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Ion Association versus Ion Interaction Models in ExaminingElectrolyte Solutions: Application to Calcium Hydroxide SolubilityEquilibriumM. Isabel Menendez and Javier Borge*

Departamento de Qumica Fsica y Analtica, Universidad de Oviedo, C/Julian Clavera 8, 33006 Oviedo, Spain

*S Supporting Information

ABSTRACT: The heterogeneous equilibrium of the solubility of calciumhydroxide in water is used to predict both its solubility product fromsolubility and solubility values from solubility product when inert salts, inany concentration, are present. Accepting the necessity of including activitycoecients to treat the saturated solution of calcium hydroxide (and byextension, all electrolyte solutions) and the inadequacy of any general ornonspecic equation (such as the DebyeHuckel limiting law or Daviesequations) to calculate activity coecients of 1:2 electrolytes, our resultsuncover (i) how the inclusion of ion pairs in the last mentioned equationsmakes them adequate for low ionic strength solutions and (ii) the need ofsophisticated models, such as the ion-interaction Pitzer equations, forcalculating the activity coecients when ionic strength is high. In addition,we have developed a set of MATLAB scripts and propose the use of thefree code PHREEQC version 3 to perform all the calculations described in the article. The tasks proposed here can becomplemented with the experimental determination of the solubility of calcium hydroxide in water along with the experimentalchecking of its solubility in aqueous salt solutions during several lab sessions. Experimental guidance is provided as SupportingInformation.

KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Computer-Based Learning, Textbooks/Reference Books,Aqueous Solution Chemistry, Precipitation/Solubility, Thermodynamics

Most electrolyte solutions cannot be adequately repre-sented by ideal and dilute ideal solutions due to thepresence of electric charge: they are the examples of nonidealsolutions, where the use of activities instead of concentrations isnecessary. Activity coecients are introduced as correctionfactors that transform concentrations into activities. Thethermodynamic representation of electrolyte solutions (thermo-dynamic model) requires (i) an itemization of the solutespresent (speciation or chemical model, answer to what speciesare there in the solution?) and (ii) a methodology to estimateactivity coecients (activity model, answer to how can we dealwith their nonideality?). There are as many thermodynamicmodels as combinations can be formed between chemical andactivity models. We present the full spectrum of thermody-namic models and apply some of them to study of the solubilityof calcium hydroxide in water and in aqueous salt solutions ofany concentration. The models and experiments are appro-priate for upper-level physical chemistry students.

THERMODYNAMIC MODELSChemical Models

Speciation of electrolyte solutions is controversial from the verybeginning.1 Arrheniuss original proposal was to assume thatthe strong and weak electrolytes do not dissociate completely.

However, some years later, total dissociation of strongelectrolytes was recognized and, almost simultaneously, theion pair notion (an association of two solvated ions of oppositesign that do not form a true chemical bond between them) wasintroduced. This latter idea was initially ranked as a cunningargument to reproduce, with greater accuracy, experimentaldata related to the electrolyte solutions (electrical conductivitymeasurements, basically). However, in a few years, exper-imental evidence conrmed their existence. Figure 1 showsdierent chemical models for the particular case of calcium

Published: December 5, 2013

Figure 1. (A) The simple chemical model (only free ions); (B) onepossible intermediate model (free ions and some ion pairs); (C) thecomplex model (free ions and all possible ion pairs).

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2013 American Chemical Society andDivision of Chemical Education, Inc. 91 dx.doi.org/10.1021/ed4004589 | J. Chem. Educ. 2014, 91, 9195

hydroxide saturated solution in presence of an inert salt (a saltthat does not present any ion common to those produced inthe dissolution of Ca(OH)2) represented by NaCl: from thoseconsidering just solvated isolated ions (Figure 1A) to thosewhere isolated ions coexist with all possible ion pairs (Figure1C) passing through intermediate cases where only some ionpairs are considered (Figure 1B).Activity Models

DebyeHuckel theory2 is the simplest approach for calculatingactivity coecients in electrolyte solutions. Its foundations andlimitations can be found elsewhere.3,4 Essentially, the activitycoecient of an ion, i, is a simple function of the charge of theion, zi, the temperature, T, the nature of the solvent (throughits relative permittivity, ), and the ionic strength of thesolution, I. The activity coecient of a neutral species is unity.Figure 2A outlines the modus operandi of the DebyeHuckel

theory, which yields activity coecients valid only for very low(0.01 mol kg1) ionic strengths. Many other semiempiricalfunctions based on the same parameters have been described inthe literature to estimate activity coecients at higher ionicstrengths (the documentation accompanying the computerprogram Aqueous Solutions5 contains an exhaustive enumer-ation of such functions). The Davies equation (valid for ionicstrengths up to 0.1 mol kg1), originally proposed in 19386 andrevised in 1962,7 is one of the most popular equations.Henceforth, activity coecients calculated via DebyeHuckel,Davies, or similar formulas will be referred to as the ionicstrength activity model.There is another model, opposite to the above one, to

calculate activity coecients in electrolyte solutions. It treats,one-by-one, the interactions of each solute species, B, with allothers. This proposal has its origin in the principle of thespecic interactions of ions,8 and Pitzer equations9,10 are apractical realization of this new procedure. Figure 2B outlinesthe modus operandi of the Pitzer methodology, which will bereferred to as the ion interaction activity model.

As previously stated, a thermodynamic model is anycombination of a chemical model and a model to calculateactivity coecients. Scientic literature classies thermody-namic models in four major groups (see Table 1).11

Note that the application of the simplest possiblecombination (free ions + ionic strength activity model) isvery restricted (only to extremely dilute solutions). The mixedmodel is the most complex (note that the most sophisticateddescriptions are used for both chemical and activity models)and will not be implemented in this article.The usefulness of a thermodynamic model is based on the

quality with which it reproduces the available experimental data.Dierent thermodynamic models are used to calculate thesolubility product of calcium hydroxide (eq 1) fromexperimental measurements of its solubility (K(T) = f (s))and the solubility from the solubility product (s = f1(K(T))),where the function f is the mathematical representation of thethermodynamic model. K(T) is the standard equilibriumconstant (see the The standard equilibrium constant in theSupporting Information).

++ Ca(OH) (s) Ca (aq) 2OH (aq)2 2 (1)It has been established that the determination of solubility

products from solubility measurements is not an adequatemethod to get accurate solubility products, mainly due to theneed of using a particular model to evaluate activity coecientsof the species involved.14 However, throughout the history ofthis Journal, many articles devoted to the study of solubilityequilibrium have appeared1523 with the goal of obtainingequilibrium constants from solubility measurements. In mostof them, authors encourage the use of activities instead ofconcentrations (Ramette15 advocates this idea with specialvehemence) and they follow nearly the same outline: choose aparticular chemical compound, measure its solubility in water,and apply dierent formulas (mainly the DebyeHuckel lawand the Davies equation) to calculate activity coecients. Thepedagogical value of this approach is indisputable, but it is notpossible to obtain good solubility products without usingadvanced thermodynamic models, as we shall show later.The reverse relationship between solubility product and

solubility, expressing the solubility as a function of the solubilityproduct, is also interesting not only from a pedagogicalviewpoint but also for its numerous applications in real-lifeproblems where solubility is strongly aected by the presenceof other ions. Real-life problems are more appealing forundergraduate learners as they allow them to appreciate thecapability of physical chemistry to solve such practical issues.To the best of our knowledge, only the paper of Willey23 dealswith the prediction of solubility from solubility products at saltaqueous solutions, a complex system more similar to real onesthan simple pure aqueous solutions. Again, the application ofadvanced thermodynamic models is mandatory to obtaincorrect values of the solubility.

Figure 2. (A) Modus operandi of the DebyeHuckel theory (ionicstrength activity model) and (B) the Pitzer methodology (ioninteraction activity model). In (A) white circles represent ions andempty circles neutral molecules. In this model, the activity coecientof species i (with charge zi) in a particular solvent () and at aparticular temperature (T) only depends on I, a global property notaected by the nature of the ions but only on their charge (zj) andconcentration (mj). In (B) green and orange circles represent ions ofopposite charge, so black and white lines correspond to repulsive andattractive interactions, respectively, modulated, in each case, by thespecic nature of the ions. In short, ion interaction activity modelsperform a microscopic description of the interactions much moredetailed than that performed by the ionic strength activity models.

Table 1. Classication of Thermodynamic Models

Thermodynamic Model Chemical Model Activity Model

Basic Free ions Ionic strengthAssociation Free ions + ion pairs Ionic strengthInteraction Free ions Ion interactionMixed Free ions + ion pairs Ion interaction

Journal of Chemical Education Article

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vaioHighlight

PREDICTING SOLUBILITY PRODUCT FROMSOLUBILITY

As the solubility product of Ca(OH)2 and its solubility in waterat several temperatures are available in the literature, studentsshould look for this information to assess the quality of theirresults. The NIST Critically Selected Stability Constants of MetalComplexes Database24 is a source of outstanding quality to getvalues of K(T). The value of the solubility product ofCa(OH)2 is presented (at 298.15 K) as log10K = 5.29(10),which implies K(298.15 K) = 5.13 106. Note that theuncertainty associated with log10K generates a range ofpossible values for K(298.15 K) comprised between 4.07 106 and 6.46 106. The NBS Tables of ChemicalThermodynamic Properties25 provides rG(298.15 K) =30.422 kJ mol1 for the solubility equilibrium of Ca(OH)2,where rG is the standard reaction Gibbs energy (see Thestandard equilibrium constant in the Supporting Information).Then, K(298.15 K) = 4.6794 106, a value within the rangepredicted by the NIST Database.Like any physical quantity derived from an experiment, there

is not one single value for the solubility of calcium hydroxide inwater at 298.15 K. The classic study, par excellence, on thissubject was carried out by Johnston and Grove26 in 1931. Theproposed value was 19.76 103 mol kg1. It seems thatseveral factors such as the aggregation state of the solid(crystalline solids do not behave like amorphous ones; particlesize is also a critical factor) or the preparation method of thesaturated solution inuence the experimental value. It is worthmentioning that the solubility of calcium hydroxide in waterdecreases with temperature, a fairly infrequent behavior. In theSupporting Information, a procedure for the experimentaldetermination of calcium hydroxide solubility in water in astudent lab is outlined. The time needed to carry out thisprocedure is 4 h.

Ideal Solution Model

Although out of Table 1, the simplest way to obtain a rstapproximation to the solubility product comes from the idealsolution model where solutesolvent interactions are verysimilar to solventsolvent interactions and the size and shapeof solute and solvent molecules are also similar. Theseconditions mean that enthalpy and volume of mixing arezero, which are never fullled by electrolyte solutions. Theapplication of the ideal solution model to the saturated solutionof Ca(OH)2 yields a solubility product of 3.09 10

5 (seePredicting solubility product from solubility: ideal solutionmodel in the Supporting Information):

= = K T p m m m( , , ) ( / ) 3.09 10mB

B5B

(2)

where the index B refers to species in solution, m is thestandard molality, and B denotes the stoichiometric number.Taking as right value that proposed by the NIST Database(5.13 106), the percentage relative error of Km is 501.59%.This large error clearly indicates that the ideal solution model isinadequate to represent the saturated solution of Ca(OH)2.

Association Model

In water, ions coming from Ca(OH)2 dissolution can form theCaOH+ ion pair. The formation of the CaOH+ ion pair insaturated solutions of calcium hydroxide through eq 3 has beenexperimentally veried.

+ + +Ca (aq) OH (aq) CaOH (aq)2 (3)The NIST Database result for the equilibrium constant of eq 3is presented (at 298.15 K) as log10K = 1.30(10). This impliesK(298.15 K) = 20.0. Note that the uncertainty associated withlog10K generates a range of possible values for K(298.15 K)comprised between 15.8 and 25.1. Using the NBS Tables,rG(298.15 K) = 7.576 kJ mol1. Then, K(298.15 K) =21.25, a value within the range predicted by the NIST Database.In this case, it is necessary to consider simultaneously the two

equilibria: solubility (eq 1) and association (eq 3). Theapplication of the association model (using the Davies equationto calculate the activity coecients) yields a solubility productof 5.57 106 (Predicting solubility product from solubility:association model in the Supporting Information):

= = K T p m m( , ) ( / ) 5.57 10mB

B B6B

(4)

where B is the activity coecient (molality scale) of thesubstance B. Using the NIST Database value for the solubilityproduct (5.13 106), the percentage relative error is 8.61%,which means that ionic association models generate accuratesolubility product values in water.Interaction Model

For the present case, Pitzer model generates a solubilityproduct of 6.90 106.

= = K T p m m( , ) ( / ) 6.90 10mB

B B6B

(5)

The percentage relative error, with respect to the NISTDatabase value (5.13 106), is 34.48% (although this valuefalls to 6.79% if the comparison is made with the upper limitproposed by the NIST Database). For the problem we aresolving now, that is, prediction of Ca(OH)2 solubility productfrom its solubility in water, values provided by Pitzer model aresimilar to those oered by the ion association method.A MATLAB script (s2k.m) for the calculation of Km(T,p), or

Km(T,p,m), for the three cases just described has been includedin the Supporting Information.

PREDICTING SOLUBILITY FROM SOLUBILITYPRODUCT

Up to now we aimed to predict the solubility product ofcalcium hydroxide from its solubility in water. Now we proceedto a more applied issue where the aim is the prediction ofCa(OH)2 solubility, not only in water but in aqueous saltsolutions, of any concentration, based on its solubility product.This procedure is customary in professional practice, such as inpharmaceutical industry to know how the solubility of a drugchanges in the presence of other species, and is also importantin the study of the variation of the solubility of calciumcarbonate in the presence of carbon dioxide (the solubility ofcalcium carbonate in fresh and salt water is a crucial parameterfor the development of all living beings with external shells).Terms salting-in and salting-out stand for the solubilization andprecipitation, respectively, of a substance due to salt addition towater. For our particular system, in 1931 Johnston and Groveexperimentally studied the variation of Ca(OH)2 solubility inpresence of dierent salts and at dierent concentrations. Aprocedure for the experimental determination of calciumhydroxide solubility in aqueous salt solutions in a studentslab is outlined in the Supporting Information.



Recently, the freely distributed program PHREEQC version3 has been published as a tool for chemical speciation.27 It is aprofessionally oriented code ready to run under both Linux andMicrosoft Windows operating systems, which allows the use ofthe two theoretical approaches tested in this work (associationand interaction models). The software was designed to handlemuch more complex systems than Ca(OH)2 in presence ofNaCl, but it is easy to use to solve the teaching problems and,at the same time, provides a professional experience to theundergraduate students.Association Model

We focus on the Davies semiempirical equation combined withthe ion pair formation to predict Ca(OH)2 solubility. The caseswhere the inert salt NaCl is present in the solution at dierentconcentrations is examined. The NaCl salt makes possible theformation of up to four ion pairs between its ions and thosecoming from Ca(OH)2 dissolution: CaOH

+, CaCl+, NaOH,and NaCl (association among more than two ions and chemicalreactions yielding new species are avoided). A literature searchindicates that these four ion pairs have been detected and theirassociation constants (obtained from NIST Database24) are20.0, 2.51, 1.26, and 0.50, respectively. It is worth mentioningthat dierent literature sources display dierent values for theassociation constants of the weakest pairs. Particularly, for thepair NaOH, Koryta et al.28 give a value of 0.20 and the value, bydefault, provided by the program PHREEQC version 3 is 1010.Calculations are performed using PHREEQC version 3 andNIST equilibrium constants (input les have been included inthe Supporting Information).Figure 3 shows four continuum lines corresponding to

dierent chemical models tested for the solution, along with the

experimental data (blue dots) obtained by Johnston and Grove.As CaOH+ ion pair shows an association constant much largerthan those for the remaining ion pairs, we start our test of theassociation models by including only this one. The red line inFigure 3 indicates that the association model with the strongestion pair does not deviate very much from experimental

measurements. Yellow, green, and purple lines were obtainedconsidering an additional ion pair each time. It is evident thatresults worsen considerably, mainly at high ionic strengths.When all possible ion pairs are taken into account, sixsimultaneous equilibria coexist in the solution (includingcalcium hydroxide solubility equilibrium and the autoionizationof water), so chemical complexity is high and the solution ofthe six coupled equations is strongly dependent on the values ofthe standard equilibrium constants used. As mentioned above,completely dierent values for the association constants of theweakest pairs can be found in the literature and choosing oneor the other fully determines the outcome. Calculations withequilibrium constants other than NIST values (not displayed)show dramatic changes in yellow, green, and purple lines andindicate that the only stable fact is that neglecting the eect ofthe weakest ion pairs (in this case the most uncertain) yieldsthe best results.The correct use of association models involves the

knowledge of accurate association constants of all the possibleion pairs. If dubious values are employed, calculated results maybe not fully trustful. To corroborate this idea, we include twonew calculations in the Supporting Information. As the numberof dierent ion pairs in a solution is increased, the use of theassociation model is more and more complicated.Interaction Model

Next step in the prospect of calculating accurate Ca(OH)2solubilities at high ionic strengths is the use of the specicinteractions Pitzers model. PHREEQC version 3 code providesthe Ca(OH)2 solubility at dierent NaCl concentrationsdisplayed in the dashed blue line in Figure 3. Pitzer resultsare close to experimental data and to those obtained by the bestassociation model. However, Pitzer results could be obtained ina straight way, so we propose that this model is adequate topredict calcium hydroxide solubility in salt aqueous solutionswhen the ionic strength is high.

CONCLUSIONSThis article presents a step forward in the pedagogical approachof physical chemistry to real situations. Both the DebyeHuckel limiting law and any of its semiempirical extentions areinstructional models useful to get acquainted with thecalculation of activity coecients in electrolyte solutions, butuseless to solve real problems. This article describes twopossible alternatives to the academic treatments: a relativelysimple one (ionic association theory) and a more advanced one(specic ion interaction theory). The rst alternative is subjectto the tentative selection of ion pairs relevant in the solution,whereas the second alternative only needs the parameters of theions in the solution. Calcium hydroxide, a sparingly solublesubstance, has been used to illustrate all concepts presentedthrough the prediction of its solubility product from itssolubility and the reverse process, the calculation of itssolubility even at high ionic strengths. Computational toolsare provided to perform all calculations described.

ASSOCIATED CONTENT*S Supporting InformationThe standard equilibrium constant; experimental determinationof calcium hydroxide solubility in water and in aqueous saltsolutions: chemicals, hazards, instrumentation and lab guide;predicting solubility product from solubility: ideal solutionmodel; predicting solubility product from solubility: association

Figure 3. Variation of the solubility of Ca(OH)2 in the presence ofNaCl. Blue dots are the experimental data obtained by Johnston andGrove. The solid lines are the theoretical results obtained by applyingthe association model (red, CaOH+; yellow, CaOH+ + CaCl+; green,CaOH+ + CaCl+ + NaOH; purple, CaOH+ + CaCl+ + NaOH +NaCl). The dashed blue line is the theoretical result obtained byapplying the interaction model.



model; MATLAB script to calculate calcium hydroxidesolubility product from solubility values; PHREEQC version3 input les; predicting solubility from solubility product;association model; inuence of equilibrium constant of theweakest ion pairs. This material is available via the Internet athttp://pubs.acs.org.

AUTHOR INFORMATIONCorresponding Author

*E-mail: [email protected]

The authors declare no competing nancial interest.

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