The Partition of Salts (I) Between Two Immiscible Solution Phases and (Ii) Between the Solid Salt Phase and Its Saturated Salt Solution

ChemTexts (2020) 6:17

https://doi.org/10.1007/s40828-020-0109-0 (0123456789().,-volV)(0123456789().,-volV)

LECTURE TEXT

The partition of salts (i) between two immiscible solution phases and (ii) between the solid salt phase and its saturated salt solution

Fritz Scholz1 · Heike Kahlert1 · Richard Thede1

Received: 17 December 2019 / Accepted: 2 March 2020 / Published online: 24 April 2020 © The Author(s) 2020

Abstract The partition of salts between two polar immiscible solvents results from the partition of the cations and anions. Because electroneutrality rules in both phases, the partition of cations is affected by that of anions, and vice versa. Thus, the partition of a salt is determined by the chemical potentials of cations and anions in both phases, and it is limited by the boundary condition of electroneutrality. Whereas the partition of neutral molecules does not produce a Galvani potential difference at the interface, the partition of salts does. Here, the equations to calculate this Galvani potential difference are derived for salts of the general composition CatðzCatÞþAnðzAnÞÀ and for uni-univalent salts CatþAnÀ. The activity of a mCat mAn speciﬁc ion in a particular phase can thus be purposefully tuned by the choice of a suitable counterion. Finally, the distribution of a salt between its solid phase and its saturated solution is also presented, together with a discussion of the Galvani potential difference across the interface of the two phases. Graphical abstract

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Keywords Partition · Partition constant · Ion · Salt · Electrolyte · Galvani potential difference · Ion adsorption · Fajans · Precipitation · Titration

List of symbols

Latin symbols a Activity Fritz Scholz and Heike Kahlert equally contributed as authors. A Neutral compound A & Fritz Scholz AnðzAnÞÀ Anion An with z negative charges [email protected] B, C,... Compounds B, C, etc. & Heike Kahlert c Molar concentration [email protected] c* Standard concentration (1 mol L−1) ðzCatÞþ Cation Cat with z positive charges 1 Institut fu¨r Biochemie, Universita¨t Greifswald, Felix- Cat Hausdorff-Str. 4, 17489 Greifswald, Germany DCM Dichloromethane 123 17 Page 2 of 12 ChemTexts (2020) 6:17

− e Electron Da;b/ Difference of inner electric potentials of the E Electrolyte phases phase a and b, i.e. the Galvani potential E Electric potential difference between these two phases. It is f Activity coefficient defined as Da;b/ ¼ /b À /a F Faraday constant (9.64853383(83)9 ÀÀ 0 Formal potential of ion transfer Ds;sol/ ; 104 C mol−1) c i g Free energy (Gibbs energy) Other symbols G Molar free energy (molar Gibbs energy) ÀÀ Plimsoll symbol, as superscript indicating standard D ÀÀ Standard Gibbs energy of transfer of A from a!bGA quantities phase a to phase b KIP Formation constant of ion pairs (association Introduction constant) K ; ; Partition constant of species A pðAÞ T p The partition of neutral compounds between immiscible Ksp Solubility product solvents is very well covered in textbooks of general, i Chemical species i physical and analytical chemistry; however, the partition of IP Ion pair salts can be found only in special monographs, where it is ln Natural logarithm (logarithm to the base e, with often presented in a form which is not easily understand- e being the Euler constant) able for students. Here, we provide an introductory text log Decadic logarithm (logarithm to the base 10) discussing salt partition on the basis of the thermodynamics M Metal phase as taught in Bachelor courses. n Amount of substance (unit: mol) of atoms, The partition of salts is of high importance in many molecules, ions or electrons fields of science: ion transfer (e.g. of drugs) through nb Nitrobenzene membranes, ion transfer catalysis in organic synthesis, and Ox Oxidized form of a redox pair extraction of metal ions in analytical chemistry are just the p Partition most common examples. p Pressure When the partition of compounds between liquid phases À pic Picrate (2,4,6-trinitrophenolate) is treated, usually the phrase ‘partition between two R Universal gas constant −1 −1 immiscible liquid phases’ is used. However, completely (8.31446261815324 J K mol ) immiscible liquids do not exist. That phrase just points to Red Reduced form of a redox pair those systems where the mutual miscibility is so small that s Indicates a solid phase (here a salt phase) the phases can be considered as pure phases, at least for a sol Indicates a solution phase simplified treatment. T Absolute temperature TBA? Tetrabutylammonium ion TEA? Tetraethylammonium ion ? Partition of neutral compounds TMA Tetramethylammonium ion between two immiscible solution phases TPA? Tetraphenylarsonium ion − TPB Tetraphenylborate ion The Nernst distribution law describes the partition of a w Water neutral compound A between two immiscible liquid phases z Charge of an ion, number of exchanged a and b electrons Aa Ab: ðEquilibrium IÞ Greek symbols An example is the distribution of iodine between water a Phase alpha (Greek letter alpha) and tetrachloromethane. Walter Nernst published the b Phase beta (Greek letter beta) respective equation in 1891 [1]. In modern terms, it states l Chemical potential (Greek letter mu in italics) that the ratio of activities [2] of the compound in the two l Electrochemical potential phases is constant at constant temperature (T) and pressure m Stoichiometric number (Greek letter nu in (p): italics) / Inner electric potential (Greek letter phi in KpðAÞ;T;p ¼ aA; b aA; a ð1Þ italics) KpðAÞ;T;p is called the partition constant of A, and the upright p in the subscript stands for partition. In

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É l l Da bG equilibrium, the chemical potentials A; a and A; b of the aA; b ! i À 2:303RT : KpðAÞ;T;p ¼ ¼ 10 ð10Þ distributed compound A must be the same in both phases: aA; a l l : A; a ¼ A; b ð2Þ Remember, the chemical potential is the partial deriva- z )+ z )- tive of the Gibbs energy over the amount (unit: mol) of Partition of a salt Catð Cat Anð An mCat mAn substance (atoms, molecules or ions) (here of A), at con- between two immiscible solution phases stant temperature, pressure and constant amounts (unit: mol) of all other compounds (e.g. B, C, etc.): For a very simple reason, the partition of a salt between o immiscible solvents is more complex than that of neutral l g : A ¼ o ð3Þ compounds: the two constituents of a salt, i.e. cations and nA T;p;n ;n ;... B C anions, are on one side free in solution, but on the other This means that the chemical potential is a measure of side, their transfer to the other phase is restricted by the the ability to produce or consume work. Since a system condition of electroneutrality in both bulk phases. The does not produce or consume work when it is in equilib- electroneutrality is the boundary condition for the partition. rium, the chemical potentials of A in the phases a and b Only in the interfacial region (double layer region), where must be equal at equilibrium, as otherwise a ﬂux of A from the two phases meet, is the condition of electroneutrality one phase to the other would occur. Since the chemical violated. This causes the occurrence of an interfacial lÀÀ potential difference called Galvani potential difference. A potential of A can be split into a standard term A (the salt CatðzCatÞþAnðzAnÞÀ partitions between two polar immis- chemical standard potential) and an activity-dependent mCat mAn term: cible solvent phases a and b according to ÀÀ z z l l m ðzCatÞþ m ðzAnÞÀ m ð CatÞþ m ð AnÞÀ: A ¼ A þ RT ln aA ð4Þ Cat;aCata þ An;aAna Cat;bCatb þ An;bAnb it follows from the condition (2) that ðEquilibrium IIÞ lÀÀ lÀÀ ; A; a þ RT ln aA; a ¼ A; b þ RT ln aA; b ð5Þ Since the stoichiometry of the salt is the same in both lÀÀ lÀÀ : m m m m m m A; a À A; b ¼ RT ln aA; b À RT ln aA; a ð6Þ phases, i.e. Cat;a ¼ Cat;b ¼ Cat and An;a ¼ An;b ¼ An, one can write The difference of chemical standard potentials is a m m a Cat a An constant: ÀÁCatðzCatÞþ;b AnðzAnÞÀ;b : K ðzCatÞþ ðzAnÞÀ ¼ m m ð11Þ p Catm Anm ; T;p Cat An a ; b a ; b Cat An a z a z lÀÀ lÀÀ A : A : Catð CatÞþ;a Anð AnÞÀ;a A; a À A; b ¼ RT ln ¼ 2 303RT log ð7Þ aA; a aA; a Equilibrium II indicates that the salt is assumed to be Since the partition constant has been deﬁned as completely dissociated in both phases, i.e. no ion pairs þ À KpðAÞ;T;p ¼ aA; b aA; a (1), the following equation results exist in the two phases. Ion pairs, e.g.½ Cat An ionpair of a from (7) and (1): uni-univalent salt, form when the electrostatic attraction

ÀÀ ÀÀ ÀÀ ÀÀ l Àl l Àl between the opposite charged ions is not overcome by the aA; b A; a A; b A; a A; b RT 2:303RT : KpðAÞ;T;p ¼ ¼ e ¼ 10 ð8Þ free energy of solvation of the single ions (see the example a ; a A of ½TBAþpicÀ at the end of this paper). This happens

Equation (8) nicely shows that KpðAÞ;T;p is larger than 1 especially in solvents which have a low dielectric constant, ÀÀ ÀÀ ÀÀ ÀÀ i.e. which are rather nonpolar. Neglecting ion pairs is when l ; a [ l ; b, and smaller than 1 for l ; a\l ; b. A A A A certainly a simpliﬁcation, as ion pairs are in principle Equation (8) can also be written using the standard Gibbs ubiquitous in electrolyte solutions. However, if the liquid D ÀÀ a b energy of transfer a!bGA of A from phase to phase , phases have a high polarity, their formation may be neg- deﬁned as ligible. When one liquid phase is water (a polar solvent), D ÀÀ lÀÀ lÀÀ : and the other solvent is also polar, like nitrobenzene, ion a!bGA ¼ A; b À A; a ð9Þ pairing can be neglected in both phases. Further, the par- Because a Gibbs energy difference always refers to a tition could be affected by complex formation or any other certain direction of the reaction, the arrow in the subscript chemical equilibria involving the partitioning ions. All indicates here the transfer from phase a to b. With Eq. (9) these side reactions are excluded in the present treatment. it is possible to write Eq. (8) as follows: Figure 1 shows a scheme of the Equilibrium II as the result of the partition equilibria of the cations and anions.

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Although the two bulk phases maintain electroneutrality by having equal numbers of negative and positive charges, at the interface of the two phases a Galvani potential difference builds up if the two partition constants

aCatðzCatÞþ;b KpðCatðzCatÞþÞ; T;p ¼ ð12Þ aCatðzCatÞþ;a and

aAnðzAnÞÀ; b KpðAnðzAnÞÀÞ; T;p ¼ ð13Þ aAnðzAnÞÀ; a do not have identical values. When these constants differ from each other, which is the case for almost all salts, a minute charge separation occurs across the interface, resulting in the Galvani potential difference: Fig. 1 The partition of the ions of a salt having the formula ðzCatÞþ ðzAnÞÀ Catm Anm between two phases is the result of the partition Da;b/ ¼ /b À /a; ð14Þ Cat An equilibria of anions and cations where /a and /b are the inner electric potentials of the two minimised, so that the Galvani potential difference phases. The determination of the two equilibrium constants between the phases a and b is the major contribution. The would need to know the activity of each single kind of ion, ÀÁ calculation of the partition constant K ðzCatÞþ ðzAnÞÀ p Catm Anm ; T;p i.e. of the cations and the anions. The situation at the Cat An interface shown in Fig. 1 is to some extent similar to that at of the salt can be attempted by calculation of the interface of an electrode, where an oxidation and z z KpðCatð CatÞþÞ; T;p and KpðAnð AnÞÀÞ; T;p , as the three constants are reduction can proceed: quantification of a redox equilib- related as follows: rium at an electrode needs to define the zero point of the m m a Cat a An Galvani potential difference. For the basics of interfacial ÀÁCatðzCatÞþ; b AnðzAnÞÀ;b K ðzCatÞþ ðzAnÞÀ ¼ m m p Catm Anm ; T;p a Cat a An and especially electrochemical thermodynamics, we sug- Cat An CatðzCatÞþ;a AnðzAnÞÀ;a m m gest the papers by Lańg [3] and Inzelt [4]. In the case of Cat An ¼ K ðz Þþ K ðz ÞÀ : redox equilibria the convention is that the Galvani potential pðCat Cat Þ; T;p pðAn An Þ; T;p difference at the platinum|solution interface of the standard ð16Þ hydrogen electrode is zero. In the case of ion partition equilibria, the commonly accepted convention (there are also other conventions) is that tetraphenylarsonium (TPA?) − Ion partition between two immiscible cations and tetraphenylborate (TPB ) anions have identical solution phases partition constants [5], because they are equally bulky ions and both are single charged: Since ions are charged particles, the transfer from one À : KpðTPAþÞ;T;p ¼ KpðTPB Þ;T;p ð15Þ phase to another phase involves electric work, when the two phases have different inner electrical potentials. This Partition of this salt produces a zero cell voltage (see the work, expressed as a molar quantity, is the charge of the end of this text) of the electrochemical cell: ion z multiplied by the charge of one mole unit charges ⁞ ⁞ a b ⁞ ⁞ i M1 |E1 salt bridge 1 phase | phase salt bridge 2 (the Faraday constant), multiplied by the difference of E2 |M2 / / inner electricÀÁ potentials a and b between the two phases, M1 is the metal phase of reference electrode 1, E1 is the i.e. ziF /b À /a . The sum of chemical and electrical work electrolyte of reference electrode 1, M2 is the metal phase for the transfer of ions from infinity (outside the respective of reference electrode 2, E2 is the electrolyte of reference electrode 2, the vertical bar | symbolises a phase boundary, phases) to the inner of the respective phases is the so-called l l and the dashed vertical bar stands for a junction between electrochemical potential, here i; a and i;b: l l / ; miscible solutions. When the two reference electrodes are i; a ¼ i; a þ ziF a ð17Þ identical, the measured cell voltage is the Galvani potential l ; b ¼ l ; b þ z F/b: ð18Þ difference across the interface of a with b defined as i i i Da;b/ ¼ /b À /a, plus the diffusion potentials at the junctions of the different solutions. The latter can be

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The equilibrium condition for ions is the equality of Table 1 Standard Gibbs energies of transfer of ions from water to m l m l D ÀÀ electrochemical potentials, i.e. i; a i; a ¼ i; b i; b, since the nitrobenzene w!nbGi , and the standard Galvani potential differ- D /ÀÀ D /ÀÀ electric work cannot be neglected: ences w;nb i (corresponding to a;b i in this text, i.e. D /ÀÀ /ÀÀ /ÀÀ m l m / m l m / : w;nb i ¼ i;nb À i;w) i; a i; a þ i; aziF a ¼ i; b i; b þ i; bziF b ð19Þ ÀÀ −1) ÀÀ Ion D G (kJ mol D ; / (mV) References When the partition is not affected by any chemical w!nb i w nb i reactions, the stoichiometry of the salt remains unaffected K? 22.65 −235 [13] in both phases, and for the cations and anions the relation Rb? 19.80 −205 [13] ? mi; a ¼ mi; b, must hold true: Tl 19.30 −200 [13] l / l / : Cs? 17.80 −184 [13] i; a þ ziF a ¼ i; b þ ziF b ð20Þ TMA? 9.60 −99 [13] This allows the calculation of the Galvani potential TEA? −0.50 5[13] difference caused by partition of the ions, either cations or TBA? −8.20 85 [13] anions: TPB− −35.90 −372 [11] / / l l −3.00 −31 ziF b À ziF a ¼ i; a À i; b ð21Þ Picrate [13] ClOÀ 8.00 83 [11] l ; a À l ; b 4 i i − Da;b/ ¼ /b À /a ¼ : ð22Þ i z F I 18.80 195 [11] i À ClO3 25.40 263 [13] l − Since the chemical potential of a species i is i ¼ Cl 31.40 325 [11] lÀÀ i þ RT ln ai (Eq. 4), one can write Eq. (22) as follows: 1 ÀÀ ÀÀ Da;b/ l ; a l ; b ; i ¼ i; a þ RT ln ai À i; b À RT ln ai ð23Þ ÀÀ ÀÀ Da bG Da bG ziF ! i ! i À À : KpðiÞ; T;p ¼ e RT ¼ 10 2 303RT ð28Þ lÀÀ À lÀÀ D / i; a i; b RT ai; a : a;b i ¼ þ ln ð24Þ (For comparison, see also Eqs. (8–10)). Alternatively, z F z F a ; b i i i one can give the relations lÀÀ ÀlÀÀ The term i; a i; b defines the standard potential of ion ziF D /ÀÀ ; ziF ln KpðiÞ; T;p ¼ a;b i ð29Þ transfer (cf. Eq. 9): RT z F ÀÀ z F ÀÀ i Da;b/ i Da;b/ RT i 2:303RT i : lÀÀ À lÀÀ D ÀÀ KpðiÞ; T;p ¼ e ¼ 10 ð30Þ ÀÀ i; a i; b a!bGi Da;b/ 25 i ¼ ¼À ð Þ ÀÀ z F z F a ; a Da bG i i Since ln 1 ¼ ln i ¼ ! i , it follows with KpðiÞ; T;p ai; b RT ÀÀ ÀÀ ÀÀ ÀÀ since Da bG ¼ l À l , which is the standard Gibbs Da bG ! i i; b i; a D / D / ! i RT Eq. (26) that a;b i is zero: a;b i ¼À þ a b ziF ziF energy of transfer of the ions from phase to , Eq. (24) ÀÀ ÀÀ Da bG Da bG can be written as follows: ln 1 ¼À ! i þ ! i ¼ 0, which was the condi- KpðiÞ; T;p ziF ziF tion to define the chemical equilibrium constant. D / D /ÀÀ RT ai; a a;b i ¼ a;b i þ ln In Table 1 standard Gibbs energies of ion transfer and ziF ai; b ÀÀ the corresponding standard Galvani potential differences Da!bG RT ai; a ¼À i þ ln ð26Þ are given for selected ions. ziF ziF ai; b The hydrophilicity of cations increases with increasing D ÀÀ ÀÀ ÀÀ a!bG RT 1 D G and with decreasing Da;b/ . The hydrophilicity ¼À i þ ln : w!nb i i ziF ziF Kp ðiÞ; T;p D ÀÀ of anions also increases with increasing w!nbGi , but ÀÀ ai; b D / The K ; ; used here is defined in Eq. (1)as , but with increasing a;b i . The latter is a result of the nega- pðiÞ T p ai; a one needs to remember that this is the chemical equilib- tive charge of the anions. In the case of the transfer of ions D ÀÀ rium constant for which equality of the inner electric from water to nitrobenzene, increasing values of w!nbGi potentials of the two phases is a prerequisite. Hence, imply that fewer ions are transferred to nitrobenzene. ÀÀ a ; b Da bG i ! i ; ln KpðiÞ; T;p ¼ ln ¼À ð27Þ ai; a RT

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z )+ ð Cat ðzAn)- D /ÀÁD /ÀÁ Partition of Cat An between two zCat a;b ðzCatÞþ ðzAnÞÀ þ jjzAn a;b ðzCatÞþ ðzAnÞÀ m m Catm Anm Catm Anm Cat An Cat An Cat An immiscible solution phases ÀÀ ÀÀ ¼ z Da;b/ þ jjz Da;b/ Cat CatðzCatÞþ An AnðzAnÞÀ

RT a ðzCatÞþ;a RT a ðzAnÞÀ; a Now we need to consider the interplay of cation and anion þ ln Cat À ln An ; partition: The chemical interaction of the ions with the two F aCatðzCatÞþ; b F aAnðzAnÞÀ; b solvents is always different for the cations and anions (with ð37Þ the exception of very few salts, like the TPA?TPB− mentioned above). Therefore, the partition of cations is affected and hence by that of anions, and vice versa. The difference of inner D /ÀÁ ðÞzCat þ jjzAn a;b ðzCatÞþ ðzAnÞÀ Catm Anm D / Cat An electric potentials a;b i is not zero, as in the theoretical ÀÀ ÀÀ case of single ion partition discussed in connection with ¼ z Da;b/ þ jjz Da;b/ Cat CatðzCatÞþ An AnðzAnÞÀ ð38Þ Eq. (27). The Galvani potential differences caused by the RT aCatðzCatÞþ;aaAnðzAnÞÀ; b D / D / þ ln ; cations a;b CatðzCatÞþ and anions a;b AnðzAnÞÀ must be equal F aCatðzCatÞþ; baAnðzAnÞÀ; a to the overall Galvani potential difference ÀÁ ÀÁ D / Da;b/ ðz Þþ ðz ÞÀ a;b ðzCatÞþ ðzAnÞÀ , as only one potential difference can Cat An Catm Anm Catm Anm Cat An Cat An ÀÀ ÀÀ exist at one interface. z Da;b/ þ jjz Da;b/ Cat CatðzCatÞþ An AnðzAnÞÀ D / D / D /ÀÁ ¼ ð39Þ a;b ðzCatÞþ ¼ a;b ðzAnÞÀ ¼ a;b ðzCatÞþ ðzAnÞÀ Cat An Catm Anm ðÞzCat þ jjzAn Cat An RT a ðzCatÞþ;aa ðzAnÞÀ; b ð31Þ þ ln Cat An : ðÞzCat þ jjzAn F aCatðzCatÞþ; baAnðzAnÞÀ; a Now, Eq. (26) can be formulated for the cations Although Eq. (39) is reminiscent of the Nernst equa- D / D /ÀÁ a;b ðzCatÞþ ¼ a;b ðzCatÞþ ðzAnÞÀ Cat Catm Anm Cat An tion, the argument of the logarithm is not the partition m m Cat An a ðz Þþ ð32Þ a a ÀÀ RT Cat Cat ;a ÀÁCatðzCatÞþ; b AnðzAnÞÀ;b D / : m m ¼ a;b ðz Þþ þ ln constant K ðzCatÞþ ðzAnÞÀ ¼ Cat An ¼ Cat Cat p Catm Anm ; T;p a a z F a ðz Þþ Cat An ðz Þþ ðz ÞÀ Cat Cat Cat ; b Cat Cat ;a An An ;a mCat mAn K ðz Þþ K ðz ÞÀ For the anions Eq. (26) follows pðCat Cat Þ; T;p pðAn An Þ; T;p (cf. Eq. 16), but a a D / D /ÀÁ the term CatðzCatÞþ;a AnðzAnÞÀ; b in Eq. (39) is the ratio of the a;b ðzAnÞÀ ¼ a;b ðzCatÞþ ðzAnÞÀ a a An Catm Anm CatðzCatÞþ; b AnðzAnÞÀ; a Cat An K ðz ÞÀ a ðz ÞÀ ð33Þ pðAn An Þ; T;p ÀÀ RT An An ; a partition constants: . ¼ Da;b/ þ ln K ðz Þþ AnðzAnÞÀ pðCat Cat Þ; T;p zAnF aAnðzAnÞÀ; b Taking into account that the activity ai is deﬁned as 1 or ai ¼ fici cÃ, where fi is the activity coefﬁcient of i, ci is the Ã D / D /ÀÁ molar concentration of i, and c is the standard concen- a;b ðzAnÞÀ ¼ a;b ðzCatÞþ ðzAnÞÀ An Catm Anm À1 Cat An tration (1 mol L ), Eq. (39) assumes the form a ðz ÞÀ ð34Þ D /ÀÀ RT An An ; a : D /ÀÁ ¼ a;b À ln a;b ðzCatÞþ ðzAnÞÀ AnðzAnÞÀ Catm Anm jjzAn F aAnðzAnÞÀ; b Cat An D /ÀÀ D /ÀÀ zCat a;b z þ jjzAn a;b z D /ÀÁ Catð CatÞþ Anð AnÞÀ To solve the equations for a;b ðzCatÞþ ðzAnÞÀ , Catm Anm ¼ Cat An ðÞzCat þ jjzAn 40 Eq. (32) has to be multiplied by zCat, and Eq. (34)byjjzAn : RT c ðzCatÞþ;ac ðzAnÞÀ; b ð Þ þ ln Cat An D /ÀÁD /ÀÀ ðÞzCat þ jjzAn F cCatðzCatÞþ; bcAnðzAnÞÀ; a zCat a;b ðzCatÞþ ðzAnÞÀ ¼ zCat a;b ðz Þþ Catm Anm Cat Cat Cat An RT fCatðzCatÞþ;afAnðzAnÞÀ; b RT a ðzCatÞþ;a : þ ln Cat : ð35Þ þ ln ðÞzCat þ jjzAn F fCatðzCatÞþ; bfAnðzAnÞÀ; a F aCatðzCatÞþ; b

ðzCatÞþ ðzAnÞÀ D /ÀÁD /ÀÀ From the stoichiometry of the salt Catm Anm it jjzAn a;b ðzCatÞþ ðzAnÞÀ ¼ jjzAn a;b ðz ÞÀ Cat An Catm Anm An An Cat An follows that in the bulk of the two phases the conditions RT aAnðzAnÞÀ; a m m : Cat Cat À ln ð36Þ c ðzCatÞþ; b ¼ c ðzAnÞÀ; b and c ðzCatÞþ; a ¼ c ðzAnÞÀ; a Cat mAn An Cat mAn An F a ðzAnÞÀ; b An hold true. With these relations follows Now, Eqs. (35) and (36) can be summed up:

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D /ÀÁ Inserting Eqs. (45) and (46) in Eq. (44) gives a;b ðzCatÞþ ðzAnÞÀ Catm Anm Cat An z F ÀÀ ÀÀ ÀÁm Cat D /ÀÀ D / D / ln K ðzCatÞþ ðzAnÞÀ ¼ Cat a;b ðz Þþ zCat a;b ðz Þþ þ jjzAn a;b ðz ÞÀ p Catm Anm ; T;p Cat Cat ¼ Cat Cat An An Cat An RT ðÞzCat þ jjzAn zAnF ÀÀ m þ m Da;b/ : Cat ð41Þ An AnðzAnÞÀ c ðzAnÞÀ; ac ðzAnÞÀ; b RT RT mAn An An þ ln m Cat 47 ðÞzCat þ jjzAn F c ðzAnÞÀ; bc ðzAnÞÀ; a ð Þ mAn An An

RT fCatðzCatÞþ;afAnðzAnÞÀ; b Since m ¼ jjz and m ¼ z , and z in the second þ ln : Cat An An Cat An ðÞzCat þ jjzAn F fCatðzCatÞþ; bfAnðzAnÞÀ; a term on the right side has a negative sign, it finally follows ÀÁzCatjjzAn F D /ÀÀ D /ÀÀ ln K ðzCatÞþ ðzAnÞÀ ¼ a;b ðz Þþ À a;b ðz ÞÀ Since the second term on the right side of Eq. (41) p Catm Anm ; T;p Cat Cat An An Cat An RT equals zero, the final equation is ÀÁzCatjjzAn F D /ÀÀ D /ÀÀ : log K ðzCatÞþ ðzAnÞÀ ¼ a;b ðz Þþ À a;b ðz ÞÀ ÀÁ p Catm Anm ; T;p : Cat Cat An An D / Cat An 2 303RT a;b ðzCatÞþ ðzAnÞÀ Catm Anm Cat An ð48Þ ÀÀ ÀÀ z Da;b/ þ jjz Da;b/ Cat CatðzCatÞþ An AnðzAnÞÀ ¼ ð42Þ It is interesting to remember that Eq. (48) has the same ðÞzCat þ jjzAn structure as that for calculating the equilibrium constant of RT fCatðzCatÞþ;afAnðzAnÞÀ; b a redox reaction m Red m Ox m Ox m Red with þ ln : 1 1þ 2 2 1 1þ 2 2 À ðÞzCat þ jjzAn F fCatðzCatÞþ; bfAnðzAnÞÀ; a the two redox equilibria Ox1 þ n1e Red1 and À Ox2þn2e Red2, where the relation m1n1 ¼ m2n2 ¼ z Equation (40) can be also applied to the solubility of a holds, and the equilibrium constant is salt, when this is considered as the result of distributing the zF ÀÀ ÀÀ F ln K ¼ E = À E = or log K ¼ z : cations and anions between the solid salt phase and the RT Ox2 Red2 Ox1 Red1 2 303RT saturated solution: see Sect. “Distribution of ions between ÀÀ ÀÀ E = À E = [6]. These equations are similar the solid salt phase and the salt saturated solution”. Ox2 Red2 Ox1 Red1 For very diluted systems, the activity coefficients because in both cases two interdependent equilibria deter- approach unity, and then the last term in Eq. (42) also mine the overall equilibrium. þ À vanishes, and the overall Galvani potential difference In the case of a uni-univalent salt Cat An , it follows D / D /ÀÁ for a;b ðÞCatþAnÀ a;b ðzCatÞþ ðzAnÞÀ can be calculated according to Catm Anm Cat An ÀÀ ÀÀ Da;b/ þ þ Da;b/ À D /ÀÀ D /ÀÀ D / Cat An zCat a;b z þ jjzAn a;b z a;b þ À ¼ ð49Þ D /ÀÁCatð CatÞþ Anð AnÞÀ : ðÞCat An 2 a;b ðzCatÞþ ðzAnÞÀ ¼ Catm Anm Cat An ðÞzCat þ jjzAn aCatþ; baAnÀ;b and for KpðÞ CatþAnÀ ; T;p ¼ ð43Þ aCatþ;aaAnÀ;a To calculate the partition constant of the salt F ÀÀ ÀÀ ln K þ À ; ; ¼ Da;b/ þ À Da;b/ À ÀÁ pðÞ Cat An T p RT Cat An K ðzCatÞþ ðzAnÞÀ , the partition constants of cations and p Catm Anm ; T;p Cat An F ÀÀ ÀÀ log K þ À ; ; ¼ Da;b/ þ À Da;b/ À : anions must be used in Eq. 16: pðÞ Cat An T p 2:303RT Cat An mCat mAn a z a z ð50Þ ÀÁCatð CatÞþ; b Anð AnÞÀ;b K ðzCatÞþ ðzAnÞÀ ¼ m m p Catm Anm ; T;p a Cat a An Cat An CatðzCatÞþ;a AnðzAnÞÀ;a Figures 2 and 3 illustrate how a lipophilic anion, here m m Cat An ; TPBÀ can affect the partition of a hydrophilic cation, here ¼ KpðCatðzCatÞþÞ; T;p KpðAnðzAnÞÀÞ; T;p Kþ; whereas KþClÀ has a partition constant þ À ÀÁ þ : m log KpðK ClÀÞ;T;p ¼À9 5, the partition constant of K TPB ln K ðzCatÞþ ðzAnÞÀ ¼ Cat ln K ðzCatÞþ ; ; p Catm Anm ; T;p pðCat Þ T p Cat An : ð44Þ is log KpðKþTPBÀÞ;T;p ¼ 2 3, i.e. the potassium concentration m ln K ðz ÞÀ : þ An pðAn An Þ; T;p in nitrobenzene is much larger when KþTPBÀ is parti- þ À For the cations, we can formulate Eq. (29) according to tioned than in the case of K Cl . Figure 4 illustrates how the chemical affinities of the zCatF ÀÀ ln K ðz Þþ ¼ Da;b/ ð45Þ anions and cations for the two phases conflict with the pðCat Cat Þ; T;p CatðzCatÞþ RT necessity of electroneutrality in each phase. The result is a and for the anions compromise, a trade-off reflecting the (mostly) different affinities of the anions and cations, so that electroneutrality zAnF D /ÀÀ : ln K ðzAnÞÀ ¼ a;b z ð46Þ in the bulk phases is maintained. One may say that the pðAn Þ; T;p RT Anð AnÞÀ cations partition in such way that they—to some extent

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Fig. 2 a Standard Galvani potentials of ion transfer of Kþ, TPB K Cl ClÀ and TPBÀ 0.372 0.235 0 0.325 w,nb ion (tetraphenylborate), b standard (A) Galvani potentials of salt partition of KþClÀ and KþTPBÀ, and (see Eq. 43) c log partition constants of KþClÀ and KþTPBÀ, for the system water–nitrobenzene 0.304 0 0.045 w,nb (Cat An ) (B) KTPB KCl

K log 9.5 0 2.3 p (Cat An ), Tp , (C)

KCl KTPB

(A) Phase Phase Water

(B) DCM Na pic Hpic TBA pic Phase Phase

K Cl TPB Fig. 5 Partition of sodium picrate (NaþpicÀ), picric acid (HþpicÀ) and tetrabutylammonioum picrate (TBAþpicÀ) between water and a a Fig. 3 Illustration of the partition of KCl between water (phase ) dichloromethane (DCM) and nitrobenzene (phase b), b KTPB between water and nitrobenzene interface (if the afﬁnities are not equal, as in the case of the example TPA?TPB−) and (ii) to different concentrations of the salt in both phases (if the afﬁnities are not identical). The tuning of partition constants of salts by choosing

Dear Plus, but I do appropriate combinations of cations and anions is illus- Come over, beloved not like your trated in Fig. 5: picric acid (2,4,6-trinitrophenol) is a rel- Minus, we need your phase! charge! atively strong acid in aqueous solution (pKa around 0.4). Although the picrate anion (2,4,6-trinitrophenolate) is rather hydrophobic, the partition of sodium picrate Phase α Phase β (NaþpicÀ) between water and dichloromethane (DCM) is such that the salt almost completely stays in water. Fig. 4 Cartoon illustrating how the different affinities of cations and aðNaþpicÀÞ;DCM The partition constant is Kp NaþpicÀ ;w DCM ¼ anions for the two phases conflict with the condition of electroneu- ð Þ À aðNaþpicÀÞ;water trality in the bulk phases ¼ 2:5 Â 10À5 [7]. This is illustrated by Fig. 5, where the aqueous phase is strongly yellow, the colour of picrate —“respect” the affinity of the anions, and vice versa. This ions, whereas the DCM phase is colourless. The same compromise leads (i) to a potential difference at the result is obtained for the partition of picric acid (see Fig. 5)

123 ChemTexts (2020) 6:17 Page 9 of 12 17 because the transfer of protons to the DCM phase is the anion and cation. This is reasonable in the case of very strongly disfavoured. However, when the sodium ions are bulky organic ions, e.g. tetraphenylarsonium (TBAþ) substituted by the rather hydrophobic tetrabutylammonium cations and tetraphenylborate (TPBÀ) anions. The latter (TBAþ) cation, the partition strongly favours the presence salt was ﬁrst proposed by Grunwald et al. [5]. On the basis of picrate in DCM. Figure 5 shows for the partition of of this extrathermodynamic assumption, it is possible to

þ À À TBA pic a yellow DCM phase and a colourless aqueous build up a system of consistent KpðCatþÞ;T;p and KpðAn Þ;T;p phase. In fact, the partition of picrates is not as simple as data from analysis of the salt concentrations in the two presented here: in DCM, the formation of ion pairs has to phases, when different salt combinations are used, among be taken into account. The formation constant of the ion which are salts with TBAþ cations and TPBÀ anions. pairs ½TBAþpicÀ in water-saturated DCM is rather large: The experimental determination of individual ion par- : : log KIP½TBAþpicÀ;DMC ¼ 3 82 Æ 0 06. The formation con- tition constants can also be achieved with the help of stant of ½TBAþpicÀ in water is comparably small electrochemistry. Dating back to ﬁrst experiments by : : (log KIP½TBAþpicÀ;w ¼ 1 033 Æ 0 008). The partition con- Nernst and Riesenfeld [9], now voltammetric techniques stant of TBAþpicÀ for water–DCM is [10] are available with which the individual transfer of ions : : across liquid–liquid interfaces can be measured, and from log Kp TBAþpicÀ ;w DCM ¼ 5 52 Æ 0 03 (all data from [8]), ð Þ À the characteristic potentials of the voltammograms, the which is clearly a result of the hydrophobic nature of standard potentials of the individual ions can be estimated, TBAþ, as well as of the favourable ion pair formation in of course always relying on an extrathermodynamic DCM. assumption, like the mentioned Grunwald assumption [11–14]. These measurements need a four-electrode Experimental considerations concerning potentiostat, and they are rather limited with respect to the the partition of a salt between two solvent systems. There exists another electrochemical immiscible solution phases approach, where an electrochemically generated redox probe drives an ion transfer across a liquid–liquid interface [15–17]. For this, a three-electrode potentiostat is sufﬁcient The partition constants of neutral molecules are experi- and the number of solvent systems is considerably larger. A mentally accessible without any principal constraints. It is complete discussion of the experimental approaches to just necessary to have analytical techniques at hand, measure the individual transfer of ions (i.e. ion partition) allowing the determination of the concentration of the cannot be given here, and the interested reader should partitioned compound in both phases. For salts, this can be consult the mentioned sources. done in the same way; however, the individual partition constants of the anions and cations are not so easily accessible. Since the individual partition constants are a Distribution of ions between the solid salt measure of the interaction of the individual ions with each phase and the salt saturated solution of the two solvents, e.g. with water and the organic solvent, The solubility of a sparingly soluble salt can be considered it is important to know these data. They provide informa- as the result of distributing the cations and anion between tion about the hydrophilicity/hydrophobicity and the solid salt phase (s) and the saturated solution (sol): oleophilicity/oleophobicity of the ions! þ À þ À Let us consider a uni-univalent salt, for which Cats +Ans Catsol +Ansol ðEquilibrium IIIÞ aCatþ; baAnÀ; b cCatþ; bcAnÀ; b K þ À ; ; ¼ is experimentally pðCat An Þ T p aCatþ; aaAnÀ; a cCatþ; acAnÀ; a accessible by measuring the concentrations c þ À ; a and þ À ðCat An Þ Of course, Cats and Ans build up the solid phase þ À þ À cðCat An Þ; b. According to Eq. (44) the relation for a uni- fgCat An s. Equilibrium III has the following equilibrium univalent salt is constant: aCatþ;solaAnÀ;sol þ þ À : K þ À ¼ : ð51Þ ln KpðCat AnÀÞ;T;p ¼ ln KpðCat Þ;T;p þ ln KpðAn Þ;T;p pðCat An Þ;T;p aCatþ;saAnÀ;s

When KpðCatþAnÀÞ;T;p is determined by measuring the salt The exact quantity of the activities of the ions in the concentrations in both phases, the partition constant of solid are not known; however, it is clear that they are either the cation must be known to calculate that of the constant. The solubility product KspðCatþAnÀÞ;T;p of the salt is anions, or vice versa. This is a serious problem which can À ; be solved only by making for some salts the extrathermo- KspðCatþAnÀÞ;T;p ¼ aCatþ;solaAn ;sol ð52Þ dynamic (i.e. not strictly based on thermodynamics) and this is the relation between the solubility product and assumptions that the partition coefﬁcients are identical for the partition constant: 123 17 Page 10 of 12 ChemTexts (2020) 6:17

þ þ þ À : It can be also shown that D ; / þ À is zero, when KspðCat AnÀÞ;T;p ¼ KpðCat AnÀÞ;T;paCat ;saAn ;s ð53Þ s sol ðÞCat An the following relations are considered: Defining the two For the cations, Eq. (26) can be written as follows: aCatþ;sol aAnÀ;sol partition constants Kp;Catþ ¼ and Kp;AnÀ ¼ and aCatþ;s aAnÀ;s ÀÀ RT aCatþ;s D ; / þ D ; / taking into account that the activities in the solid are s sol Cat ¼ s sol Catþ þ ln F aCatþ;sol a þ pconstantffiffiffiffiffiffiffi (although not necessarily equal)pffiffiffiffiffiffiffi gives Cat ;sol ¼ D /ÀÀ RT RT : ¼ s;sol þ þ ln a þ; À ln a þ; K ¼ K ; þ a þ; and a À; ¼ K ¼ K ; À a À; . Cat F Cat s F Cat sol sp p Cat Cat s An sol sp p An An s Using the relation between equilibrium constants and the ð54Þ ÀÁ formal Galvani potentials yields RT ln K ; þ a þ; ¼ pffiffiffiffiffiffiffi ÀÁp Cat Cat s The first and second terms on the right side of the ÀÀ 0 D ; / ; À À; F s sol c;Catþ ¼ RT ln Ksp and RT ln Kp An aAn s ¼ equation can be combined to the formal potential of the 0 pffiffiffiffiffiffiffi 0 ÀÀ ÀÀ ÀÀ RT ÀFDs;sol/ ; À ¼ RT ln Ksp. This means that the relation D ; / D ; / þ c An cation transfer s sol c;Catþ ¼ s sol Catþ þ F ln aCat ;s (the ÀÀ 0 ÀÀ 0 0 D / D / ÀÀ F s;sol þ ¼ÀF s;sol ; À holds true, i.e. D ; / c;Cat c An subscript c of s sol c;Catþ stands for ‘conditional’ because ÀÀ 0 ÀÀ 0 D ; / D ; / À formal potentials relate to fixed conditions), and thus: s sol c;Catþ þ s sol c;An ¼ 0. Hence, in Eq. (59) both terms of the right side are zero and thus Ds;sol/ þ À is D / D /ÀÀ 0 RT : ðÞCat An s;sol Catþ ¼ s;sol ; þ À ln aCatþ;sol ð55Þ c Cat F also zero. The partition constants Kp;Catþ and Kp;AnÀ are defined for a zero potential difference, as was done in the For the anions, Eq. (26)is case of the partition constants of ions between two liquid D / D /ÀÀ RT aAnÀ;s phases; see discussion following Eq. (26). Hence, it is not s;sol AnÀ ¼ s;sol AnÀ À ln F aAnÀ;sol surprising that in the end a zero potential difference results, ÀÀ RT RT but it shows that the entire description is consistent. D / À À : ¼ s;sol þ À ln aAn ;s þ ln aAn ;sol Cat F F If the concentration of the cations or anions in the ð56Þ solution is changed by addition of a salt containing the same cation or anion (e.g. by addition of silver nitrate or The formal potential of the anion transfer is ÀÀ 0 ÀÀ sodium chloride to the saturated solution of silver chlo- D / D / RT À s;sol c;AnÀ ¼ s;sol AnÀ À F ln aAn ;s. With that follows ride), a potential difference at the salt|solution interface ÀÀ 0 ÀÀ 0 0 RT Ds;sol/ þDs;sol/ ÀÀ c;Catþ c;AnÀ D ; / À ¼ D ; / ; À þ ln a À; : ð57Þ builds up. Since the term in Eq. (59)is s sol An s sol c An F An sol 2 zero, but the anion and cation concentrations are no longer Like at the liquid|liquid interface, also here only one equal, this equation can be used to calculate the potential Galvani potential difference can be established: difference at the salt|solution interface: D / D / D / s;sol Catþ ¼ s;sol AnÀ ¼ s;sol ðÞCatþAnÀ . Summing up D / RT cAnÀ;sol RT fAnÀ;sol Eqs. (55) and (57), and substituting activities with the s;sol ðÞCatþAnÀ ¼ ln þ ln ð60Þ 2F c þ; 2F f þ; products of activity coefficients and concentrations allows Cat sol Cat sol D / one to calculate s;sol ðÞCatþAnÀ : When anion addition is considered, the concentration ÀÀ 0 ÀÀ 0 cCatþ;sol can be substituted by D ; / þ þ D ; / À D / s sol c;Cat s sol c;An s;sol ðÞCatþAnÀ ¼ Ksp : 2 cCatþ;sol ¼ ð61Þ ð58Þ c À; RT f À; RT c À; An sol þ ln An sol þ ln An sol ; 2F fCatþ;sol 2F cCatþ;sol Inserting Eq. (61) in Eq. (60) gives ÀÀ 0 ÀÀ 0 2 D ; / D ; / À RT c À; RT f À; s sol c;Catþ þ s sol c;An An sol An sol D / Ds;sol/ þ À ¼ ln þ ln : ð62Þ s;sol CatþAnÀ ðÞCat An ðÞ 2 2F Ksp 2F fCatþ;sol RT cAnÀ;sol : þ ln ð59Þ Equation (62) nicely shows that the potential difference 2F c þ Cat ;sol D / / at the salt|solution interface s;sol ðÞCatþAnÀ ¼ sol;ðÞCatþAnÀ For the saturated solution of the pure sparingly soluble / À s;ðÞCatþAnÀ becomes positive. In the case of addition of salt, the concentration of the cations equals the concen- the cations to the solution (e.g. addition of silver nitrate to tration of the anions (no excess of one ion sort) in the the saturated solution of silver chloride), the potential solution. difference is D / In the equilibrium, s;sol ðÞCatþAnÀ must be zero, because RT Ksp RT fAnÀ;sol D ; / þ À ; in both phases an excess of cations or anions does not exist. s sol ðÞCat An ¼ ln 2 þ ln ð63Þ þ 2F cCatþ;sol 2F fCat ;sol

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precipitate, which serves as indication of the equivalence point. In Fig. 6 deviations between concentrations and activities are not taken into account. At this point, it is interesting to note the similarities of the solid salt|solution interface with the processes under- lying the function of the glass electrode: whereas in the case of the silver halides, the cations and anions are cap- able of partitioning, in the case of the glass electrode, only the cations, i.e. the protons, can partition and the anions, i. e. the “surface silicate groups”, are unable to partition because they are immobile. This results in the well-known pH response of the Galvani potential difference at glass|- solution interfaces [21–23].

Conclusions

In this lecture text the partition of salts between two immiscible solution phases is presented, and the equation describing the Galvani potential difference across the liquid|liquid interface is derived. The meaning of standard potentials of transfer of single ions from one solution phase to the other is discussed on the basis of an extrathermodynamic assumption. Further, some experimental details of the determination of the standard potentials of ions are mentioned. Finally, the equilibrium of a solid salt phase with its saturated solution phase is presented on the basis of ion partition equilibria, and the resulting surface charge of Fig. 6 Top: dependence of the potential difference at the salt|solution D / / / the solid phase is discussed in connection with adsorption interface s;sol ðÞCatþAnÀ ¼ sol;ðÞCatþAnÀ À s;ðÞCatþAnÀ on the concentration of added anions or cations in the solution. Bottom: same indicators. potential difference as function of the logarithm of concentration In this text, the structure of the double layers at the interfaces could not be considered.

Acknowledgements Open Access funding provided byProjekt DEAL. i.e. the potential difference becomes negative. Figure 6 We are very thankful for many helpful suggestions provided by PD. illustrates the dependencies (62) and (63) for the case of the Dr. Gudrun Scholz, Nico Heise, Javier Roman Silva, and Karup- pasamy Dharmaraj. silver halides AgCl, AgBr and AgI. Since the interfacial D / Open Access potential difference is deﬁned as s;sol ðÞCatþAnÀ ¼ This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, / ; þ À À / ; þ À , a positive sign of sol ðÞCat An s ðÞCat An adaptation, distribution and reproduction in any medium or format, as D / s;sol ðÞCatþAnÀ means a negatively charged surface of the long as you give appropriate credit to the original author(s) and the silver halides with respect to solution, and vice versa. In the source, provide a link to the Creative Commons licence, and indicate case of titrations of halides with silver ions, the sign of if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless D / s;sol ðÞCatþAnÀ changes at the equivalence point. Before the indicated otherwise in a credit line to the material. If material is not D / included in the article’s Creative Commons licence and your intended equivalence point s;sol ðÞCatþAnÀ has a positive sign, i.e. use is not permitted by statutory regulation or exceeds the permitted the solid AgX is negatively charged. After the equivalence use, you will need to obtain permission directly from the copyright D / point, when silver ions are in excess, s;sol ðÞCatþAnÀ has a holder. To view a copy of this licence, visit http://creativecommons. negative sign, i.e. the surface of the silver halides is posi- org/licenses/by/4.0/. tively charged. This is used in the Fajans methods [18–20] of indication using adsorption indicators, where, as an example, ﬂuorescein anions are used as they are adsorbed References on the positively charged surface of silver halides. The 1. Nernst W (1891) Verteilung eines Stoffes zwischen zwei adsorption changes the colour of the silver halide Lo¨sungsmitteln und zwischen Lo¨sungsmittel und Dampfraum.

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