A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 20
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Mobility usually carries dimensions of cm 2 V" 1 s" 1 (i.e., cm/s perV/cm). When a field of strength % is applied to an ion, it will accelerate under the forceimposed by the field until the frictional drag exactly counterbalances the electric force.Then, the ion continues its motion at that terminal velocity.
This balance is represented inFigure 2.3.4.The magnitude of the force exerted by the field is \z-\ e%, where e is the electroniccharge. The frictional drag can be approximated from the Stokes law as 6ттг]ги, where rjis the viscosity of the medium, r is the radius of the ion, and v is the velocity. When theterminal velocity is reached, we have by equation and rearrangement,The proportionality factor relating an individual ionic conductivity to charge, mobility,and concentration turns out to be the Faraday constant; thus(2.3.10)Direction of movementDrag forceч—УElectric forceFigure 2.3.4 Forces on a chargedparticle moving in solution under theinfluence of an electric field.
The forcesbalance at the terminal velocity.2.3 Liquid Junction Potentials67The transference number for species / is merely the contribution to conductivity made bythat species divided by the total conductivity:(2.3.11)For solutions of simple, pure electrolytes (i.e., one positive and one negative ionicspecies), such as KC1, CaCl2, and HNO 3 , conductance is often quantified in terms of theequivalent conductivity, Л, which is defined by(2.3.12)where C e q is the concentration of positive (or negative) charges. Thus, Л expresses theconductivity per unit concentration of charge. Since C\z\ = C e q for either ionic species inthese systems, one finds from (2.3.10) and (2.3.12) thatЛ = F(u+ + u-)(2.3.13)where u+ refers to the cation and u- to the anion.
This relation suggests that Л could beregarded as the sum of individual equivalent ionic conductivities,Л = Л+ + A_(2.3.14)Ai = Fu{(2.3.15)hence we findIn these simple solutions, then, the transference number tx is given byл(2.3.16)or, alternatively,(2.3.17)Transference numbers can be measured by several approaches (31, 32), and numerousdata for pure solutions appear in the literature. Frequently, transference numbers are measured by noting concentration changes caused by electrolysis, as in the experiment shownin Figure 2.3.3 (see Problem 2.11). Table 2.3.1 displays a few values for aqueous solutionsat 25°C.
From results of this sort, one can evaluate the individual ionic conductivities, Aj.Both Aj and t-x depend on the concentration of the pure electrolyte, because interactions between ions tend to alter the mobilities (31-33). Lists of A values, like Table 2.3.2, usuallygive figures for AOi, which are obtained by extrapolation to infinite dilution. In the absenceof measured transference numbers, it is convenient to use these to estimate t\ for pure solutions by (2.3.16), or for mixed electrolytes by the following equivalent to (2.3.11),(2.3.18)In addition to the liquid electrolytes that we have been considering, solid electrolytes, such as sodium /3-alumina, the silver halides, and polymers like polyethylene68Chapter 2.
Potentials and Thermodynamics of CellsTABLE 2.3.1 Cation Transference Numbersafor Aqueous Solutions at 25°CbConcentration, CeqElectrolyte0.010.050.10.2HC1NaClKC1NH4C1KNO3Na 2 SO 4K 2 SO 40.82510.39180.49020.49070.50840.38480.48290.82920.38760.48990.49050.50930.38290.48700.83140.38540.48980.49070.51030.38280.48900.83370.38210.48940.49110.51200.38280.4910aFrom D. A. Maclnnes, "The Principles of Electrochemistry," Dover, New York, 1961, p. 85 and referencescited therein.^Moles of positive (or negative) charge per liter.oxide/LiClO4 (34, 35), are sometimes used in electrochemical cells. In these materials,ions move under the influence of an electric field, even in the absence of solvent. For example, the conductivity of a single crystal of sodium /3-alumina at room temperature is0.035 S/cm, a value similar to that of aqueous solutions.
Solid electrolytes are technologically important in the fabrication of batteries and electrochemical devices. In some ofthese materials (e.g., a-Ag2S and AgBr), and unlike essentially all liquid electrolytes,TABLE 2.3.2 Ionic Properties at InfiniteDilution in Aqueous Solutions at 25°CIonA0,cm2n"1equiv"lflи, cm 2 sec" 1 V" 1 *H+K+Na +Li +NH^349.8273.5250.1138.6973.459.5019876.3478.476.8571.4440.968.079.844.48101.0110.53.6257.6195.1934.0107.616.1662.057.9128.137.967.4044.247.058.274.6101.0471.145ka2+OH~СГBr"I"NO3OAc"СЮ4kstitHCO3|Fe(CN)^|Fe(CN)^aX 10" 3X 10" 4X 10" 4X 10" 4X 10~4X 10~4X 10" 3X 10~4X 10" 4X 10" 4X 10" 4X 10" 4X 10" 4X 10" 4X 10" 4X 10~3X 10" 3From D.
A. Maclnnes, "The Principles of Electrochemistry,"Dover, New York, 1961, p. 342^Calculated from AQ.2.3 Liquid Junction Potentials69WWWWWWWWWWWWVWNA-JFigure 2.3.5 Experimental system fordemonstrating reversible flow of charge through acell with a liquid junction.there is electronic conductivity as well as ionic conductivity. The relative contribution ofelectronic conduction through the solid electrolyte can be found by applying a potential toa cell that is too small to drive electrochemical reactions and noting the magnitude of the(nonfaradaic) current. Alternatively, an electrolysis can be carried out and the faradaiccontribution determined separately (see Problem 2.12).2.3.4Calculation of Liquid Junction PotentialsImagine the concentration cell (2.3.3) connected to a power supply as shown in Figure2.3.5.
The voltage from the supply opposes that from the cell, and one finds experimentally that it is possible to oppose the cell voltage exactly, so that no current flows throughthe galvanometer, G. If the magnitude of the opposing voltage is reduced very slightly,the cell operates spontaneously as described above, and electrons flow from Pt to Pt' inthe external circuit. The process occurring at the liquid junction is the passage of anequivalent negative charge from right to left. If the opposing voltage is increased from thenull point, the entire process reverses, including charge transfer through the interface between the electrolytes. The fact that an infinitesimal change in the driving force can reverse the direction of charge passage implies that the electrochemical free energy changefor the whole process is zero.These events can be divided into those involving the chemical transformations at themetal-solution interfaces:(2.3.19)Н + (/3) + e(Pt') ^± ^Н 2(2.3.20)and that effecting charge transport at the liquid junction depicted in Figure 2.3.6:t+H+(a) + Г-СГ08) ^± t+ H + 08) + /- С Г (a)(2.3.21)Note that (2.3.19) and (2.3.20) are at strict equilibrium under the null-current condition;hence the electrochemical free energy change for each of them individually is zero.
Ofcourse, this is also true for their sum:H + 08) + e(Pt')H + (a)(2.3.22)(a2)LCI-Figure 2.3.6 Reversible charge transfer through the liquid junction inFigure 2.3.5.70Chapter 2. Potentials and Thermodynamics of Cellswhich describes the chemical change in the system.
The sum of this equation and thecharge transport relation, (2.3.21), describes the overall cell operation. However, since wehave just learned that the electrochemical free energy changes for both the overall processand (2.3.22) are zero, we must conclude that the electrochemical free energy change for(2.3.21) is also zero. In other words, charge transport across the junction occurs in such away that the electrochemical free energy change vanishes, even though it cannot be considered as a process at equilibrium. This important conclusion permits an approach to thecalculation of junction potentials.Let us focus first on the net chemical reaction, (2.3.22). Since the electrochemicalfree energy change is zero,^H+ + Jg = /Zg+ + jS*FE = F(0 P t ' - <£pt) = £&+ - ДЙ+Е = ^1п^(2.3.23)(2.3.24)+ (фР-фа)(2.3.25)The first component of E in (2.3.25) is merely the Nernst relation for the reversible chemical change, and ф@ — фа is the liquid junction potential.
In general, for a chemically reversible system under null current conditions,^cell = ^Nernst + Щ(2.3.26)hence the junction potential is always an additive perturbation onto the nernstian response.To evaluate E}, we consider (2.3.21), for whicht+JL^ + t_ J&- = t+^H+ + t_ /Zg r(2.3.27)Thus,- - lib-) = 01J Lft c\-(2.3.28)P -фа)\= 0 (2.3.29)JActivity coefficients for single ions cannot be measured with thermodynamic rigor (30,36, 37-38); hence they are usually equated to a measurable mean ionic activity coefficient.Under this procedure, ag+ = ogj- = ax and a^+ = a^x- = a2. Since t+ + t- = 1, wehaveа- ф ) = (t+ - t_) Щ- In ^(2.3.30)for a type 1 junction involving 1:1 electrolytes.Consider, for example, HC1 solutions with ax = 0.01 and a2 = 0.1.
We can see fromTable 2.3.1 that t+ = 0.83 and t- = 0.17; hence at 25°C(\^ y ) - -39.1 mV/(2.3.31)For the total cell,E = 59.1 log ^ + £j = 59.1 - 39.1 = 20.0 mV(2.3.32)thus, the junction potential is a substantial component of the measured cell potential.In the derivation above, we made the implicit assumption that the transference numbers were constant throughout the system. This is a very good approximation for junctions2.3 Liquid Junction Potentials «I 71of type 1; hence (2.3.30) is not seriously compromised. For type 2 and type 3 systems, itclearly cannot be true.
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