Approximation of Spherical Polyatomic Thermochemical Radii of General Formula MXnzHugo Solk-Correa' CINVESTAV-IPN-MERIDA, Apdo. Postal 73 Cordemex. 97310 MBrida. YucaGn, Mexico Jacobo Gmez-Lara lnstituto de Quimica UNAM, Circuito Exterior, C. U., 04510 Mexico, D. F., MBxico
+ Tx). AS real bonds have an intermediate character, the calculated internuclear distance for either of these two extreme cases is not the real one. Instead, it is possible to obtain a "corrected radius" taking into account the intermediate character of the M-X bond according to
Kapustinskii's equation is used to obtain an approximate lattice energy value of ionic compounds whose structural parameters are not known, as this semiempirical equation depends on the internuclear distance do only, that is, the sum of cationic and anionic radii2
whered,, = r , + r., in picometers,r,,r, = cationir and anionic radii, I , = numl~rrof ions in the simplest formula unit (e.g., v = ? t i ~CuC1.and r I , = 3 for CuliX0~Ll.Z-.Z. . " - , , ,. =moduli of the cation and anion charges, Uo = energy of the crystal lattice, in kJ mol-I. Values of ionic radii for crystals formed by monoatomic ions are available from manv sources3 includine most of the periodic tables for s t u d e n t ' s ~ s e However, .~ wceu a crystalline compound is formed by polyatomic ions, such as (NH.dC1, Na(BH4), or (Me4N) (ClOa), the term "ionic radius"-cationic or anionic-loses its original meanine. When the lattice energy of such ionic compounds is availahre from thermochemical data,2 the Kapustinskii equation gives the empirical internuclear distance, and, if one of the ionic radii is known, then i t is possible to calculate the other one.5 This em~iricalradius is known as a "thermochemical radius". Here, we present a method to calculate the approximate thermochemical radii for polyatomic ions, of general formula MX:- (e.e., BF;, COf, PO?-, etc.), where M is a central metallic or &nmGallic Lon and x is aperipheralatom singly or multiply bonded to M. The M-X bond lies between two limitingcases: (1) ionic, where, the internuclear distance is the sum of the ionic radii ( r h r;) and (2) covalent where the internuclear distance is the sum of the covalent radii (&
+
where i. = corrected radius, r = ionic radius, F = covalent radius, and c = covalent character of M-X single bond, obtained using Pauling's electronegativity values.6 A convenient approximation is suggested by Barbe,7 xx-XM e=l-p-xx
-x,
(3)
xx
'
Author to whom correspondence should be addressed: present address: Hda. de 10s Portales No. 84, Fracc. Prados del Rosario, 02410 Mexico, D.F., MCxico. Kapustinskii, A. F. Quart Revs. 1956, 10, 283. Dean. J., Ed. Lange's Handbook of Chemistry; McGraw-Hill: New York, 1979. Jenkins, H. D. 6.. Ed. Handbook of ChemishyandPhys ics. 59th ed.: CRC: Boca Raton. FL. 1979. Huheev. J. E. lnoroanic Chemistry. ~rincipiesof ~truchrreand ~eactivifyf 'Harper 8 Row: 1972: n 7 3 ~ New ~- Yark. ~- ~~. "Table of Periodic Properties of the Elements"; Sargent-Welch Catalog No. 5-18806; 1979. Jenkins, H. D. 6.; Thakur, K. P. J. Chem. Educ. 1979, 56, 576. 'The reference in footnote 4 contains a table of "Percent lonic Character of a Sinqle Bond" as function of the difference in electronegativity. In this case, the applied formula is % I.C. = 1 6 1 ~ ~ 13.5 (Ax)2,proposed by Hannay. N. 6.; Smith C. P. J. Am. Chem. Soc. ~
'
~
-
~
-
.
7
+
1946, 68, 171.
Barbe. J. J. Chem. Educ. 1983, 60,640.
Estlmatlon of Thermochemlcal Radll (In Picometers) Using Eqs 2 and 5 Mb
ion
XM'
Xxa
c
BF; ClO, Cloy NO; NO, Br0, 10; 10; MnO; CO?? SO!Cr0:BiOi
2.04 3.16 3.16 3.04 3.04 2.96 2.66 2.66 1.55 2.55 2.58 1.66 2.02 2.19 1.90
3.98 3.44 3.44 3.44 3.44 3.44 3.44 3.44 3.44 3.44 3.44 3.44 3.44 3.44 3.44
0.5126 0.9186 0.9186 0.8837 0.8837 0.8605 0.7732 0.7732 0.4506 0.7413 0.7500 0.4825 0.5872 0.6366 0.5523
PO:Si0:-
r 66 99 99 70 70 114 133 133 117 77 104 117 152 110 117
20 34 26 13 11 47 62 50 80
15 29 52 74 34 41
Elenronsgativio,valuss taken from reference in fwtnote 2. a Covalent and Ionic radii taken *om Laoge's Handbook of WKlmish(ree fannote 3).
942
Journal of Chemical Education
Reported
Xb
i 37.9 88.8 87.6 57.5 . 57.1 96.6 104.4 99.6 87.5 49.1 71.2 67.1 100.9 64.8 64.2
-~
64 66 66 66 66 66 66 66 66 66 66 66 66 66 66
136 140 140 140 140 140 140 140 140 140 140 140 140 140 140
i
re
(Kapustinskiil
117.1 77.5 77.5 82.2 82.2 85.2 95.8 95.8 125.0 99.3 98.4 122.8 114.5 110.0 117.4
217.7 195.0 194.1 177.5 177.2 213.6 236.8 233.0 270.0 198.2 214.4 250.2 263.9 227.8 239.2
228 200 236 155 189 191 182 249 240 185 230 240 266 238 240
% error -4.5 -2.5 -17.75 14.5 -8.9 11.8 30.1 -6.4 12.5 7.1 -8.5 4.2 -1.5 -4.3 -0.3
~
Furthermore, if we consider MX:-ion to he ideally spherical. with M nucleus in the center and X ions tanzent to the inside of the sphere, then the sphere radius must b e the sum of corrected radius of M plus twice the corrected radius of X: = i~ +2i~
(4)
This sphere radius overestimates the real thermochemical ionic radius. We have found empirically that the thermochemical radius is only about 80%of the radius of the sphere calculated by the means referred t o above so that6
+
r,, = 0.8(iM 2ix)
(5)
The tahle shows the results of thermochemical radii calculation for several polyatomic ions following the approach and the comparison of the values obtained in this work with previous reported v a l ~ e s . ~
From the table one may conclude the following:
. When the ionic charge is equally distributed, the deviations are I
the smallest. For instance, the error in the case of is only 0.370, whereas for BiO5 it is 1.5%. for CrOf it is 4.290,and for MnO; it is 12.5%. Our mlrulntion cixcs the same thrrmoehrmirnl radii lor 1 l X f and MI;;,r c m (see C103 and CIO;, NO; and NO,, 10,'and IO;I.' 'l'hrreime. oneofthere \.slurs 13 closer 10 the thermorhemied radius. Our calculation takes into account ionic radius contraction only when the charge increases, but it does not account for symmetry changes. 3 The method is not valid for ions such as OH- or CN- as these cases do not present spherical symmetry. Ions containing more than two elements, such as HCO,, RCO.0-, CNO-, CNS-, and the like, do not follow this model. NH; bears the charge on the central atom, whereas for the proposed model, the charge is equally distributed.
Volume 64
Number 11
November 1987
943
