942. 28. Bogaard, M. P.; Peterson, J.; Rae, A. D. Acta Crystallogr., Sect. B 1981, 37, 1357. 29. Hine, F.; Innta, S. Bull. Chem. Soc. Jpn. 1968, 41, 71. 30. Robbiani, R.; Franklin, J. L. J. Am. Chem. Soc. 1979, 101, 3709. 31. Novoa, J. J.; Mota, F.; Alvarez, S. J. Phys. Chem. 1988, 92, 6561. 32. Ogawa, Y.; Takahashi, O.; Kikuchi, O. J. Mol. Struct. (Theochem) 1998, 424, 285; 1998, 429, 187. 33. Scaife, D. B.; Tyrrell, H. J. V. J. Chem. Soc. 1958, 386. 34. Ramette, R. W.; Sandford, R. W. Jr. J. Am. Chem. Soc. 1965, 87, 5001. 35. Topol, L. E. Inorg. Chem. 1971, 10, 736; Danovich, D.; Hruˇsák, J.; Shaik, S. Chem. Phys. Lett. 1995, 233, 249. Sharp, S. B.; Gellene, G. I. J. Phys. Chem. A 1997, 101, 2192. Sato, H.; Hirata, F.; Myers, A. B. J. Phys. Chem. A 1998, 102, 2065. 36. Nelson, P. G. What Is Chemistry, That I May Teach It? Department of Chemistry, University of Hull: Hull, 1981; p 22.
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