The electrostatic interaction among a neutral and finite set of point charges is based on the sum of their pairwise charge products,     , yet many analyses yield terms which simply contain a sum of the squares of the separate charges, corresponding to the ionic strength, 1/2 Σ   2  . This submission collects together a number of important instances of this result and explains their equivalence. In effect, the ionic strength-like terms provide conveniently calculated coulomb sums for systems with finite collections of charges

Glasser, Leslie

English

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Chemical Thermodynamics and Thermal Analysis 6 (2022) 100059 Contents lists available at ScienceDirect Chemical Thermodynamics and Thermal Analysis journal homepage: www.elsevier.com/locate/ctta The Equivalence of the Charge Interaction Sum and the Ionic Strength Leslie Glasser Curtin Institute for Computation, Discipline of Chemistry, Curtin University, GPO Box U1987, Perth, WA 6845, Australia a b s t r a c t The electrostatic interaction among a neutral and finite set of point charges is based on the sum of their pairwise charge products, 𝑧 𝑖 𝑧 𝑗 , yet many analyses yield terms which simply contain a sum of the squares of the separate charges, corresponding to the ionic strength, 1 2 ∑𝑚 𝑖 𝑧 2 𝑖 . This submission collects together a number of important instances of this result and explains their equivalence. In effect, the ionic strength-like terms provide conveniently calculated coulomb sums for systems with finite collections of charges. 1 q  s  u  t  t  t  i  i  r E  t  s  g  c  w  r [  c  i  c  i𝐼  w  T  t  u  t  aΓ   s  i Hr  k  s  t  s e  a  d  e s  t  e  e  t  s  e  c𝑈  hR2(. Introduction It has often been observed, but seldom noted and perhaps neverueried that, while the electrostatic interaction among a neutral finiteet of point charges is based on the sum of their pairwise charge prod-cts, 𝑧 𝑖 𝑧 𝑗 , based on Coulomb’s Law, yet analysis yields terms (such ashe ionic strength, 1 2 ∑𝑚 𝑖 𝑧 2 𝑖 ) which are simply a sum of the squares ofhe separate charges. It is the purpose of this brief contribution to collectogether a number of important instances of this result and to explaints occurrence, independent of the mathematical complexities in whicht may be cloaked in particular instances of its appearance. A generaleview of ionic strength applications was published in 2004. [1-2] Perhaps the earliest well-known appearance of such a sum is inwald’s accelerated convergence method [3] (1921) for calculating lat-ice energies. Three terms appear in the Ewald sum: one arises as amoothed sum of the short-range interactions of the contents of the ori-in unit cell with the direct lattice, a second from a corresponding re-iprocal lattice sum, while the third arises from the charge interactionsithin the origin unit cell itself – this has the form ∑𝑧 2 𝑖 , where the z’sepresent the charges of the n species within the unit cell. Ionic strength, I , was introduced (also in 1921) by Lewis and Randall4] as a purely empirical quantity which correlated (surprisingly suc-essfully) the thermodynamic activities of different strong electrolytesn dilute solution; in their words "In dilute solutions the activity coeffi-ient of a given strong electrolyte is the same in all solutions of the sameonic strength". They defined ionic strength on a molality basis as:  = 1∕2 ∑𝑚 𝑖 𝑧 2 𝑖 (1)here m i represents the molality of ions of type i with integer charge, z i .he factor 1 2 is introduced in order that the ionic strength of a 1:1 elec-rolyte ( e.g. , NaCl) equals its molality. In practice, it is more common tose a related term, Γc , also called ional concentration, where concentra-E-mail address: l.glasser@curtin.edu.au ttps://doi.org/10.1016/j.ctta.2022.100059 eceived 9 March 2022; Accepted 7 April 2022 667-3126/© 2022 The Authors. Published by Elsevier B.V. This is an open access ar http://creativecommons.org/licenses/by-nc-nd/4.0/ ) ion, c, replaces molality; the two expressions are proportional to onenother in sufficiently dilute solution: [5] 𝑐 = 1∕2 ∑𝑖 𝑐 𝑖 𝑧 2 𝑖 (2)Lewis and Randall refer admiringly to earlier work by Ghosh [6] ontrong electrolytes which uses the same charge-squared factor as appearsn their ionic strength term. A related term, ∑𝑛 𝑖 𝑧 2 𝑖 , appeared independently in Debye andückel’s (1923) development of a theory of electrolytes [7] where n i epresents the number of ions per cm 3 . Later, in 1924, Debye [8] ac-nowledged the prior definition by Lewis and Randall of the ionictrength, and the afore-mentioned proportionality of these two quan-ities to one another in dilute solution. He then proceeded to use ionictrength in his treatment of ionic activity. The Madelung energies of ionic solids relate to their geometry andlectrostatic interactions, [9] with the electrostatic term consisting of sum of pairwise ion charge products. In 1955 Templeton [10] intro-uced a related factor, S ( = 2 I ), as a normalizing factor for Madelungnergies but this seems to have been applied [11] only occasionally. In 1995, Glasser [12] noted the appearance of a similar ionictrength-like term (now including the factor 1 2 ) in the Kapustinskii equa-ion, [13] an equation which provides good estimates of the lattice en-rgies, U , of binary ionic crystalline solids. By detailed calculation, hestablished that the sum of the pairwise charge interactions was equal tohe sum of the squares of the individual charges in a unit cell. On the ba-is of this recognition, it proved possible to generalize the Kapustinskiiquation beyonf its purely binary form to predict the lattice energies ofomplex ionic crystals. The generalized equation has the form:  = − 𝐴 < 𝑟 > (1 − 𝜌< 𝑟 > )𝐼 (3a)ticle under the CC BY-NC-ND license L. Glasser Chemical Thermodynamics and Thermal Analysis 6 (2022) 100059  b  s  ca𝑈   H  s  t  p   g  o  i  w  t  b  i  e 1∑t∑  d∑ t∑ ∑  k𝑛  t∑ o∑  c  t  t  ( c  t  tDA t  e  E  iR       [          where I = 1 2 ∑𝑛 𝑖 𝑧 2 𝑖 n i = number of ions of species i in the formula unit < r > = weighted mean cation-anion Goldschmidt radius sum A = conversion factor between charge and energy units; 121.4 kJ mol − 1 nm 𝜌/ < r > = a Born-Mayer correction term for ion repulsion, where 𝜌 is commonly assigned the value 0.0345 nm. The mean radius is a rather nebulous quantity but can be substitutedy the cube-root of the formula unit volume, V m 1/3 , having the neces-ary dimension of length. Actual use of the revised equation [14] is mostonveniently in terms of a linear fit to data sets, with fitted constants 𝛼nd 𝛽 = 2 𝐼( 𝛼𝑉 1∕3 + 𝛽) (3b)Such a relation between the continuum approximation of the Debye-ückel theory and the necessarily discrete ion treatment of the Kapustin-kii equation was earlier foreshadowed by Fuoss and Accascina [15] inheir remarks on the likely form of the radial distribution function as oneroceeds from dilute electrolyte solutions to fused salts to ionic crystals.Appearing in these various situations, ionic strength (in its variousuises) seems simply to be a correlation factor of somewhat obscurerigin. This does not mean, however, that the ionic strength concepts underplayed; it is of importance in all electrolyte experimentationhere, for example, in order to reduce extraneous electrolyte concen-ration effects, solutions are generally made up to constant ionic strengthy adding inactive strong electrolyte to the active ion constituents, andt remains as an essential term in the important Debye-Hückel theory oflectrolytes. We here provide a more fundamental interpretation of this quantity..1. Analysis For an arbitrary but finite neutral system of n charges: 𝑛 𝑖 =1 𝑧 𝑖 = 0 (4) hen: 𝑛 𝑖 =1 𝑧 𝑖 ⋅𝑛 ∑𝑗=1 𝑧 𝑗 = 0 (5)Expanding this product, and separating it into diagonal and off-iagonal elements: 𝑛 𝑖 𝑧 𝑖 ⋅𝑛 ∑𝑗 𝑧 𝑗 = 𝑛 ∑𝑖 =1 𝑧 2 𝑖 + 𝑛 ∑𝑖 𝑛 ∑𝑗≠𝑖 𝑧 𝑖 𝑧 𝑗 = 0 (6)hus, rearranging: 𝑛 𝑖 𝑛 ∑𝑗≠𝑖 𝑧 𝑖 𝑧 𝑗 = − 𝑛 ∑𝑖 𝑧 2 𝑖 (7)By symmetry, since z i z j = z j z i , the above equation becomes: 𝑛 𝑖 𝑛 ∑𝑗>𝑖 𝑧 𝑖 𝑧 𝑗 = 1∕2 𝑛 ∑𝑖 𝑛 ∑𝑗≠𝑖 𝑧 𝑖 𝑧 𝑗 = −1∕2 𝑛 ∑𝑖 𝑧 2 𝑖 (8)We may now classify the ions into t types, with n k ions of each type,, such that:  = 𝑡 ∑𝑘 =1 𝑛 𝑘 (9)2 hen: 𝑛 𝑖 𝑛 ∑𝑗>𝑖 𝑧 𝑖 𝑧 𝑗 = −1∕2 𝑡 ∑𝑘 𝑛 𝑘 𝑧 2 𝑘 (10)r, briefly: 𝑗>𝑖 𝑧 𝑖 𝑧 𝑗 = −1∕2 𝑡 ∑𝑘 𝑛 𝑘 𝑧 2 𝑘 (11)The system of n charges considered here might consist of: the chargesontained in a kilogram of solvent (so that the last term corresponds tohe ionic strength, I ); or the charges in a liter of solution (leading tohe ional concentration, Γc ); or the charges in a formula unit of a solidleading to the generalised Kapustinskii term, I ). Thus, the ionic strength, and its related terms, simply provideonveniently-calculated sums of systems of charges. This accords withhe only specific comment, [16] rather than “hand-waving ” generalisa-ions, that I have found on the nature of ionic strength. eclaration of Competing Interest There are no conflicts of interest. cknowledgements I wish to acknowledge, with appreciation, that the form of the deriva-ion of Eq. (11) given above was supplied to me by an anonymous ref-ree in support of my calculations and intuitive identification whereq. (11) was developed [12] but its proof was not provided, nor werets implications explored. eferences [1] M.E. Sastre de Vicente, The Concept of Ionic Strength Eighty Years after Its Intro-duction in Chemistry, J. Chem. Educ. 81 (2004) 750–753 . [2 L. Glasser, Reaction to "the concept of ionic strength eighty years after its introduc-tion in chemistry", J. Chem. Educ. 82 (2005) 212 . [3] B.P. Van Eijck, J. Kroon, Coulomb Energy of Polar Crystals, J. Phys. Chem. B 101(1997) 1096–1100 . [4] G.N. Lewis, M. Randall, The activity coefficient of strong electrolytes.1, J. Amer.Chem. Soc. 43 (1921) 1112–1154 . [5] T. Solomon, The Definition and Unit of Ionic Strength, J. Chem. Educ. 78 (2001)1691 . [6] J. Ghosh, The abnormality of strong electrolytes: I-IV, J. Chem. Soc. 113 (1918)449–458 627-638, 707-715, 790-788 . [7] P. Debye, E. Hückel, Zur Theorie der Elektrolyte, Phys. Zeits. 24 (1923) 185–206 . [8] P. Debye, Osmotic constitutive equation and activity of diluted strong electrolytes,Phys. Zeits. 25 (1924) 97–107 . [9]] L. Glasser, Solid-State Energetics and Electrostatics: MADELUNG Constants andMadelung Energies, Inorg. Chem. 51 (2012) 2420–2424 . [10] D.H. Templeton, Prediction of Madelung Constants, J. Chem. Phys. 23 (1955)1826–1829 . [11] J.B. Torrance, P. Lacorre, C. Asavaroengchai, R.M. Metzger, Why are some oxidesmetallic, while most are insulating? Physica C 182 (1991) 351–364 . [12] L. Glasser, Lattice Energies of Crystals with Multiple Ions - a Generalised Kapustin-skii Equation, Inorg. Chem. 34 (1995) 4935–4936 . [13] A.F. Kapustinskii, Lattice energy of ionic crystals, Quart. Rev. Chem. Soc. 10 (1956)283–294 . [14] H.D.B. Jenkins, H.K. Roobottom, J. Passmore, L. Glasser, Relationships among IonicLattice Energies, Molecular (Formula Unit) Volumes, and Thermochemical Radii,Inorg. Chem. 38 (1999) 3609–3620 . [15] R.M. Fuoss, F. Accascina, in: Electrolytic Conductance, 106, Interscience, New York,1959, pp. 101–102 . [16] S. Glasstone, in: An Introduction to Electrochemistry, van Nostrand, Princeton,1942, p. 140 . 

The Equivalence of the Charge Interaction Sum and the Ionic Strength

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The Equivalence of the Charge Interaction Sum and the Ionic Strength

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