A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 41
Текст из файла (страница 41)
127.146Chapter 4. Mass Transfer by Migration and DiffusionP 4.4 DIFFUSIONAs we have just seen, it is possible to restrict mass transfer of an electroactive speciesnear the electrode to the diffusive mode by using a supporting electrolyte and operating ina quiescent solution. Most electrochemical methods are built on the assumption that suchconditions prevail; thus diffusion is a process of central importance.
It is appropriate thatwe now take a closer look at the phenomenon of diffusion and the mathematical modelsdescribing it (16-19).4.4.1 A Microscopic View—Discontinuous Source ModelDiffusion, which normally leads to the homogenization of a mixture, occurs by a "randomwalk" process. A simple picture can be obtained by considering a one-dimensional random walk. Consider a molecule constrained to a linear path and, buffeted by solvent molecules undergoing Brownian motion, moving in steps of length, /, with one step beingmade per unit time, r.
We can ask, "Where will the molecule be after a time, ft" We cananswer only by giving the probability that the molecule will be found at different locations. Equivalently, we can envision a large number of molecules concentrated in a line att = 0 and ask what the distribution of molecules will be at time t. This is sometimes calledthe "drunken sailor problem," where we envision a very drunk sailor emerging from a bar(Figure 4.4.1) and staggering randomly left and right (with a stagger-step size, /, one stepevery т seconds). What is the probability that the sailor will get down the street a certaindistance after a certain time tlIn a random walk, all paths that can be traversed in any elapsed period are equallylikely; hence the probability that the molecule has arrived at any particular point is simplythe number of paths leading to that point divided by the total of possible paths to all accessible points.
This idea is developed in Figure 4.4.2. At time т, it is equally likely thatthe molecule is at +/ and -/; and at time 2т, the relative probabilities of being at +2/, 0,and —2/, are 1, 2, and 1, respectively.The probability, P(m, r), that the molecule is at a given location after m time units (m= t/т) is given by the binomial coefficient(4A1)^h$"where the set of locations is defined by x = (—m + 2r)/, with r = 0, 1 , .
. . m. The meansquare displacement of the molecule, Д 1 , can be calculated by summing the squares of thedisplacements and dividing by the total number of possibilities (2 m ). The squares of thedisplacements are used, just as when one obtains the standard deviation in statistics, because movement is possible in both the positive and negative directions, and the sum ofthe displacements is always zero.
This procedure is shown in Table 4.4.1. In general, Д 2 isgiven by~~-4/-3/-2/-iо*I+/|+2/\+3/(4.4.2)|+4/|Figure 4.4.1 The onedimensional random-walkor "drunken sailorproblem."4.4 Diffusion-5/-41-31-21JОт+/IL11Ii11IiIiA1 >1т13T11IA1 уIifA1 у1Y/iA1 уiiAAii+31I+41I+51IiIiIiIiIIiIiI1V 14A4 2 уЧ З у4 4 у-4/i+21IiVA4 6 уiV1A4i\ 4-2/i 147+2/V 14AJiV 14.Figure 4.4.2 (a) Probabilitydistribution for a one-dimensionalrandom walk over zero to four timeunits.
The number printed over eachallowed arrival point is the numberof paths to that point, (b) Bar graphshowing distribution at t = AT. Atthis time, probability of being atx = 0 is 6/16, at x = ± 2/ is 4/16,and at JC = ± 4/is 1/16.I+41where the diffusion coefficient, D, identified as /2/2r, is a constant related to the step sizeand step frequency.2 It has units of Iength2/time, usually cm2/s. The root-mean-square displacement at time t is thus= V2Dt(4.4.3)This equation provides a handy rule of thumb for estimating the thickness of a diffusion layer (e.g., how far product molecules have moved, on the average, from an electrodein a certain time).
A typical value of D for aqueous solutions is 5 X 10~6 cm2/s, so that adiffusion layer thickness of 10~4 cm is built up in 1 ms, 10~3 cm in 0.1 s, and 10~2 cm in10 s. (See also Section 5.2.1.)As m becomes large, a continuous form of equation 4.4.1 arises. For No molecules located at the origin at t = 0, a Gaussian curve will describe the distribution at some laterTABLE 4.4.1 Distributions for a Random Walk Processtn»к = 2°)От4T2(4(8(16(тт2IT2T3TД====2')22)23)24)с0±/(1)0(2), ± 2/(1)±/(3), :t 3/(1)0(6), ± 2/(4),±4/(1)m0SДF = ISA200/22г/S/224/264/2m 2mnl\ = m2 / )2/23/24/2ml2al = step size, 1/r = step frequency, t = mr = time interval.bn = total number of possibilities.CA = possible positions; relative probabilities are parenthesized.2This concept of D was derived by Einstein in another way in 1905.
Sometimes D is given as//2/2, where/isthe number of displacements per unit time (= 1/r).148Chapter 4. Mass Transfer by Migration and Diffusiontime, t. The number of molecules, N(x, t), in a segment Ax wide centered on position x is(20)«*•»-**,(=£).4.4.4,A similar treatment can be applied to two- and three-dimensional random walks, wherethe root-mean-square displacements are (4Dt)l/2 and (6Dt)l/2, respectively (19, 21).It may be instructive to develop a more molecular picture of diffusion in a liquid byconsidering the concepts of molecular and diffusional velocity (21).
In a Maxwellian gas,a particle of mass m and average one-dimensional velocity, vx, has an average kinetic energy of l/2rnv2. This energy can also be shown to be 4T/2, (22, 23); thus the average molecular velocity is vx = (6T/m)m. For an O2 molecule (m = 5 X 10~ 23 g) at 300 K, onefinds that vx = 3 X 104 cm/s. In a liquid solution, a velocity distribution similar to that ofa Maxwellian gas may apply; however, a dissolved O 2 molecule can make progress in agiven direction at this high velocity only over a short distance before it collides with amolecule of solvent and changes direction. The net movement through the solution by therandom walk produced by repeated collisions is much slower than vx and is governed bythe process described above.
A "diffusional velocity," u^, can be extracted from equation4.4.3 asvd = A/t = (2D/t)l/2(4.4.5)There is a time dependence in this velocity because a random walk greatly favors smalldisplacements from a starting point vs. large ones.The relative importance of migration and diffusion can be gauged by comparing udwith the steady-state migrational velocity, v, for an ion of mobility щ in an electric field(Section 2.3.3). By definition, v = uY%, where % is the electric field strength felt by theion. From the Einstein-Smoluchowski equation, (4.2.2),v = \z{\ FDfflRT(4.4.6)When v « L?d, diffusion of a species dominates over migration at a given position andtime.
From (4.4.5) and (4.4.6), we find that this condition holds when2D\112RTl\z{\Fwhich can be rearranged to(2Д/)1/2%«2 -Щ-(4.4.8)where the left side is the diffusion length times the field strength, which is also the voltagedrop in the solution over the length scale of diffusion. To ensure that migration is negligible compared to diffusion, this voltage drop must be smaller than about 2RTl\z-^F, whichis 51.4/|z}| mV at 25°C. This is the same as saying that the difference in electrical potentialenergy for the diffusing ion must be smaller than a few 4T over the length scale of diffusion.4.4.2Fick's Laws of DiffusionFick's laws are differential equations describing the flux of a substance and its concentration as functions of time and position.
Consider the case of linear (one-dimensional) diffusion. The flux of a substance О at a given location x at a time t, written as JQ{X, t), is the4.4 Diffusion149net mass-transfer rate of O, expressed as amount per unit time per unit area (e.g., mol s lcm~ 2 ). Thus Jo(x, i) represents the number of moles of О that pass a given location persecond per cm 2 of area normal to the axis of diffusion.Fick's first law states that the flux is proportional to the concentration gradient,дСо/дх:-Jo(x, t) = DodC0(x, t)dx(4.4.9)This equation can be derived from the microscopic model as follows.
Consider location x,and assume NQ(X) molecules are immediately to left of x, and NQ(X + Дл:) molecules areimmediately to the right, at time t (Figure 4.4.3). All of the molecules are understood tobe within one step-length, Ax, of location x. During the time increment, Д*, half of themmove Ax in either direction by the random walk process, so that the net flux through anarea A at x is given by the difference between the number of molecules moving from leftto right and the number moving from right to left:No(x)Jo(x,t) = ±Multiplying by Дх /AxriveNo(x + Ax)(4.4.10)Atand noting that the concentration of О is C o = No/AAx, we de--Jo(x, i) =2At(4.4.11)AxFrom the definition of the diffusion coefficient, (4.4.2), Do = Ax2/2At, and allowing Axand Д По approach zero, we obtain (4.4.9).Fick's second law pertains to the change in concentration of О with time:(4.4.12)This equation is derived from the first law as follows.
The change in concentration at a location x is given by the difference in flux into and flux out of an element of width dx (Figure 4.4.4).J(x, t) - J(x + dx, t)dxdCo(x, i)dt(4.4.13)Note that J/dx has units of (mol s l cm 2)/cm or change in concentration per unit time, asrequired. The flux at x + dx can be given in terms of that at x by the general equationJ(x + dx, t) = J(x, t) +(4.4.14)dxNo (x + Ax)No (x + Ax)N0(x)^~~— ^22Figure 4.4.3in solution.Fluxes at plane x150Chapter 4. Mass Transfer by Migration and DiffusiondxJ0JQ (x + dx, t)(X, t)Figure 4.4.4 Fluxes into andout of an element at x.x + dxand from equation 4.4.9 we obtain_ dJ(x, t)dxx, t)ddx'(4.4.15)dxCombination of equations 4.4.13 to 4.4.15 yields(4.4.16)When Do is not a function of x, (4.4.12) results.In most electrochemical systems, the changes in solution composition caused by electrolysis are sufficiently small that variations in the diffusion coefficient with x can be neglected.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
