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Specific ion interaction theory
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<h2 id="background">Background</h2> <p>This theory arises from the need to derive activity coefficients of solutes when their concentrations are too high to be predicted accurately by the <a href="/facts/Debye%25E2%2580%2593H%25C3%25BCckel_theory/TlWlVo9a">Debye–Hückel theory</a>. Activity coefficients are needed because an <a href="/facts/Equilibrium_constant/yMTuuBbX">equilibrium constant</a> is defined in chemical <a href="/facts/Thermodynamics/q4hIx3yP">thermodynamics</a> as the ratio of <a href="/facts/Activity_(chemistry)/kGUf85XI">activities</a> but is usually measured using <a href="/facts/Concentration/wqBmz5od">concentrations</a>. The protonation of a <a href="/facts/Monobasic_acid/FySU18n1">monobasic</a> <a href="/facts/Acid/FySU18n1">acid</a> will be used to simplify the presentation. The <a href="/facts/Equilibrium_chemistry/mkjCWwnx">equilibrium</a> for <a href="/facts/Protonation/q5Gf9z0M">protonation</a> of the <a href="/facts/Conjugate_base/hohgQ07M">conjugate base</a>, A− of the acid HA, may be written as: </p> H + + A − ↽ − − ⇀ HA {\displaystyle {\ce {H+ + A- <=> HA}}} <p>for which the association constant <i>K</i> is defined as: </p> K = { HA } { H + } { A − } {\displaystyle K={\frac {{\ce {\{HA\}}}}{{\ce {\{H^+\}\{A^{-}\}}}}}} <p>where {HA}, {H+}, and {A–} represent the <a href="/facts/Thermodynamic_activity/kGUf85XI">activity</a> of the corresponding chemical species. The role of water in the association equilibrium is ignored as in all but the most concentrated solutions the activity of water is constant. <i>K</i> is defined here as an <i>association</i> constant, the reciprocal of an <a href="/facts/Acid_dissociation_constant/t4YyRtKC">acid dissociation constant</a>. </p><p>Each activity term { } can be expressed as the product of a concentration [ ] and an <a href="/facts/Activity_coefficient/ImxE0u7t">activity coefficient</a> γ. For example, </p> { H A } = [ H A ] × γ H A {\displaystyle \{HA\}=[HA]\times \gamma _{HA}} <p>where the square brackets represent a concentration and γ is an activity coefficient. Thus the equilibrium constant can be expressed as a product of a concentration ratio and an activity coefficient ratio. </p> K = [ HA ] [ H + ] [ A − ] × γ HA γ H + γ A − {\displaystyle K={\frac {{\ce {[HA]}}}{{\ce {[H^+][A^{-}]}}}}\times {\frac {\gamma _{{\ce {HA}}}}{\gamma _{{\ce {H^+}}}\gamma _{{\ce {A^-}}}}}} <p>Taking the logarithms: </p> log K = log K 0 + log γ HA − log γ H + − log γ A − {\displaystyle \log K=\log K^{0}+\log \gamma _{{\ce {HA}}}-\log \gamma _{{\ce {H^+}}}-\log \gamma _{{\ce {A^{-}}}}} <p>where: </p> K 0 = [ HA ] [ H + ] [ A − ] {\displaystyle K^{0}={\frac {{\ce {[HA]}}}{{\ce {[H^+][A^{-}]}}}}} at infinite <a href="/facts/Dilution_(equation)/6XFJA3b5">dilution</a> of the <a href="/facts/Solution_(chemistry)/MGchhMQV">solution</a> <p><i>K</i>0 is the hypothetical value that the equilibrium constant <i>K</i> would have if the solution of the acid HA was infinitely diluted and that the activity coefficients of all the species in solution were equal to one. </p><p>It is a common practice to <a href="/facts/Determination_of_equilibrium_constants/yJWXGK1x">determine</a> equilibrium constants in solutions containing an electrolyte at high <a href="/facts/Ionic_strength/72GKfhmE">ionic strength</a> such that the activity coefficients are effectively constant. However, when the ionic strength is changed the measured equilibrium constant will also change, so there is a need to estimate individual (single ion) activity coefficients. <a href="/facts/Debye%25E2%2580%2593H%25C3%25BCckel_equation/VkxVxlAA">Debye–Hückel</a> theory provides a means to do this, but it is accurate only at very low concentrations. Hence the need for an extension to Debye–Hückel theory. Two main approaches have been used. SIT theory, discussed here and <a href="/facts/Pitzer_equations/TkuVEuTM">Pitzer equations</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="development">Development</h2> <p>SIT theory was first proposed by <a href="/facts/Br%25C3%25B8nsted/pw6sL4oc">Brønsted</a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> in 1922 and was further developed by Guggenheim in 1955.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> Scatchard<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> extended the theory in 1936 to allow the interaction coefficients to vary with ionic strength. The theory was mainly of theoretical interest until 1945 because of the difficulty of determining equilibrium constants before the <a href="/facts/Glass_electrode/KB1Io0tM">glass electrode</a> was invented. Subsequently, Ciavatta<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> developed the theory further in 1980. </p><p>The activity coefficient of the <i>j</i>th ion in solution is written as γj when concentrations are on the <a href="/facts/Molal/Saa8Jj8w">molal</a> <a href="/facts/Concentration/wqBmz5od">concentration</a> scale and as <i>y</i>j when concentrations are on the <a href="/facts/Molar_concentration/P2WL5b9q">molar concentration</a> scale. (The molality scale is preferred in thermodynamics because molal concentrations are independent of temperature). The basic idea of SIT theory is that the activity coefficient can be expressed as </p> log γ j = − z j 2 0.51 I 1 + 1.5 I + ∑ k ϵ j k m k {\displaystyle \log \gamma _{j}=-z_{j}^{2}{\frac {0.51{\sqrt {I}}}{1+1.5{\sqrt {I}}}}+\sum _{k}\epsilon _{jk}m_{k}} (molalities) <p>or </p> log y j = − z j 2 0.51 I 1 + 1.5 I + ∑ k b j k c k {\displaystyle \log y_{j}=-z_{j}^{2}{\frac {0.51{\sqrt {I}}}{1+1.5{\sqrt {I}}}}+\sum _{k}b_{jk}c_{k}} (molar concentrations) <p>where <i>z</i> is the electrical charge on the ion, <i>I</i> is the ionic strength, ε and <i>b</i> are interaction coefficients and <i>m</i> and <i>c</i> are concentrations. The summation extends over the other ions present in solution, which includes the ions produced by the background electrolyte. The first term in these expressions comes from Debye–Hückel theory. The second term shows how the contributions from "interaction" are dependent on concentration. Thus, the interaction coefficients are used as corrections to Debye–Hückel theory when concentrations are higher than the region of validity of that theory. </p><p>The activity coefficient of a neutral species can be assumed to depend linearly on ionic strength, as in </p> log γ = k m I {\displaystyle \log \gamma =k_{m}I\,} <p>where <i>k</i>m is a <a href="/facts/Sechenov/WGpF7ERd">Sechenov</a> coefficient.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>In the example of a monobasic acid HA, assuming that the background electrolyte is the salt NaNO3, the interaction coefficients will be for interaction between H+ and NO3−, and between A− and Na+. </p> <h2 id="determination-and-application">Determination and application</h2> <p>Firstly, equilibrium constants are determined at a number of different ionic strengths, at a chosen temperature and particular background electrolyte. The interaction coefficients are then determined by fitting to the observed equilibrium constant values. The procedure also provides the value of <i>K</i> at infinite dilution. It is not limited to monobasic acids.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> and can also be applied to metal complexes.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> The SIT and Pitzer approaches have been compared recently.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> The <a href="/facts/Bromley_equation/BTrECFd7">Bromley equation</a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> has also been compared to both SIT and <a href="/facts/Pitzer_equations/TkuVEuTM">Pitzer equations</a>.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> It has been shown that the SIT equation is a practical simplification of a more complicated hypothesis,<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> that is rigorously applicable only at trace concentrations of reactant and product species immersed in a surrounding electrolyte medium. </p> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.acadsoft.co.uk/aq_solutions.htm">SIT program</a> A PC program to correct stability constants for changes in ionic strength using SIT theory and to estimate SIT parameters with full statistics. Contains an editable database of published SIT parameters. It also provides routines to inter-convert MolaRities (c) and MolaLities (m), and lg K(c) and lg K(m).</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Guggenheim, E.A.; Turgeon, J.C. (1955). "Specific interaction of ions". Trans. Faraday Soc. 51: 747–761. doi:10.1039/TF9555100747. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Ciavatta, L. (1980). "The specific interaction theory in the evaluating ionic equilibria". Ann. Chim. (Rome). 70: 551–562. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Pitzer, K.S. (1973). "Thermodynamics of electrolytes, I. Theoretical basis and general equations". J. Phys. Chem. 77 (2): 268–277. doi:10.1021/j100621a026. <a href="https://escholarship.org/uc/item/15n9h7zx" target="_blank">https://escholarship.org/uc/item/15n9h7zx</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Pitzer, K.S. (1991). Activity coefficients in electrolyte solutions. Boca Raton, Fla: CRC Press. ISBN 0-8493-5415-3. <a href="0-8493-5415-3" target="_blank">0-8493-5415-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Brønsted, J.N. (1922). "Studies on solubility IV. The principle of the specific interaction of ions". J. Am. Chem. Soc. 44 (5): 877–898. Bibcode:1922JAChS..44..877B. doi:10.1021/ja01426a001. <a href="https://zenodo.org/record/1428790" target="_blank">https://zenodo.org/record/1428790</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Guggenheim, E.A.; Turgeon, J.C. (1955). "Specific interaction of ions". Trans. Faraday Soc. 51: 747–761. doi:10.1039/TF9555100747. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Scatchard, G. (1936). "Concentrated solutions of strong electrolytes". Chem. Rev. 19 (3): 309–327. doi:10.1021/cr60064a008. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Ciavatta, L. (1980). "The specific interaction theory in the evaluating ionic equilibria". Ann. Chim. (Rome). 70: 551–562. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Setchenow, I.M. (1892). "Action de l'acide carbonique sur les solutions des sels d'acides forts. Étude absorptiométrique". Ann. Chim. Phys. 25: 226–270. <a href="https://gallica.bnf.fr/ark:/12148/bpt6k348958/f268.item.r=Setchenow" target="_blank">https://gallica.bnf.fr/ark:/12148/bpt6k348958/f268.item.r=Setchenow</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Crea, F.; De Stefano, C.; Foti, C.; Sammartano, S. (2007). "Sit parameters for the dependence of (poly)carboxylate activity coefficients on ionic strength ...". J. Chem. Eng. Data. 52: 2195–2203. doi:10.1021/je700223r. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Ciavatta, L. (1990). "The specific interaction theory in equilibrium analysis. Some empirical rules for estimate interaction coefficients of metal ion complexes". Ann. Chim. (Rome). 80: 255–263. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Elizalde, M. P.; Aparicio, J. L. (1995). "Current theories in the calculation of activity coefficients—II. Specific interaction theories applied to some equilibria studies in solution chemistry". Talanta. 42 (3): 395–400. doi:10.1016/0039-9140(95)01422-8. PMID 18966243. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Bromley, L.A. (1973). "Thermodynamic properties of strong electrolytes in aqueous solutions". AIChE J. 19 (2): 313–320. Bibcode:1973AIChE..19..313B. doi:10.1002/aic.690190216. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Foti, C.; Gianguzza, A.; Sammartano, S. (1997). "Comparison of equations for fitting protonation constants of carboxylic acids in aqueous tetramethylammonium chloride at various ionic strengths". Journal of Solution Chemistry. 26 (6): 631–648. doi:10.1007/BF02767633. S2CID 98355109. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>May, Peter M.; May, Eric (2024). "Ion Trios: Cause of Ion Specific Interactions in Aqueous Solutions and Path to a Better pH Definition". ACS Omega. 9 (46): 46373–46386. doi:10.1021/acsomega.4c07525. PMC 11579776. PMID 39583676. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11579776" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11579776</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>