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Home , Dynamic equilibrium, Relative permittivity

Electrochemistry course Electrolyte - reminder ACME Faculty, EHVE course Liquid or solid that conducts electricity B.Sc. Studies, II year, IV semester by means of its ions. Ions can move when they have freedom Leszek Niedzicki, PhD, DSc, Eng. of movement. That freedom can be provided by molten (ionic liquid) , specific structure of solid enabling ionic mobility or (most commonly) of ions in the solution by Fundamentals of ionics molecules (and as a result - shielding them from counter-ions and causing dissociation).

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Solvation once more Dynamic equilibrium Disturbance of solvent structure by an ion: • It is a phenomenon observed when on a large scale (e.g. billions of billions of molecules) a statistical equilibrium A – I solvation layer (directly coordinated by a cation) is observed, i.e. mean value of a given parameter is B – II and further solvation layers (attracted steady, but individual molecules often change their electrostatically by a cation and can interact with other solvent state. molecules – e.g. through the hydrogen bonds) • In practice dynamic equilibrium is defined C – solvent structure disturbed by the cation as an equilibrium of two opposite processes, which presence in the vicinity occur at the same rate (in a given conditions). In case D – original solvent structure of solvation solvent molecules are all the time C+ joining and leaving solvation layer (e.g. are knocked A B out of it). However, mean solvent molecules C in solvation layer of a given ion stays the same. D 3 4

Dynamic equilibrium Solvent • In dissociation or solvation case dynamic • Solvent in the electrolyte formation process is equilibrium forms because solvent molecules required to solvate ions (shields them against and ions are bumping on each other association or crystal formation) and dissociate compound into ions (strength of interaction with part (and at the vessel walls) all the time (due to chaotic of the compound tears it from the other part moves, vibrations, etc. ). Thus, constant exchange of the compound at the ionic bond) . of molecules in solvation layers is taking place. • The measure of how solvent is eager to interact Due to that, “puncture” of solvation layer with ions and how good its molecules are by the counter-ion can happen, if it will be shielding ions against interaction with other not full or/and counter-ion would bump ions (counter-ions) is constant (relative with proper angle and momentum. ) .

5 6 Dielectric constant (ε) Dielectric constant Dielectric constant of a material (in that case solvent/solution) can be measured with two methods: ε is dimensionless, because it is ratio of material • permittivity to ε – permittivity of free space. method – requires measurement of a 0 capacitance of a with between It means, that if certain ions attract each other its plates. Next, material to be measured is placed in vacuum with a certain force, then in a medium between those plates and the capacitance (solvent/solution) with a relative permittivity (dielectric measurement is repeated. Dielectric constant is constant) ε, they would attract each other ε times calculated from the capacitance formula:

more weakly. C = ε ε 0 A/d , where: Salt as a material with low ε value is decreasing ε – dielectric constant of a material between plates; overall dielectric constant of a solution (when ε0 – permittivity of free space; considered for next ions/ addition to that solution) . A – capacitor plates surface (surface of the cross-section of a capacitor) ; d – distance between plates. 7 8

Dielectric constant Dielectric constant material ε (at 20°C) material ε (at 20°C) • (principle of operation is similar to that of impedance spectroscopy, but works at much higher ) – vacuum 1 glyme ~7 enable dielectric constant measurements 2 diglyme ~7 with an a.c. signal. 25 PEO 5 At the THz order of magnitude relaxation DMF 36 effects of material allows to determine dielectric AN 37 SiO 3.9 constant of a material (as a value of extrapolation at 0) . 2 In practice it is possible to conduct PC 63 PE 2.3 that measurement at the lower frequencies as well. 80 PP 2.2

Dielectric constant is temperature-dependent H2SO 4 ~90 ~5 (for some materials differences are of magnitude order for 50°C EC 100 ~4 temperature change) . salts (including ionic liquids) 5-15 9 10

Solvent-ion interactions Ionic activity Increase of ionic above infinitesimal Dielectric constant is not the only way to concentration (any non-zero concentration) cause ions measure interactions of solution components, to meet and interact with each other as well as there are secondary parameters such as change their activity (theoretical) into non-ideal as donor number and acceptor number. Also one. It means, that parameters deviate from those the temperature dependence of dielectric calculated by theoretical formulae upon ionic constant is important – in case of some concentration increase. Theoretical equations describe ideal system exclusively (even those taking materials it is not changing much, others show the concentration into account are taking concentration value enormous change over only few centigrades. as it was group of individual ions not interacting with each other and as the solvent would interact with ions to a maximum extent). 11 12 Ionic activity Ionic activity Activity coefficient γ can be calculated from the In reality ionic activity depends empirical formula, where the simplest and basic on concentration, but it is not the same formula (enriched with additional elements and parameters when as concentration. Concentration instead complexity increases for use with higher and higher ) is: 2 1/2 1/2 of the ionic activity can be used for extremely log γi = -A z i I /(1+I ) diluted solutions only, where deviations where: A = 0.5091 (water at 25°C) ; I – ionic strength; from theory are negligible. In practice (for useful z – ionic charge (its electrovalence); Ionic strength can be calculated from the formula: solutions) ionic activity is used: 2 I = ½ Σ (z i ci) ai = c i γi where: c – concentration. It is the sum of all ions where a – ionic activity; c – concentration; present in the solution. For solutions of one binary salt γ – activity coefficient; I = c. 13 14

Ionic activity Ionic activity

Example for NaCl (z Na =1, z Cl =1, so I=c) . For 0.01 mol/kg concentration: 1/2 1/2 log γ = -0.5091∙0.1/(1+0.1) = -0.05091/1.1 log γNa = -A∙z Na ∙I /(1+I ) Na 1/2 1/2 γ = 10^(-0.04628) = 0.899 log γNa = -0.5091∙1∙c /(1+c ) Na Thus, deviation is higher than 10%. For 0.0001 mol/kg concentration: For 1 mol/kg concentration (much more complex log γNa = -0.5091∙0.01/(1+0.01) = -0.005091/1.01 formulae should be used for such concentration for better results):

γNa = 10^(-0.00504) = 0.988 log γNa = -0.5091∙1/(1+1) = -0.5091/2 Thus, result is quite close to 1 (1.2% deviation). γNa = 10^(-0.2546) = 0.556 Thus, deviation is higher than 44%.

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Ionic activity Phenomena in the solution

In practice ionic activity is used everywhere • Salt/acid/base dissolution when theory considers ionic concentration – pH • Dissociation (partial for weak electrolytes) calculation, determining half-cell potential • Associations formation from the Nernst formula and other similar • equations, e.g. : Acid-base equilibria • Complex formation + pH = -log(a H3O ) 0 E = E + R∙T/(z∙F) ∙ ln(a ox /a red )

17 18 of hardly soluble compounds Solubility product • + - 2 • Any equilibrium process constant is described by For small concentrations Ksp = [M ][X ] = c . the following formula: In order to simplify the formula while taking K = Π[product] / Π[substrate] into account potential interactions with other ([] means molar concentration; Π – product of all reagents of a given type) salts with the common ions, maximum • In practice, for the higher concentrations (where it solubility of hardly soluble salts is given by this has any notable effect) , [x] is x’s activity. K (solubility of soluble salts is • In case of solubility of hardly soluble salts: sp + - given in g/100g) . MnXm(s) = nM + mX the equilibrium constant formula is: Due to the form of the formula for K sp this + n - m Ksp = [M ] [X ] /[M nXm(s) ] quantity has been named the solubility Where activity of a substance in a standard state product (Ksp ). (in case of salt – solid state) is equal to 1. 19 20

Solubility product Solubility product -16 K AgI = 1.5∙10 -16 (water, at 25°C) Ksp AgI = 1.5∙10 (water, at 25°C) sp [Ag +][I -] = 1.5∙10 -16 c2 = 1.5∙10 -16 c = 1.22∙10 -8 [Ag +][I -] = 1.5∙10 -16 Maximum concentration of AgI solution in water If a solution in which we try to dissolve AgI already contains any Ag + ions (e.g. already dissolved AgNO in a given at 25°C is 1.22∙10 -8M. 3 solute), than the solubility of AgI will be lower, e.g. for a 0.01M AgNO 3 solution: -12 + 2 - -12 + - 2 -16 Ksp Ag 2CrO 4 = 4.1∙10 [Ag ] [CrO 4 ] = 4.1∙10 [Ag ]∙[I ] = (0.01+c)∙c = c + 0.01c = 1.5∙10 (2c) 2∙c = 4c 3 = 4.1∙10 -12 c = (4.1∙10 -12 /4) 1/3 We can calculate that c = 1.5∙10 -14 M, c = 1.008∙10 -4M – maximum concentration which means solubility of AgI in water that already -14 of Ag 2CrO 4 solution in water at 25°C contains 0.01M AgNO 3 is 1.5∙10 M (while solubility without any previous Ag + or I - content was 1.22∙10 -8M) .

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Solubility product Solubility product If one adds soluble salt to the concentrated solution of a hardly soluble salt with a common ion, then the Salt effect is an increase in solubility of a hardly worse soluble salt precipitates: soluble salt upon addition of a soluble salt -6 -10 Ksp CuCl = 10 Ksp AgCl = 1.6∙10 without common ions with the first one. Concentrated solution of AgCl has a concentration The effect takes place because addition of large -5 of 1.26∙10 M. amount of ions increases ionic strength Upon addition of CuCl to obtain 10 -4M concentration (of CuCl) AgCl precipitates: of a solution, thus decreasing ionic activity [Ag +][Cl -] = 1.6∙10 -10 x∙(10 -4+1.26∙10 -5) = 1.6∙10 -10 coefficients in a solution. Thanks to that, solubility x∙0.0001126 = 1.6∙10 -10 x=1.421∙10 -6. product is lower (as solubility product is a product Precipitate weight will be approximately of activities, not concentrations!) and more of a hardly 1.26∙10 -5M - 1.421∙10 -6M = 1.118∙10 -5M AgCl. soluble salt can be dissolved. (multiplied by amount of a solution) 23 24 Acids and bases Solubility product cont’d dissociation in water • Apart from the salt effect, also addition of acid + • Acids dissociate into the proton (H 3O ions) effects in higher solubility of hardly + - (in H3O form, as proton immediately soluble bases (dissociating into OH ions). It is due combines with water molecule) and to the neutralization reaction, thanks to which the acid radical (anion). If acid has more than one OH - ions amount decreases and more of the base hydrogen atoms that can dissociate as protons, can dissolve (however, it works only with the dissociated ions, then it happens in stages (protons dissociate one at a so it works as much as high dissociation degree of a compound is) . stage, with each stage having its own equilibrium). • Hydrolysis of a compound also increases • Bases dissociate into the hydroxide anion (OH -) solubility, although the resulting solution have and the cation (usually metal cation). different compound dissolved in it (as hydrolysis (similarly to acids, if base has more than one OH- group, then they is a reaction with water) dissociate in stages - one group at a time) .

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Hydrolysis Weak electrolytes dissociation

Dissociation degree: α = c ion /c substance • If ion is strongly interacting with water Equilibrium constant of CA = C + + A - dissociation is molecules of solvating layer and that described by a formula: K = [C +][A -]/[KA] attraction is stronger than strength of O-H As amount of ions from the dissociation degree is bond in water, then the hydrolysis takes place. by definition α∙c, then amount of undissociated • Cations draw OH - part to them (cation draws substance is (1-α)∙c. Thus: the negative charge stronger than proton can do it in this K = α∙c∙α∙c/((1-α)∙c) = α2∙c/(1-α) situation). In case of weak electrolytes – they do not dissociate • Anions draws H + to them (pulling it from the rest of fully and dissociation degree is very low, even much the water molecule – OH -). below 1%. Thus for very low dissociation degrees is: K ≈ α2∙c.

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Weak electrolytes dissociation Weak electrolytes dissociation Dissociation of acids and bases goes similarly • + - to that of salts, although in case of weak acids Ka = [H 3O ][A ]/[HA] and weak bases it runs according to the following • K = [BH +][OH -]/[B] equations: b - + AH + H 2O = A + H 3O In practice Ka and Kb are used to describe and + - B + H 2O = BH + OH compare acids and bases strength (respectively) . It means that theoretically, when calculating acidity In order to simplify the notation of such a low constant (Ka) and basicity constant (Kb), one should take into account water concentration as well. values (often high negative powers of 10) , it is Despite that, as weak acids and weak bases are transformed with p operator (-log(x)) and noted as: dissociating to a small degree, ionic concentrations pK and pK . are so small in comparison to water concentration, a b -5 that there is no need to take water into account (its E.g. Ka = 1.73∙10 is pK a = 4.76 percent change is negligible when compared to that of ions). 29 30