CHAPTER 2. MASS TRANSFER I. General Mass Transfer Equation Nernst-Planck Equation: J i (x) = -D i [C i (x)/x] [(z i F)/(RT)][(x)/x] + C i v(x) J i (x) = flux of species i (mols -1 cm -2 ) at distance x from the surface D i = diffusion coefficient (cm 2 /s) C i (x)/x = concentration gradient at distance x (x)/x = potential gradient z i = charge of species i C i = concentration of species i v(x) = velocity (cm/s) with which a volume element in solution moves along the axis The terms on the right-hand side of the equation represent the contributions of diffusion, migration, and convection, respectively, to the total mass transfer.
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CHAPTER 2. MASS TRANSFER
I. General Mass Transfer Equation
Nernst-Planck Equation:Ji(x) = -Di[Ci(x)/x] [(ziF)/(RT)][(x)/x] + Civ(x)Ji(x) = flux of species i (mols-1cm-2) at distance x from the surfaceDi = diffusion coefficient (cm2/s)Ci(x)/x = concentration gradient at distance x(x)/x = potential gradientzi = charge of species iCi = concentration of species iv(x) = velocity (cm/s) with which a volume element in solution moves along the axis
The terms on the right-hand side of the equation represent the contributions of diffusion, migration, and convection, respectively, to the total mass transfer.
II. Migration
Migration of an ionic species occurs in response to an applied electrical field.FIGURE 2-1. “Bard” 2nd Ed, Fig. 4.3.1 (p. 141).The contribution of migration to the total flux can be made negligible if a large excess (a 50-100 fold excess is sufficient) of an unreactive electrolyte, called supporting electrolyte is added to the solution. The reason for this is easily understood from a simple example:
Consider the reduction of Cu2+ to Cu0 in a solution of Cu(NO3)2 containing no other source of ions. The transfer of each Cu2+ ion to the electrode is accompanied by transfer of two NO3
ions to maintain the local electrostatic neutrality. The Cu2+ is reduced to Cu0 and an electrostatic imbalance is produced near the electrode surface because of the remaining negatively charged NO3
ions. In response to the potential field produced by the charge imbalance, Cu2+ can migrate by electrostatic attraction to the electrode surface (i.e., a migrational flux); alternatively, NO3
can migrate away from the electrode. If a large excess of an inert electrolyte is added to the solution, such as NaNO3, the Na+ ions will migrate in the preference to Cu2+ because of their relative excess concentration.Thus, the addition of an excess supporting electrolyte decreases the contribution of migration to mass of the electroactive species and product molecules, and simplifies the mathematical treatment of electrochemical systems in the mass transfer equations.
Moreover, the supporting electrolyte serves the important function of decreasing the cell resistance and, in analytical applications, it may decrease or eliminate matrix effects.
III. Faraday's Equation
The Faraday’s law of electrolysis relates the electrical current in a submerged electrode to the chemical equivalents of the dissolved species which are reacting at the electrode.
I = nF(dN/dt) (1)where dN/dt = the number of moles of depolarizer that reaches the electrode in
unit time and is subjected to electrochemical changeF = the faraday (96,500 C)n = number of electron taken up or delivered by a single molecule of depolarizer during the electrode process
IV. Diffusioni/. Fick's First Law of Diffusion
Flux, Ji(x,t) = (dN/dt)/A = Di[Ci(x,t)/x] where A = area (2)The diffusion is determined by the concentration gradient at the electrode surface, i.e., (C/x)x=0, which is time dependent.Thus, I = nF(dN/dt) = nFAD(C/x)x=0 (3)FIGURE 2-2. “Diffusion.
J(x) J(x + x) = x(C/t) where C = average concentrationLet x 0, C C(x),then [J(x + x) J(x)]/x J/x C/t = J/x
ii/. Fick's Second Law of DiffusionJi(x,t)/x = Ci(x,t)/t = Di[2Ci(x,t)/x2] (4)
The solution of this partial differential equation with appropriate initial and boundary conditions can be obtained by means of Laplace transformation.
iii/. Nernst’s Concept of the Diffusion LayerIn the Nernst diffusion layer treatment one assumes that a stagnant layer of thickness exists near the electrode surface.FIGURE 2-3. “Bard” Fig. 1.4.5 (p. 32).Outside this layer, convective transport maintains the concentration uniform at the bulk concentration C*. Within the layer mass transfer occurs only by diffusion. Hence, in the interfacial region (i.e., x = 0),
I = nFAJx=0 = nFAD(dC/dx)x=0
The concentration gradient is thus(C/x)0x = (C* - Cs)/
Hence, I = nFAD(C* Cs)/ (5)As Cs = 0,
I = Id, the limiting diffusion current,Id = nFADC*/
This same concept can be used to give an approximate treatment of the time-dependent buildup of the diffusion layer either in a stirred solution or in an unstirred solution where the diffusion layer continues to grow with time.FIGURE 2-4. “Bard” Fig. 1.4.6 (p. 33).Consider what happens when a potential step of E is applied to an electrode immersed in a solution containing a species O. The volume of the diffusion layer is A(t), the current flow causes a depletion of O, and the amount of O electrolyzed by this current is given by
Moles of O electrolyzed in diffusion layer (C* - Cs)A(t)/2 = 0t (Idt/nF) (6)
From Equations (5) and (6),(C* - Cs)(Ad(t)/dt)/2 = I/nF = AD(C* - Cs)/
or d(t)/dt = 2D/(t) (7)Since (t) = 0 at t = 0, the solution of Equation (7) is
(t) = 2(Dt)½ (8)and I = (nFAD½/2t½)(C* - Cs) (9)Thus this approximate treatment predicts a diffusion layer that grows with t½ and a current that decays with t-½ under a potential step.As t , or dC/dx 0, i.e., Id 0.Therefore it is not practical to obtain limiting currents with planar electrodes in unstirred solution under a potential step.
iii/. Diffusion to Stationary Electrodes - Semi-Infinite Linear DiffusionInitial and boundary conditions:
at t = 0, x 0; C(x,0) = C*at t > 0, x = 0; C(0,t) = Cs
By Laplace transformation,C(x,t) = (C* - Cs)erf[x/2D½t½] (10)where erf = mathematical error function
For the current towards a planar stationary electrode,I = nFAD[(C* - Cs)/(Dt)½] (12)
When Cs 0, I Id,Id = nFAD[C*/(Dt)½] (Cottrell equation) (13)
Thus the current decreases with time according to Equation (12), I = kt-½.FIGURE 2-5. “Bard 2nd Ed” Fig. 1.41 (p. 30).FIGURE 2-6. "Bard" Fig. 5.2.1 (p. 144).If one draws a tangent to the curve of C vs x, then the intersection of the tangent with the straight line C = C* occurs at a distance, = (Dt)½. The quantity is called the differential thickness of the diffusion layer and is a function of time. It is a measure of the thickness of the region of the solution that has been exhausted by the diffusion process. Rate of transport, J D(C/x)x=0 D[(C* - Cs)/]For Cs 0,
Maximum rate of transport, Jd D(C*/)
iv/ Diffusion to Stationary Electrodes - Spherical DiffusionFick's second law in spherical coordinates has the form:
C/t = D[2C/r2 + (2/r)(C/r)]where r = radial distance from the electrode center
Initial and boundary conditions:at t = 0, r r0; C(r,0) = C*at t > 0, r = r0; C(r0,t) = 0where r0 is the radius of the spherical electrode
By Laplace transform,C(r,t) = C*(r0/r)erf{(r r0)/[2(Dt)½]} + C*(1 + r0/r)
When r r0, the solution of the equation resembles that for diffusion towards a planar electrode.
The reason for this curious nonzero limit is that the growth of the depletion region fails to affect the concentration gradient at the surface.FIGURE 2-7. Spherical diffusion.
V. Convective Diffusion
i/. Flow Patterns in the Presence of ConvectionIn order to understand the effect of stirring, it is necessary to develop a picture of liquid flow patterns in a stirred solution containing a small planar electrode.FIGURE 2-8. “Skoog” Fig. 22-8 (p. 543).Three types of flow can be identified.1. Turbulent Flow- In which liquid motion has no regular pattern, occurs in the bulk of the solution away from the electrode.2. Laminar Flow- As the surface is approached, a transition to laminar flow takes place. In laminar flow, layers of liquid slide by one another in a direction parallel to the electrode surface.3. Stagnant Solution- At distance from the surface of the electrode, the rate of laminar flow approaches zero as a result of friction between the liquid and the electrode, giving a thin layer of stagnant solution called the Nernst diffusion layer.
ii/. Nernst ConceptNernst assumed that at the phase boundary a layer of the solution of thickness remains stagnant when the solution is stirred.
(C/x)x=0 = (C* - Cs)/A steady-state concentration gradient is formed in this steady-state diffusion layer. The quantity is a function of the rate of stirring of the solution.FIGURE 2-9. Nernst concept.Nernst concept has been shown to be incorrect!
iii/. Prandtl Boundary LayerConsider a simple plate along which a liquid flows, the distribution of velocities as a function of the distance from the plate is shown in the figure.FIGURE 2-10. Prandtl boundary layer.The speed, which is zero at the surface, gradually increases until it becomes constant and equal to u0 at a certain distance. The layer, in which the uniform motion of the liquid is disturbed, is termed the Prandtl boundary layer (or hydrodynamic boundary layer), yh.
Average boundary layer thickness, yh(x) = (2/3)(L/Re½)where Re Reynolds numbers = u0L/
= / = kinematic viscosity = density of the element = coefficient of viscosityL = a characteristic length
The hydrodynamic Prandtl layer is formed at Reynolds numbers greater than one. At largeReynolds numbers, where instead of laminar streaming, turbulent streaming develops. For a rotating electrode, the onset of turbulence occurs for rotational velocities characterized by a Reynolds number of approximately 105. The Reynolds number of a rotating electrode is given by
Re = r2/where r = total radius of the electrode plus the nonactive shroud
For a rotating electrode, the thickness of the solution layer near the electrode in which the axial flow has reached 80% of its maximum value is given by
yh = 3.6(/)½
where is the angular velocityThis roughly represents the thickness of the layer dragged by the rotating electrode.In general, yh >> .
It is interesting to note that, in a convective system, the diffusion layer ultimately approaches the steady-state value characterized by (t) = 0.FIGURE 2-11. "Bard" Fig. 1.4.7 (p. 34).
iv/. Concentration Profiles in Stirred SolutionWithin the static diffusion layer, mass transport takes place by diffusion alone, just as was the case with the unstirred solution. With the stirred solution, however, diffusion is limited to a narrow layer of liquid, which even with time, cannot extend out indefinitely into the solution.FIGURE 2-12. “Skoog” Fig. 22-9 (p. 543).As a consequence, steady, diffusion-controlled currents are realized shortly after application of a potential step.FIGURE 2-13. “Skoog” Fig. 22-5 (p. 540).
VI. Current-Potential Curves
An electrode reaction is reversible if the shape of the current-potential (I - E) curve can be described on the basis of the Nernst equation. Reversibility is expected when the rate of electron transfer at the electrode surface is fast relative to the rate of convective-diffusionalmass transfer. Hence, the electrode reaction is at equilibrium relative to the concentrations of species at the electrode surface and the Nernst equation can be applied accurately.
For the case of an electrode in a uniformly stirred solution, values of I will be considered as independent of time. By an arbitrary convention, electrode current observed is a positive quantity when the net reaction is cathodic (reduction). The current is negative when the net reaction is anodic (oxidation).
Application of Equation (3) for the cathodic reaction and solving for COs yields
COs = CO* I/[nFA(D/)O] (16)
The largest rate of mass transfer of O occurs when COs = 0. The value of the current under
these conditions is called the limiting current, Il, wherecathodic limiting current, Il,c = nFA(D/)OCO* (17)
From Equations (16) and (17),CO
s = (Il,c I)/[nFA(D/)O] (18)
In a similar manner, CRs is obtained with recognition that the polarity of anodic current is
opposite that of cathodic current.CR
s = (Il,a I)/[nFA(D/)R] (19)where Il,a is the anodic limiting current
Substitution of Equations (18) and (19) into the Nernst equation produces an equation describing the shape of a reversible I - E curve.
E = E' + (RT/nF)In[(D/)O/(D/)R] + (RT/nF)In[(Il,c I)/(I Il,a)]A satisfactory approximation in many situations is that (D/)O = (D/)R
and E E' + (RT/nF)In[(Il,c I)/(I Il,a)] (20)
There are several interesting experimental situations which will help to illustrate the significance of Equation (20).i/. Case I: CR* = 0 and, therefore, Il,a = 0.E1/2 is defined as the value of E when I = Il,c/2.At the E1/2 value, (Il,c I)/I = 1 and E1/2 = E'.Doubling CO* results in a doubling of Il,c but does not change the value of E1/2.FIGURE 2-14. “Johnson” Fig. 4 (p. IV-15).
ii/. Case II: CO* = 0 and, therefore, Il,c = 0.E1/2 is again defined as the value of E when I = Il,a/2.Consequently, E1/2 = E'.FIGURE 2-15. “Johnson” Fig. 5 (p. IV-15).
iii/. Case III: CO* = CR* 0 and (D/)O = (D/)R.For this example, Il,c = Il,a and E = E' when I = 0.FIGURE 2-16. “Johnson” Fig. 6 (p. IV-15).
The value of E1/2 for the considerations above were shown to be approximately equal to E' and independent of the bulk concentrations of the species.This statement is not true for electrode reactions resulting in deposition of a solid.Consider the reaction
M+n + ne MThe activity of the deposited metal is unity when the amount of deposit is greater than the equivalent of a monolayer and the wave equation is
E E + (RT/nF)In{(Il,c - I)/[nFA(D/)]}The value of E1/2 is
E1/2 = E + (RT/nF)In{(Il,c/2)/[nFA(D/)]}And E1/2 is obviously a function of CM
+n*.
FIGURE 2-1. Bard et al, Electrochemical Methods, 2nd Ed.
FIGURE 2-2. Diffusion.
FIGURE 2-3. Bard et al, Electrochemical Methods, 1st Ed.
FIGURE 2-4. Bard et al, Electrochemical Methods, 1st Ed.
FIGURE 2-5. Bard et al, Electrochemical Methods, 2nd Ed.
I = 0
I > 0
I = Id
t
Diffusion layer
Bulk of Solution
FIGURE 2-6. Bard et al, Electrochemical Methods, 1st Ed.
δ→
FIGURE 2-7. Spherical diffusion.
FIGURE 2-8. Skoog et al, Principles of Instrumental Analysis, 4th Ed.
FIGURE 2-9. Nernst concept of diffusion layer when the solution is stirred.
FIGURE 2-10. Prandtl boundary layer.
FIGURE 2-11. Bard et al, Electrochemical Methods, 1st Ed.
FIGURE 2-12. Skoog et al, Principles of Instrumental Analysis, 4th Ed.
FIGURE 2-13. Skoog et al, Principles of Instrumental Analysis, 4th Ed.
