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H. D. B Jenkins and K. P. Thakur Deoarirneni of Chemistrv and Molecular Sciences Universitv of Warwick Coventry CV47AL. Warwickshire, U.K.
Reappraisal of Thermochemical Radii for
I
I
Thermochemical radii (1, 2) play an important role in thermochemistry in the sense that they can be used to estimate lattice energies, for example, uia the Kapustinskii equation (1). They have an extended role in chemistry in general (2) and most mddern textbooks on inorganic chemistry (3-7) tabulate thermochemical radii as reported by either Kapustinskii ( I ) in 1956 or Waddington (8) in 1959. Although Kapustinskii computed the values of thermochemical radii from the thermodynamic data available at the time, he was unable to list the thermochemical radii of a large numher of complex ions which have been the subject of extensive study since 1956. Such data is included in the present study. Today more elaborate and reliable calculations of lattice energy can he made (9) particularly with regard to complex salts, and this fact together with newer ancillary thermodynamic data obtained since the mid-fifties, renders the originally quoted radii out or date. I t is the purpose of our present article to recalculate these radii. Calculation of Thermochemical Radii Kapustinskii (1) has given an equation which enables one to estimate the lattice energy of a crystal (see reference (3)) provided that the thermochemical radius of the anion, F, (in nm) and of the cation F, (in nm) are known. The equation takes the form:
."
".
,
.-
..,
where p is a constant (taken to be 0.0345 nm), u is the numher of ions in the simplest formula unit or stoichiometric "molecule" (e.g. u = 2 for NaCI, u = 3 for Ca(NO&) and Z, and Z, are the moduli of the anion and cation charges and U p o is ~ in kJ mol-1. If the thermochemical radius, F,, of a cation is not known, then Goldschmidt radii (10) are substituted. In order to generate an extensive tabulation of thermochemical radii for anions. in the oresent studv the value of I ' 1 0 1 . hm heen rnktn irum the results (9.11)of accurate lattice entwv calculations of a series of dkidi metal lexcept lithium) salts'?mostly having complex ions) and used to caliu~ateradii from e m . (11. The equation used is of the form:
from which if Fc is known, we can obtain Fa (and vice versa). The omission of Li+ salts from the analysis arises from our experiences (12) in the calculation of lattice energy of such salts in that the small and highly polarizable ion tends to penetrate the envelope of the complex anions and constitutes a departure from the ionic model. In cases where cation thermochemical radii have been derived, we have employed the lattice energy data derived for their halide salts and used Goldschmidt radii for the halide anions. Thermochemical radii thus obtained have been listed in the Table. These values are compared with the original and usually quoted values. Standard deviations and values of Kapustinskii's original radii are listed where available. Our table of thermochemical radii contains the radii of 104 anions and 4 cations and these dat:~ahen u s l d in eqn. 1 1 1 regenerate lattice energiesoftl~e A t s ru within usu.ully less than 5'; uf th~.currentlyaccepred 576 / Journal of Chemical Education
Histogram shawhg the distribufion of AU (percentage1definedby equation (3) in 817 simple and complex ionic crystals. value (11) and in the case of alkali metal salts to within 2%. The figure shows the measure of agreement between currently accepted values of U p o (~1 1 ) and values obtained from substitution of our derived radii in eqn. (1). The histogram is constructed for 817 data points of AUL (percentage) defined by the equation: Upo~Uenkins)- Upo~(calc) AUL (percentage)= x lW (3) UpodJenkinsI and shows that the maioritv . of the crvstals derived from anions (and cations) concerned in this study have estimated lattice energy values to within 2% error when eqn. (1)is used with our radii of the table. This seems highly satisfactory.
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Discussion The citation of radii for planar ions (e.g. C0s2-), V-shaped ions (e.g. NO2-) and linear ions (e.g CNS-) clearly evolves from the fact that we use lattice energy data to generate radii from an equation which regards the ion as being spherical. Consequently less weight must clearly be placed on such values as having direct bearing on the absolute size of the ions concerned, although clearly some relative size features are preserved. For example, it is interesting to compare the trends in Goldschmidt radii of C1- (0.181 nm), Br- (0.196 nm) and I- (0.220 nm with the radii generated for C103- (0.171 nm), BrOs- (0.154 nm) and lo3-(0.122 nm) in the present study. While the halogen ion X- increases in size with atomic number of X, the corresponding halate ion XOs- decreases in size; this is, however, completely in conformity with the trends in lyotropic numher (13,141 of these ions and presumably therefore in the single ion hydration enthalpies. The lyotropic numher evidence (13) suggests that the Nos- ion is larger than NOzion whereas our radii (accepting the large standard deviation
Thermochemlcal Radii (in nm) of ions Calculated According to Equation (2) together with Available Kapustinskii (1956) ( 1 ) Values Radii eqn. (2)
Ions
(nm)
Radii (nm) Kapuslinskii
for NOz-) suggest the reverse, this is probably due to the na. ture o f the NO%- ion itself whose departure from spherical symmetry must he considerable (although in the cubic alkali metal nitrite structures the ion is freely rotating). However, the CIOC and C103- ions show the expected size trends. Literature Cited
Radii eqn. (2)
ION
(nm)
Radii (nm) Kapustinskii
(5, Heslo~.R. B. andJone8. H.."lnor~snicChemisir~: AGuidefo Advanced Study: Elnwier Seientilie Publishing Company. Oxford. 1976. 16, ~ ~ h ~ ~ ~ , ~ . ~ . , R~ ~ P ~ ~~ and ROW. publishers, N*W YO& 1972. (7) La~oniki,J. J.. "Modern lnorgsnic Chemistry: Marcel Dakker Inc.. New York. 1973. (81 wadding ton,^. c . . A ~ u lnom . cham. ~ ~ d i ~I , 168 ~ h(1959). ~ ~ . , (9) Jenkins, H. D. B. and Waddingtan.T. C.. Ado, Inorp Chem. Rndioihem.. to be published. (10) ~ d d ~ ~ h ~V. i M.. d tskrilter . N O P . T ~ P vidm.kop..~kad. orlo ,wet. NO^ K I . 1.2
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11) K8p~atinakii,AF., Quart. RPU..10,283 (1956). 12) Smith.D. W..J.CHEM.EDUC.,54.640l1977). (3) Dssent. W. E.."Lno~aganicEnergefies.ll Penguin Bmka, Baltimore, 1910. (4) Cotton, F. A. and Wiikinson, G.."Aduanced InomagsnieChemistry: 3rd Edition, Inlerseience, New York, 1972.
I l l ) J e n k i ~ H. . D B.. "HsndbookofChemirt~andPhmim"69th Edifion.C.R.C. Prea~. Ohio. U.S.A.. 1979. 112) Jenkins, H. D. B. andPratt, K. F..Prup. SolidSLnle Chm.. 12 (1979) in prenn. (131 Morris. D. F. C.. J. lnorp. Nuei Chrm.. 6,295 (1968). (14) Finch. A. andGerdner. P.. J. Phvs Chrm.. 69.384 119651.
Vohne 56, Number 9,Se~tember1979 I 577
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