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Burgot2012 Dependence of the Solubility Ionic Strength

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  Chapter 33 Dependence of the Solubility on the Solution’sIonic Strength and on the Presence of CommonIons: Superimposition of Several PrecipitationEquilibria As we could have predicted, solubility depends on the solution’s ionic strength. Theionic strength of the solution may be due to the presence of ions differing fromthose constituting the electrolyte whose solubility is under study. This influence isinvestigated in the first section of the chapter. The ionic strength may also be dueto the addition of ions that constitute the electrolyte itself. In this case, an effectanalogous to the preceding one occurs, but, in addition, another effect, which isfar stronger than the preceding one, arises: the common ion effect. It is due to theshift of an equilibrium. This is the subject of Sect. 33.2. Finally, we investigate thesuperimposition of two precipitation equilibria. 33.1 Influence of the Ionic Strength on the Solubility First, let’s notice that the ionic strength of the solution influences the solubility of ions. We have indeed seen that the activity of a molecular species does not changewith the ionic strength of the solution, provided the latter does not exceed about0.1mol/L. Therefore, in this section, we are interested only in its influence on theionic solubility.It is an experimental fact that the solubility of an electrolyte increases slightlywhen the ionic strength increases. Figure 33.1 gives the solubility changes of silveriodate as a function of the concentration of potassium nitrate present in the solution.Quiteevidently, theionicstrengthofthesolutionincreaseswiththeconcentrationof potassium nitrate. [To be rigorous, we must notice, however, that in this examplethe ionic strength is not due only to K +  and NO − 3  , but also to Ag +  and IO − 3  ionsresulting from the dissolution of silver iodate ( K s  = 10 − 7 . 52 )].We can easily explain the increase in solubility with the ionic strength if werememberthatsolubilityproducts, definedintermsofactivities, donotvarywiththeionic strength as other thermodynamic constants do. Expressing solubility productsin terms of concentrations and activity coefficients such as K s  = [M + ][X − ] γ  ± 2 J.-L. Burgot,  Ionic Equilibria in Analytical Chemistry,  619DOI 10.1007/978-1-4419-8382-4_33, © Springer Science+Business Media, LLC 2012  620 33 Dependence of the Solubility on the Solution’s Ionic Strength ... 0,200,180 2 4 6 8 10 12 14 16 Concentration  KNO 3 (m.mol/L) Solubilityof  AgIO 3 (m.mol/L) Fig. 33.1  Effect of the addition of potassium nitrate on the silver iodate’s solubility immediately shows that since activities decrease (at least in a first stage) when theionic strength increases, the concentrations [M + ] and [X − ] must simultaneouslyincrease for the  K s  value to remain constant.This example provides the occasion to recall that only the constants defined interms of concentrations, that is, the formal or apparent constants, exhibit values thatchange with the ionic strength of the solution. Moreover, formal solubility products K s  defined as follows: K s app  =  M + y   z  X −  z  y are seldom used for a simple reason: The values of the apparent and thermodynamicconstantsareidenticaltotheconcentrationsatwhichtheconceptofsolubilityproductis useful. Indeed, it is because these concentrations are very low.From the standpoint of physical chemistry, the measurement of the solubility  S  at different ionic strengths  I   (whose values are imposed by the addition of a foreignelectrolyte in judicious concentrations) permits us to determine both the thermody-namicsolubilityproductandthemeanactivitycoefficient.Thisdeterminationcanbecarried out in a geometric or algebraic way. Let’s consider the case of a one-to-oneelectrolyte 1.1.According to the preceding considerations, we can write K s / γ  ± 2 = S  2 ,log K s  = 2log γ  ± + 2log S. At weak ionic strengths, according to Debye–Hückel’s limiting law (see Chap. 3),log γ  ±  =−  Az + z − √  I   ,log S   = 12log K s +  Az + z − √  I   .  33.1 Influence of the Ionic Strength on the Solubility 621 Drawing the diagram of log  S   as a function of  √   I   permits us to determine  K s .Alternatively, combining two values  S  1  and  S  2 , determined at two different ionicstrengths  I  1  and  I  2 , permits us to obtain  K s  through the preceding relation. Thebetter process is probably to draw the best straight line of log  S   /  √   I  . It presents theadvantage of taking all of the experimental data (i.e., all the couple values  S  i  /   I  i ) intoaccount. For example, the solubility at null ionic strength of silver iodate and itssolubility product value have been determined in this way. They are S  (AgIO 3 ) = 1 . 733 × 10 − 4 mol/L, K s (AgIO 3 ) = S  2 , K s (AgIO 3 ) = 3 . 00 × 10 − 8 . Then the calculation of the mean activity coefficient is immediate by using a coupleof data  S  i  /   I  i . Indeed, we can write γ  ± 2 = K s S  i 2 . Itisinterestingtonoticethatthegraphoflog  γ  ± / √  I   isactuallyastraightline, infullaccordance with Debye–Hückel’s law. This fact signifies that the part of the ionicstrength due to the concentrations [IO − 3  ] and [Ag + ] is negligible compared to thatdue to the foreign electrolyte, since it is on this assumption that the straight line isdrawn. Exercise 1  The solubility at 25 ◦ C of thallium chloride in pure water is  S   = 1 . 64 × 10 − 2 mol/L.What are its solubility into a solution of potassium nitrate 2 . 5 × 10 − 2 mol/L and its solubility product value ( A = 0 . 509)?One can write for the solution in pure waterlog S   = 12log K s + Az + z − √   1 . 64 × 10 − 2  . If   S    is the desired solubility in the potassium nitrate solution, the ionic strength of the solution is I   =  S   + 2 . 50 × 10 − 2  mol/L,from which we findlog S    = 12log K s + Az + z − √   2 . 5 × 10 − 2 + S    . Combining the relations giving log  S   and log  S    giveslog  S   S   = Az + z −  √   S   + 2 . 50 × 10 − 2  −√   1 . 64 × 10 − 2  .  622 33 Dependence of the Solubility on the Solution’s Ionic Strength ... At 25 ◦ Clog   S    1 . 64 × 10 − 2  = 0 . 509  √   S   + 2 . 50 × 10 − 2  −√   1 . 64 × 10 − 2  or  S    = 1 . 80 × 10 − 2 mol/L . We notice that passing from an ionic strength equal to 1 . 64 × 10 2 mol/L to anotherequal to 4 . 14 × 10 − 2 mol/L induces a change in solubility from 1 . 64 × 10 − 2 mol/Lto 1 . 80 × 10 − 2 mol/L, that is, a change of  + 9 . 8%. The calculation of the solubilityproduct is immediate. For example, using the values found for the solubility of thallium chloride in pure water, we can writelog  1 . 64 × 10 − 2  = 12log K s + 0 . 509 √   1 . 64 × 10 − 2  , K s  = 1 . 99 × 10 − 4 . 33.2 The Common Ion Effect In this section, we consider the influence of the presence of an ion identical to one of the ions constituting the electrolyte whose solubility is under study. For example, westudy theinfluenceof thepresenceof sodium sulfatein asolution upon thesolubilityof barium sulfate in the same solution. Of course, the added common ion influencesthe solubility through its contribution to the ionic strength, as the previous sectiondescribed. This is not the phenomenon under study in this section. Here we studythe  common ion effect  , which can be far more important than the preceding one.For example, let’s study the solubility of barium sulfate. Once its precipitationoccurs, the solubility-product relation is satisfied:  Ba 2 +  SO 42 −  = K s (saturation) . If the barium or sulfate ion concentration is increased by adding barium chloride orsodiumsulfate(whicharefairlysolublethemselves), thebariumsulfateprecipitationis more pronounced since its solubility is decreased, as we can see now.If   S   is the barium sulfate’s solubility in a solution of barium chloride at concen-tration  C  , the relations that are satisfied once the saturation is attained are (by mixingactivities and concentrations)  Ba 2 +  SO 42 −  = K s ,  Ba 2 +  = S  + C ,  SO 42 −  = S. The barium sulfate’s solubility is equal here to the sulfate ion concentration at satu-ration. In other words, the solubility  S   is the maximum number of moles of barium
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