Non-ideality and Ion Association in Aqueous Electrolyte Solutions

In the Laboratory

Non-Ideality and Ion Association in Aqueous Electrolyte Solutions: Overview and a Simple Experimental Approach Margaret R. Wright and Iain L. J. Patterson School of Chemistry, University of St. Andrews, St. Andrews, Fife KY16 9ST, U.K. Kenneth D. M. Harris* School of Chemistry, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K.

Quantitative studies of aqueous electrolyte solutions almost always require consideration of non-ideality and ion association. It is vital, therefore, that students be introduced to these concepts early in their education and that they be given an experimental demonstration of the necessity to allow for these phenomena when interpreting experimental data. Here we present an overview of this topic and illustrate a simple experimental approach that can form the basis of an undergraduate laboratory experiment introducing the practical consequences of these concepts. Students are introduced to the concept of strong and weak electrolytes early in undergraduate courses in solution chemistry. For example, solid sodium chloride dissolves in water to form a strong electrolyte solution, in which the solute consists almost entirely of Na +(aq) and Cl᎑ (aq) ions. Thus, [Na+]act = [Cl᎑ ]act ≈ [NaCl]st, where the subscript “st” refers to the stoichiometric concentration, determined by considering the total number of moles of NaCl used to make up the solution. The subscript “act” refers to the actual concentration of each species present in the solution at equilibrium. In many strong electrolytes, the solute exists almost entirely as free ions; we refer to these solutes as completely dissociated (no ion association). In other cases, however, ion pairing can occur to a significant extent, and these ion pairs exist in equilibrium with the free ions. Although we will not discuss the physical nature of these ion pairs (see ref 1 for a detailed discussion), they should be regarded as physically associated single entities that move about the solution together and have lifetimes sufficiently long to survive several collisions before dissociating. They should not be confused with cases (e.g., many weak electrolytes, such as acetic acid) in which ions can associate with chemical bond formation to form a new molecule or ion. As examples of ion-pair formation in aqueous solution, tetrabutylammonium iodide dissolved in water is present as the free ions N(C4H9)4+(aq) and I᎑ (aq) in equilibrium with the ion pair [N(C4H 9) 4+I᎑ ](aq); magnesium sulfate dissolved in water is present as the free ions Mg2+(aq) and SO42᎑ (aq) in equilibrium with the ion pair [Mg2+SO42᎑ ](aq); and lanthanum hexacyanoferrate(III) dissolved in water is present as the free ions La 3+(aq) and Fe(CN)63᎑ (aq) in equilibrium with the ion pair [La3+Fe(CN)63᎑ ](aq). N(C4H9)4+(aq) + I᎑ (aq) Mg2+(aq) + SO42᎑ (aq) La3+ (aq) + Fe(CN)63᎑ (aq)

[N(C4H9) 4+I᎑ ](aq) [Mg2+SO42᎑ ](aq) [La3+Fe(CN) 63᎑ ](aq)

*Corresponding author. Tel: 0121-414-7474; Fax: 0121-4147473.

352

In this paper, we consider the effects of ion association on the properties of nonideal electrolyte solutions, focusing on how molar conductivity varies with concentration in such cases. For the concentration ranges of interest, it is necessary to take non-ideality of the solutions into account. By using an equation that gives a good account of the dependence of molar conductivity on concentration in the case of nonideal solutions for which there is no ion association, deviations from this behavior can be taken as evidence of ion association. Conductivity measurements using very basic equipment can clearly reveal such deviations, and can thus demonstrate the effects of ion association in nonideal electrolyte solutions. Such measurements can form the basis of a straightforward experiment in the undergraduate teaching laboratory, introducing students to the concepts of non-ideality and ion association and the implications of these concepts for the transport properties of ions in such solutions. For many decades, ion association in electrolyte solutions has been an area of active research interest (see refs 2–33). Several approaches, including conductivity measurements (as described in this paper), spectroscopic techniques, computer simulation, and theoretical modeling, have been used to probe the phenomenon of ion association in both aqueous and nonaqueous solvents. Although ion association influences many physicochemical properties of solutions, we focus here on its effect on the conductivity of the electrolyte solution. Background When the concentration of an electrolyte solution tends towards zero, the ions are sufficiently far apart that Coulombic interactions between them are negligible. Under these circumstances the electrolyte behaves as an ideal solution. As the concentration increases, the average distance between the ions decreases, Coulombic interactions between the ions increase, and the solution becomes progressively less ideal. The ion–solvent interactions that are present even in the ideal solution also become increasingly modified as the concentration increases, again making a progressively increasing contribution to non-ideality. As a consequence, the solvent–solvent interactions become modified—a further contribution to nonideality. The effects of non-ideality can be observed from the way in which certain properties of the solution, such as its molar conductivity, vary with concentration. The conductivity (κ) of a solution depends on the number of current-carrying species (ions) present. We focus here on the molar conductivity (Λ), which is related to the conductivity and the stoichiometric concentration (cs t) via the equation Λ= κ

c st

Journal of Chemical Education • Vol. 75 No. 3 March 1998 • JChemEd.chem.wisc.edu

(1)

In the Laboratory

For ideal electrolyte solutions, Λ is independent of cs t, whereas for nonideal electrolyte solutions, Λ is found to depend on cs t. For strong electrolytes, the Onsager limiting conductance equation (2)

proximations in the derivation. Equation 4 is often used in the less exact form

Λ = Λ o – S(cs t) 1/2

In Appendix 1, the expressions and numerical values for some of the terms used in eq 5 are given for symmetrical electrolytes (defined as those for which the charges on the cation and anion are equal). Note that Λo is generally found by extrapolation to cst = 0 of a graph of observed molar conductivity (Λ obs) vs (cs t) 1/2 (or, more accurately, a graph of [Λobs + S (cs t) 1/2 ] vs cs t ). The Fuoss–Onsager equation (eq 5) provides a good description of deviations from ideal behavior for concentrations up to ca. 10᎑2 mol dm᎑3 for 1-1 electrolytes, up to ca. 4 × 10᎑3 mol dm᎑3 for 2-2 electrolytes, and up to ca. 10᎑3 mol dm᎑3 for 3-3 electrolytes, provided that there is no ion association. Examples (4–6 ) include the 1-1 electrolytes potassium chloride and sodium bromide and 2-2 electrolytes such as the divalent cation salts of alkyl and aryl disulfonic acids. (Note that comparatively few 2-2 electrolytes are completely dissociated.) Testing the applicability of the Fuoss–Onsager equation to 3-3 electrolytes has been limited by the paucity of completely dissociated 3-3 electrolytes, although the La3+ salt of naphthalene 1,3,6-trisulfonic acid represents an important exception (6 ). For 1-1 electrolytes, comparison between experimental behavior and the Onsager equation (eq 2) has been found to be a good criterion for assessing whether the electrolyte is completely dissociated. Thus, for 1-1 electrolytes that are completely dissociated, Λ approaches the behavior predicted by the Onsager equation asymptotically from above as cst is decreased. Negative deviations, on the other hand, can be ascribed to ion association. For 2-2 and 3-3 electrolytes, graphs of Λ vs (cst )1/2 for electrolytes in which there is no ion association (4–6 ) lie

(2)

has been shown to fit the dependence of Λ on cst at very low values of cst, at which the solutions approximate ideal behavior. The parameter S is defined as S = αΛo + β

(3)

and Λo denotes the molar conductivity at infinite dilution. The parameters α and β (which are constant for any given system) are defined in Appendix 1. For higher concentrations, the effects of non-ideality become increasingly important. We now consider eqs 4 and 5, which have been developed to describe the effect of nonideality on the molar conductivity for the case in which it is assumed that there is no ion association. Using this as a basis for describing the behavior of electrolytes in which there is no ion association, we will then consider experimental deviations from the behavior described by these equations as a basis for probing the occurrence of ion association. The Fuoss–Onsager extended conductance equation (3) Λ = Λ o – S (cs t)1/2 + E cst log10[cst / mol dm᎑3]

+ J cst – F Λ cst + O {(cs t) 3/2}

(4)

has been developed to account for the interactions leading to nonideal behavior for the case in which it is assumed that there is no ion association. The parameters E and J are defined in Appendix 1. The terms {E cst log10[cst/ mol dm᎑3]} and {J cs t} take account of relaxation and electrophoretic effects on ionic mobilities, the term {᎑ F Λ cs t} describes the effect of the viscosity of the solution on ionic mobility (F is a constant), and the term O {(cst) 3/2} represents the effects of higher ap-

Λ ≈ Λo – S (cs t)1/2 + E cst log10[cs t / mol dm–3] + J cs t

150

(5)

240

148

Λ (cm 2 S mol᎑1 )


146

144 142

220

200

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136 0 0

0.04

0.08

0.12

(cst )1/2 (mol1/2 dm᎑3/2)

Figure 1. Λ vs (c st )1/ 2 for aqueous solutions of potassium chloride at 298 K. Values of Λ: ( 䊊 ) observed; (–––) from Onsager equation (eq 2); ( – – –) from Fuoss–Onsager equation ( eq 5). Crossover points: experimental not detected; theoretical ( c stx )1/ 2 = 0.013 mol 1/2 dm᎑3/ 2. Λ o = 149.9 cm 2 S mol ᎑1; S = 94.55 cm 2 S mol᎑3/ 2 dm3/ 2; E = 58.9 cm2 S mol ᎑2 dm3; J = 221.2 cm 2 S mol ᎑2 dm3; å = 3.5.

0.02

0.04

0.06

(cst )1/2 (mol1/2 dm᎑3/2)

Figure 2. Λ vs (c s t )1/ 2 for aqueous solutions of the magnesium salt of ethane 1,2-disulfonic acid at 298 K. Values of Λ: ( 䊊 ) observed; (–––) from Onsager equation (eq 2); (– – – ) from Fuoss–Onsager equation (eq 5 ). Crossover points: experimental (c stx )1/ 2 = 0.04 mol1/ 2 dm᎑3/ 2; theoretical (c stx )1/ 2 = 0.042 mol1/ 2 dm᎑3/ 2. Λ o = 238 cm2 S mol ᎑1; S = 917.7 cm2 S mol ᎑3/ 2 dm3/2; E = 6736 cm2 S mol ᎑2 dm3; J = 18, 500 cm2 S mol ᎑2 dm3; å = 5.3.

JChemEd.chem.wisc.edu • Vol. 75 No. 3 March 1998 • Journal of Chemical Education

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In the Laboratory

above the line predicted by the Onsager equation at sufficiently high cst, but then cross this line as cst is decreased and approach Λo from below. This can be attributed to the interplay of the {Ecs t log10[cs t /mol dm ᎑3]} and {J cs t} terms in the Fuoss– Onsager equation. Over the range of cst for which eq 5 is valid, {Ecs t log10[cs t /mol dm᎑3]} is always negative, whereas {Jcst} is positive. For high-charge types (2-2 and 3-3 electrolytes), the {Ecs t log10[cs t /mol dm᎑3]} term is dominant at very low values of cs t but is more than compensated by the {J cs t} term at higher cs t. Assuming that the experimental behavior is described by eq 5, the experimental Λ vs (cs t)1/2 data will cross the line defined by eq 2 when Ecs t log10 [cs t /mol dm᎑3] + Jcs t = 0 The value (denoted is therefore given by

c xst)

(6)

of cst at which this crossover occurs

log10 [c xst /mol dm᎑3] = ᎑ J/E

(7)

c sxt

Thus, the value of is characteristic of the electrolyte. For 1-1 electrolytes, the effect is sufficiently small that no crossover is detected, and the line defined by the Onsager equation appears to be the tangent to the Fuoss–Onsager curve for experimental data at sufficiently low values of cs t . Within the concentration ranges over which the Fuoss– Onsager equation is valid, this equation can be taken to define the effects of non-ideality for electrolyte solutions in which there is no ion association. It can thus be used as a baseline equation, and any deviations from the predicted behavior can be taken as evidence of ion association. In principle, the equilibrium constant for ion association could be determined from knowledge of these deviations. However, even when ion association is considerable, accurate experimental data obtained with precision conductance bridges are required to determine reliable values of such equilibrium constants. We make no attempt to derive such quantities from the data presented

Table 1. Crossover Points for Selected Aqueous Electrolyte Solutions (c xst) 1/2 Electrolyte a

Electrolyte Type

Theoretical (mol1/2 dm᎑3/2)

Observed (mol1/2 dm᎑3/2)

KCl

1-1

0.013

N(C4H9)4I

1-1

0.000033

not found not found

Mg EDS

2-2

0.042

0.04

Ca BDS

2-2

0.047

0.045

MgSO4

2-2

0.048

not found

La NTS

3-3

0.042

0.054

LaFe(CN)6

3-3

0.035

0.06

a EDS

represents ethane 1,2-disulfonate; BDS represents benzene 1,3disulfonate; NTS represents naphthalene 1,3,6-trisulfonate.

here. (A procedure that may be used to determine the equilibrium constant for the ion association from the conductance data is outlined in Appendix 2.) Nevertheless, a simple and clear qualitative demonstration of ion association can be made by observing the way in which experimental data recorded using rudimentary conductance equipment deviate from predictions based on the less exact form (eq 5) of the Fuoss– Onsager equation. Results Experimental details relating to the experiments reported here are given in Appendix 3. Conductivity data measured in aqueous solution for potassium chloride, the magnesium salt of ethane 1,2-disulfonic acid, and the calcium salt of benzene 1,3-disulfonic acid (6 ) are shown in Figures 1–3.

96

94

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200

90

88

86

84

82 180 80 0

0.02

0.04

0.06

( cst)1/2 (mol1/2 dm ᎑3/2)

Figure 3. Λ vs (cst )1/ 2 for aqueous solutions of the calcium salt of benzene 1,3-disulfonic acid (6 ) at 298 K. Values of Λ: ( 䊊 ) observed; (–––) from Onsager equation (eq 2); ( – – – ) from Fuoss– Onsager equation (eq 5). Crossover points: experimental (c stx)1/ 2 = 0.045 mol 1/ 2 dm–3/2; theoretical ( c stx)1/ 2 = 0.047 mol1/ 2 dm᎑3/ 2. Λ o = 238.22 cm2 S mol ᎑1; S = 918.26 cm2 S mol ᎑3/ 2 dm3/ 2; E = 6743 cm2 S mol ᎑2 dm3; J = 17,900 cm2 S mol ᎑2 dm3; å = 4.9.

354

0

0.04

0.08

0.12

(cst )1/2 (mol1/2 dm᎑3/2)

Figure 4. Λ vs (cst )1/ 2 for aqueous solutions of tetra- n - butyl ammonium iodide at 298 K. Values of Λ: ( 䊊 ) observed; (–––) from Onsager equation (eq 2); (– – –) from Fuoss–Onsager equation (eq 5 ). Crossover points: experimental not observed; theoretical (c stx )1/ 2 = 3.31 × 10 ᎑5 mol1/ 2 dm᎑3/ 2. Λ o = 96.3 cm2 S mol ᎑1; S = 82.29 cm2 S mol ᎑3/ 2 dm3/ 2; E = 30.48 cm2 S mol ᎑2 dm3; J = 273.1 cm2 S mol ᎑2 dm3; å = 6.0.


In the Laboratory

Comparing the Λ vs (cs t) 1/2 curve predicted by eq 5 with the observed data, it is clear that the observed behavior reflects non-ideality in the cases of 1-1 and 2-2 electrolytes that are completely dissociated as free ions. Table 1 gives observed and calculated values of the crossover concentrations (c xst ) for these electrolytes. The level of agreement further confirms that there is essentially no ion association in these electrolytes. Results for tetra-n-butyl ammonium iodide and magnesium sulfate are shown in Figures 4 and 5. The marked deviations from the behavior predicted by the Fuoss–Onsager equation provide unambiguous evidence of ion association in these cases. The negative deviations from the line defined by the Onsager equation are much greater than those predicted by the Fuoss–Onsager equation (i.e., on the basis of non-ideality, but with no ion association) and provide definite proof for the occurrence of ion association in these systems. Again, the equilibrium constants for ion association could be estimated in principle from the experimental conductivity data reported here, but improved apparatus would be required to determine these equilibrium constants accurately. For 3-3 electrolytes, deviations from ideal behavior (Onsager equation) are typically very large and could be interpreted as arising from the combination of a relatively small value of J (as a consequence of small å, see Appendix 1) and some degree of ion association. The Fuoss–Onsager equation can again be used as a baseline from which to give a qualitative demonstration of ion association. Figure 6 shows data for lanthanum hexacyanoferrate(III). Some representative data from an accurate experimental study (7) have also been added to this graph. Very large negative deviations from the Fuoss–Onsager equation are clearly observed, from which the

occurrence of ion association in this 3-3 electrolyte is clear. Note that several other techniques (both experimental approaches and computer simulation) can be used to furnish information about ion association in electrolyte solutions, and particularly to estimate values for ion association constants. Such techniques complement conductivity measurements in the investigation of ion association. However, we again emphasize that conductivity measurements allow the effect of ion association on non-ideality to be probed directly. An Experiment for the Undergraduate Laboratory We have shown how simple conductance equipment can clearly demonstrate the effects of non-ideality and ion association in aqueous electrolyte solutions. The simplicity of the experimental approach and the fact that meaningful and reasonably accurate results can be obtained by students who have little practical experience make it readily adaptable as an experiment for the undergraduate teaching laboratory. We have implemented such an experiment to study the conductivity of aqueous solutions of magnesium sulfate for concentrations in the range cs t ≈ 1 × 10᎑3 mol dm᎑3 to cst ≈ 5 × 10᎑3 mol dm᎑3 and to compare the experimental results with those predicted on the basis of the Fuoss–Onsager and Onsager equations (eqs 5 and 2). Typical results for this experiment are illustrated in Figure 5. In our experience, students can readily complete the experimental measurements and analysis of data in about 3 hours. In addition to introducing students to the concepts of ion association and non-ideality in aqueous electrolyte solutions, other specific learning objectives of this undergraduate

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200

180

0 0

0.02

0.04

0.06

0.08

( cst )1/2 (mol 1/2 dm᎑3/2 )

Figure 5. Λ vs (cs t )1/ 2 for aqueous solutions of magnesium sulfate at 298 K. Values of Λ: ( 䊊 ) observed; (––– ) from Onsager equation (eq 2); (– – – ) from Fuoss–Onsager equation (eq 5). Crossover points: experimental not observed; theoretical (c stx )1/ 2 = 0.048 mol1/2 dm᎑3 /2. Λ o = 266.1 cm2 S mol ᎑1; S = 969 cm2 S mol ᎑3/ 2 dm3/ 2; E = 7,685 cm 2 S mol ᎑2 dm3; J = 20,240 cm2 S mol᎑2 dm3; å = 5.0.

0.02

0.04

0.06

0.08

(c st) 1/2 (mol1/2 dm ᎑3/2)

Figure 6. Λ vs (cst )1/ 2 for aqueous solutions of lanthanum hexacyanoferrate(III) at 298 K. Values of Λ: ( 䊊 ) observed; (–––) from Onsager equation (eq 2); (– – –) from Fuoss–Onsager equation (eq 5); (+) data from ref 7. Crossover points: experimental (c stx )1/ 2 = 0.06 mol1/ 2 dm᎑3 /2; theoretical (c stx )1/ 2 = 0.035 mol1/ 2 dm᎑3/ 2. Λ o = 506.7 cm2 S mol ᎑1; S = 4757 cm2 S mol᎑3/ 2 dm3/ 2; E = 180,057 cm2 S mol ᎑2 dm3; J = 522,444 cm2 S mol ᎑2 dm3; å = 7.0. Values of Λ o and å are those quoted in ref 7.


355

In the Laboratory

experiment are to: a. Introduce the essential techniques for experimental measurement of conductivity, emphasizing the precautions and careful practice that should be observed and the potential pitfalls of failing to follow these procedures. b. Understand and be able to manipulate the Onsager and the Fuoss–Onsager equations in calculating the predicted Λ vs (cst )1/ 2 behavior given values of the constants (Appendix 1) appropriate for magnesium sulfate. c. Provide experience in the correct handling of units in relation to conductivity data, in both the presentation of experimental data and the manipulation of the calculated data. d. Develop good practice in the construction of graphs from experimental and calculated data. e. Understand fully the reasons why experimental Λ vs (cst )1/ 2 behavior deviates from that predicted by the Fuoss–Onsager equation (Fig. 5). f. Encourage awareness at a qualitative level of sources of experimental error and ways in which these could be reduced or eliminated by improved experimental design.

Literature Cited 1. Wright, M. R. The Nature of Electrolyte Solutions; MacMillan Education: Basingstoke and London,1988. 2. Onsager, L.; Fuoss, R. M. J. Phys. Chem. 1932, 61, 2689. 3. Fuoss, R. M.; Onsager, L. J. Phys. Chem. 1957, 61, 668; 1958, 62, 1339. 4. Bell, R. P.; Robson, M. Trans. Faraday Soc. 1965, 61, 928. 5. Atkinson, G.; Yokoi, M.; Hallada, C. J. J. Am. Chem. Soc. 1961, 83, 1570. 6. Atkinson, G.; Petrucci, S. J. Phys. Chem. 1963, 67, 337. 7. Davies, C. V.; James, J. C. Proc. R. Soc. London 1948, A195, 116. 8. McElwain, S. L.; Jelinek, A.; Rorig, K. J. Am. Chem. Soc. 1945, 83, 1578. 9. Marsh, J. K. J. Chem. Soc. 1947, 118, 1084. 10. Davies, C. W. Ion Association; Butterworths: London, 1972. 11. Friedman, H. L. Annu. Rev. Phys. Chem. 1981, 32, 179. 12. Justice, J.-C. In Comprehensive Treatise of Electrochemistry; Conway, B. E.; Bockris, J. O’M.; Yeager, E., Eds.; Plenum: New York, 1983; pp 223–337. 13. Pethybridge, A. D.; Taba, S. S. Discuss. Faraday Soc. 1977, 64, 274. 14. Katayama, S. Bull. Chem. Soc. Jpn. 1973, 46, 106. 15. Katayama, S. J. Solution Chem. 1976, 5, 241. 16. Pitzer, K. S. J. Chem. Soc., Faraday Trans. 2, 1972, 68, 101. 17. Archer, D. G.; Wood, R. H. J. Solution Chem. 1985, 14, 757. 18. Ansari, A. A.; Islam, M. R. Can. J. Chem. 1988, 66, 1720. 19. Islam, N.; Zaidi, S. B. A.; Ansari, A. A. Bull. Chem. Soc. Jpn. 1989, 62, 309. 20. Iida, M.; Iwaki, M.; Matsuno, Y.; Yokoyama, H. Bull. Chem. Soc. Jpn. 1990, 63, 993. 21. Price, W. E.; Weingartner, H. J. Phys. Chem. 1991, 95, 8933. 22. Justice, J.-C. J. Phys. Chem. 1996, 100, 1246. 23. Kalyuzhnyi, Y. V.; Holovko, M. F. Mol. Phys. 1993, 80, 1165. 24. Smith, D. E.; Dang, L. X. J. Chem. Phys. 1994, 100, 3757. 25. Wang, J.; Haymet, A. D. J. J. Chem. Phys. 1994, 100, 3767. 26. Spohn, P. D.; Brill, T. B. J. Phys. Chem. 1989, 93, 6224. 27. Gil Montoro, J. C.; Bresme, F.; Abascal, J. L. F. J. Chem. Phys. 1994, 101, 10892. 28. Friedman, H. L.; Larsen, B. J. Chem. Phys. 1979, 70, 92. 29. Altenberger, A. R.; Friedman, H. L. J. Chem. Phys. 1983, 78, 4162. 30. Zhong, E. C.; Friedman, H. L. J. Phys. Chem. 1988, 92, 1685. 31. Abascal, J. L. F.; Turq, P. Chem. Phys. 1991, 153, 79. 32. Smith, D. E.; Kalyuzhnyi, Y. V.; Haymet, A. D. J. J. Chem. Phys. 1991, 95, 9165. 33. Abascal, J. L. F.; Bresme, F.; Turq, P. Mol. Phys. 1994, 81, 143.

356

Appendix 1. Parameters Relevant to the Fuoss–Onsager Equations (Eqs 4 and 5) Parameter S S /cm2 S mol᎑3/ 2 dm3/2 = ( α Λ o + β ) /cm2 S mol᎑3/ 2 dm3/ 2 α /mol᎑1/ 2 dm3/ 2 = z 2 × 0. 2289 For 1-1 electrolytes (z = 1): α /mol ᎑1/ 2 dm3/ 2 = 0.2289 For 2-2 electrolytes (z = 2): α /mol ᎑1/ 2 dm3/ 2 = 0.9156 For 3-3 electrolytes (z = 3): α /mol ᎑1/ 2 dm3/ 2 = 2.0601 β / cm2 S mol᎑3/ 2 dm3/ 2 = z 3 × 60.24 For 1-1 electrolytes (z = 1): β / cm2 S mol᎑3/ 2 dm3/ 2 = 60.24 For 2-2 electrolytes (z = 2): β / cm2 S mol᎑3/ 2 dm3/ 2 = 481.9 For 3-3 electrolytes (z = 3): β / cm2 S mol᎑3/ 2 dm3/ 2 = 1626

Parameter E E /cm2 S mol᎑2 dm3 = (E1Λ o – E 2) / cm2 S mol᎑2 dm3 E 1 /mol᎑1 dm 3 = z6 × 0.5276 For 1-1 electrolytes (z = 1): E 1 / mol᎑1 dm3 = 0.5276 For 2-2 electrolytes (z = 2): E 1 / mol᎑1 dm3 = 33.77 For 3-3 electrolytes (z = 3): E 1 / mol᎑1 dm3 = 384.6 E 2 /cm2 S mol᎑2 dm3 = z 6 × 20.33 For 1-1 electrolytes (z = 1): E 2 / cm2 S mol᎑2 dm 3 = 20.33 For 2-2 electrolytes (z = 2): E 2 / cm2 S mol᎑2 dm 3 = 1301 For 3-3 electrolytes (z = 3): E 2 / cm2 S mol᎑2 dm 3 = 14,820

Parameter J J / cm2 S mol᎑2 dm3 = ( σ 1 Λo + σ2) / cm2 S mol᎑2 dm3 σ 1 / mol᎑1 dm3 = 0.4582 z 6 [h(b) + ln[ å] + ln[z ] – 0.0941] σ2 / cm2 S mol᎑2 dm3 = 15.48 z 6 + 18.15 z 4 å – 17.66 z 6 [ln[å] + ln[z]] h (b) = (2b 2 + 2b – 1) / b 3 b = 7.135 z 2 / å å = (distance of closest approach / m) / 10 ᎑10

Appendix 2. Determination of Equilibrium Constant for Ion Association If the experimentally determined graph of Λ vs (c s t)1/ 2 deviates from the graph expected on the assumption that there is no ion association, the equilibrium constant for the ion association can be determined, in principle, from these deviations as described below. However, accurate conductivity data are required to allow reliable estimation of the equilibrium constant. We consider, in general, the equilibrium Mn+ (aq) + X n᎑ (aq) M n+ X n᎑(aq) for which the equilibrium constant for association of the ion pair is n+

M X

K=

M

n+ act

n᎑ act

X

(A1)

n᎑ act

The fraction of ions associated as ion pairs is defined as n+

β=

amount of M present as the ion pair total amount of M

n+

present (as free ion or ion pair)

M

β=

M

n+

n+

X

n᎑ act

n+

act

+ M X

(A2)

n᎑ act

where [M n +] act + [M n+ X n᎑ ] act = [Mn+ ] s t

(A3)

with the subscripts “act” and “st” defined in the text. It follows from eqs A2 and A3 that [M n+ ]act = (1 – β )[M n+ ] s t

(A4)

Similarly, it can be shown that [ X n᎑ ] act = (1 – β ) [X n᎑ ] st

(A5)

The equilibrium constant for ion association is therefore given by

K=

β M

1–β M

n+

n+ st

st

1–β X

(A6)

n᎑ st

As [ M n+ ] st = [ X n᎑ ] s t = c s t , this equation for K can be expressed as

K=

βcst

1–β

2

c st

2

=

β 2

1 – β c st


(A7)

In the Laboratory This equation has two unknowns, K and β. The value of β can be estimated from the conductivity data by making use of the equation Λ obs = Λ ′ (1 – β )

(A 8)

where Λ′ represents the molar conductivity expected if there were no ion association (i.e., β = 0) at the value of c s t of interest and Λ obs represents the actual measured molar conductivity at this value of c s t. For any given c s t, the value of Λ ′ can be estimated from the Fuoss–Onsager extended conductance equation (eq 5), allowing the value of β to be estimated at the concentration c s t of interest. Using these values of β and c s t , the corresponding value of K can be determined from eq A 7.

Appendix 3. Experimental Details Magnesium sulfate (AR grade), potassium chloride (AR grade) and tetran-butyl ammonium iodide (>98%) were obtained commercially. The magnesium sulfate was used directly, the potassium chloride was dried at 120 °C overnight and then cooled in a desiccator, and the tetra-n-butyl ammonium iodide was dried at 120 °C for 2 h and then cooled in the dark in a desiccator. The magnesium salt of ethane 1,2-disulfonic acid was prepared by neutralization from the acid, which was synthesized by the procedure of McElwain (8). Lanthanum hexacyanoferrate (III) was prepared as described by Marsh ( 9 ). Although we are not aware of any specific hazards associated with the chemicals discussed in this paper, it is important that standard laboratory practices are followed and that (as for all experiments) students are educated in the correct procedures for safe handling of chemicals. Thus, contact of these com-

pounds with the skin and eyes should be avoided, and they should not be inhaled or ingested. Conductivity water was prepared by passing nitrogen through distilled water until the conductivity fell to a steady value of about 0.7 µS cm᎑1. In all cases, solutions were thermostatted while final volumetric adjustments were made, and all conductivity measurements were made on the same day as the solutions were prepared. The conductance bridge was manufactured in-house and can measure conductivities above 1 µS cm᎑1 to an accuracy of ±0.5%. Special procedures using the bridge capacitance compensator enabled accurate measurement of the conductivity [(0.7 ± 0.2) µS cm᎑1 ] of the water used to prepare the electrolyte solutions. This was subtracted from the mean conductivity for each electrolyte solution studied. Measurements of conductivity were made at 25 °C in a 50-mL jacketed beaker using a standard electrode with a cell constant close to unity. The beaker was fitted with a plastic cover accommodating the electrode and thermometer, and water from a thermostatted bath was circulated through the jacket. For each solution, 100 mL was prepared by dilution of a stock solution and about 25 mL was used for each measurement. Conductivities were measured in order of increasing concentration. Care was taken to ensure that the conductance cell was completely immersed in the solution and contained no air bubbles. The first reading for each solution was generally low (and was discarded), but subsequent readings were in close agreement. The cell constant was determined using standard aqueous potassium chloride solutions at 298 K ( κ = 1412 µ S cm᎑1 at c st = 0.01 mol dm᎑ 3; κ = 2767 µS cm ᎑1 at c st = 0.02 mol dm᎑ 3 ).


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Non-ideality and Ion Association in Aqueous Electrolyte Solutions

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