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Vol. 99, No.









C. W. Tobias, M. Eisenberg, a n d C. R. Wilke ~


Mechanisms of Ion Transfer

An electrode reaction proceeding at a finite rate involves
nmvement of reacting materials between the electrode and
the solution surrounding it. In absence of fluid turbulence,
ions are transferred from solution to an electrode by three
principal mechanisms: (a) migration, (b) diffusion, and (c)
convection. For a given species the rate of transfer in the y
direction per unit area perpendicular to y at any point in
the fluid nmy be expressed by the equation:
Nt = CU oy

ay + Vc


where: Nt = total rate of transfer, gram ions/em~-sec; c =

concentration of given species, gram ions/cma; U = mobility,
cruZ/see-volt; q5 = potential, volts; y = distance in direction
of transfer, cm; D = diffusion coefficient, crab/see; and
V = velocity of bulk fluid movement in direction of transfer,

The three terms on the right represent, respectively, the
contributions of migration, diffusion, and convection. When
transfer is not unidirectional corresponding equations nmy be
written for each coordinate axis.
For unsteady state processes, changes of concentration
with time must be considered, and the general transfer equation becomes (1)
Oc = _ div(Uc grad r

+ div(D grad c) -- V grad c


where t = time, sec. The potential q) must satisfy the Poisson

equation (1), and the velocity vector V the three NavierStokes equations. Thus a set of nonlinear differential equations is obtained which has been solved only f m ' a rehtively
few cases.
In a fluid in which turbulence has been induced by stirring
or pumping, transfer by diffusion is augmented by eddy
diffusion resulting from turbulent pulsations. Analytical solution of the convection-diffusion equations for such systems is
in general impossible at the present time.
Equations for Steady State Mass Transfer
The term "mass transfer" will be used to denote the transport of ions or molecules by the combined effects of diffusion,
eddy diffusion, and convection, excluding the contribution of
migration to the process. The mass transfer process is commonly designated as "forced convection" when the system
is mechanically agitated by stirring or pumping, and as
"fi'ee convection" when mixing is induced as a result of
density differences accompanying concentration differences
from point to point within the fluid. Mass transfer in the
steady state implies that the transfer rates, current density,
concentrations, temperature, and other variables associated
with cell operation are independent of time at any particular
point in the system.
~Department of Chemistry and Chemical I",ngineertng,
University of California, Berkeley, California. This work has
been supported by the Office of Naval Research.

The different mass transfer mechanisms participate to a

varying degree as material moves from the main body of
solution in contact with an electrode to the electrode-solution
interface, with the natm'e of the process at any distance from
the electrode depending on electrolyte motion, shape and
position of the cell, and other operational variables. As a
result of the participation of diffusion in the transfer process,
it may be stated in general that movement of species toward
or away from the electrode results in a, decreasing concentration of the species in the direction of transfer.
For steady state processes it is convenient to express the
rate of migration as a function of current per unit area perpendicular to the direction of transfer:
N,~ = e U 0 ~ = __PI


where: N,~ = rate of migration, gram ions/cm2-see; P =

transference number for the given species; n = valence charge
of the ion;'and F = the Faraday, amp-see/gram equivalent.
In sufficient excess of inert electrolyte the transference number of the species participating in the electrode reaction may
be made negligibly small and migration need not be considered. In the general case the rate of mass transfer may be
expressed by the relation:
N4 = Nt -- N,,, =


-- F)



Consider a cell containing at one end a vertical plane

cathode upon which metal ions are being deposited. Assume
that due to mechanical stirring the concentration is uniform
at "dl distances from the floor of the cell at any particular
dist:mee x away from the electrode. For the species deposited
a concentration profile through the main body of solution up
to the electrode-solution interface may prevail as illustrated
in Fig. 1. The exact shape of the profile will depend upon the
mass transfer mechanism predomina~ting at a w place. Asstone further that the concentration in the main body of
solution remains constant. In view of the complicated nature
of the problem it is sometimes convenient to express the
over-all mqss transfer process by equations of simple empirical
form which relate the mass transfer rate to the difference
between the mixed-mean or average concentration of the
transferred species in the nmin body of fluid and the concentration at the electrode-solution interface.
(A) Mass transfer coefficient: the rate of transfer may be
expressed by a coefficient, /eL, defined as follows:
Na = kL(cb -- c~)


where: Nd = mass transfer rate at any particular point on

the electrode, gram mole/cm2-sec; kL = mass transfer coefficient, cm/see; c~ = average concentration in bulk fluid,
gram mole/cnr~; and c~ = concentration at interface.
(B) Equivalent fihn thickness or diffusion layer: this
concept expresses the mass transfer process by the rate equation for steady state diffusion from concentration ca to cl
over a hypothetical layer of stagnant fluid of thickness ~.
The diffusional term of equation [1], i.e., Fick's first law, is





integrated for this case to give the relation which defines

the equivalent film thickness:
Nd --

D(cb -



where ~t = equivalent film thickness.

This general mathematical formulation was originated by
Nernst (2) who termed ~t the "diffusion layer," a name which
has been used widely throughout the electrochemical literature. The term "equivalent film thickness" has been used
extensively in chemical engineering writings.


December 1952

Concentration polarization.--During the passage of current,

the concentration of metal ions at the solution-electrode
interface changes from the value at rest, cb, to ci. The corresponding shift in electrode potential, AEo, the concentration
polarization, is equal to the emf of a concentration cell with
transference, in which the electrodes are in contact with
solutions of concentrations Cb and c~, respectively. If transfer
by migration of the potential determining ion is negligible,
and the ratio of ionic activity coefficients is assumed to be
fi T
~Ec = ~ h~ -






The dependence of concentration polarization on current

density for any current, I, can be found by combining equations [9] and [10]: 2
AEc = ~ - ~ l n

1 -- ~




i , , / ~I




,-- 8 - , , . I





That concentration changes in solutions of electrolytes in

the vicinity of working electrodes are of importance was
realized as early as 1879 by Weber. In his work on the diffusion of zinc sulfate (3) Weber obtained an equation for the
concentration as a function of time and distance from the
electrode using a cell model of the form of a short cylinder
closed at its ends by a cathode and anode and without liquid
circulation or stirring. For practical calculations, use of this
equation was not convenient since it contained a Fourier
series which did not converge fast enough for large distances
between the electrode and for small time periods.
In 1900, Sand (5) in an attempt to improve Weber's theory
considered a model with only one electrode at the end of a
semi-infinite cylindrical cell. Assuming no motion of the
electrolyte, and the material transfer to occur only by diffusion, with the diffusion coefficient independent of concentration, equation [2] reduces to the form:





Fro. 1. Illustration of concentration profile in the vicinity

of an electrode during electrolysis.

r = D -O~c

The relation between current density, concentration and electrode potential.--Taking the example of the deposition of an
n valent metal in the presence of a large excess of "inert"
electrolyte which takes no part in the electrode process,
equation [6] can be expressed as:

Upon integration of equation [12] with the boundary conditions as given below Sand obtained the result expressed by
equation [13].





The current density can only be increased above this

value if a consecutive electrode process is possible.
If one makes the assumption that ~t is independent of the
rate of diffusion, the concentration at the interface, c~-, can be
related to the current density by combining equations [7]
and [8],


- -




The maximum rate of deposition, or limiting current density,

will be given b y the condition when c~" = 0, and

N~ = D ~ x x ; a t t = 0, betweenx = 0



I~ -

A t x = 0,





where: c~., = concentration at any time and distance x from

the electrode; cb = original uniform concentration; D = diffusion coefficient; Na = rate of removal of ions at the electrode; and t = time.
For x = 0, i.e., at the electrode, this reduces to the simple
Co,t= c b - - 1 . 1 2 8 5 N d ~ D .


For a cathodic deposition Nd is given by equation [4].

2 It should be noted that this expression applies only for
electrodeposition. For discussions that include generalized
electrode-reactions, and the effect of migration, see Ref. (4).


V o l . 9 9 . N o . 12


By means of equation [14] Sand was able to determine

diffusion coefficients in electrolytes by observing the time
interval necessary for a given current density to cause an ion
depletion at the cathode to zero concentration, i.e., to reach
limiting current density. Sand realized that this procedure
is not rigorous on account of the variations in transference
numbers with concentration. However, on the basis of
Kohlrausch's work (6) he found that such variations result
in an error not exceeding 2.5 per cent for the systems investigated.
An extension of the work of Weber and Sand was given by
Rosebrugh and Miller in 1910 in an excellent paper on the
mathematical treatment of electrolysis with direct and successive currents (7). A cell of uniform cross section provided
with identical metal electrodes at both ends served as the
model. The electrode reactions were assumed to be identical
but opposite in sense (at the cathode, metal depositing; at
the anode, metal going into solution, both with 100 per
cent current efficiency).
Equation [12] was solved for the boundary conditions
at x = 0 (at half distance between the electrodes)
art =0


- 0

--0c = 0 .



across which mass transfer is possible only by diffusion and

migration. The magnitude of 8 depends only on hydrodynamic
conditions (r~te of stirring, rotation, etc.) and the viscosity
of the solution.
In the steady state the concentration gradient in this
"diffusion layer" is constant, and the rate of diffusion is equal
to the rate of the electrode process. Thus for the case of metal
deposition, equation [7] applies. For any particular solvent,
geometryand rate of stirring, 8 is a constant, and can be deter-






Of most interest is the relation giving the concentration at

the electrode as a function of time. Neglecting migration
D ( c b - ci)nY
1I. L

where: y -

8 m__~l. 1 e_m%t


= ,~ --~

mined experimentally if (cb - ci) and D are known. Under

limiting current conditions c~ = 0, from which it follows that


4L 2 , and L = half the length of the cell.

For yt > 0.6 equation [15] may be approximated by:

D(cb -- c d n F
' ~ 1 - - ~ e- ' t .



Similarly for vt < 0.6

D(cb -- c~)nY

FIG. 2. Laminar flow past an electrode (Eueken)

~__ 1.129 % / D t .


Equation [17] shows the concentration at the electrode to be

independent of the length of the cell. This result is identical
to equation [14] indicating that for small v t a cell of finite
length can be treated as a semi-infinite column of electrolyte.
The foregoing equations have been found to agree fairly
well with experimental findings (8, 9) for the deposition of a
metal with the cathode placed horizontally at the top of the
cell. In these experiments natural convection was suppressed.
Forced Convection

The treatment of an electrolytic cell as a slab of electrolyte

free of convection currents, in which ions are transferred only
by diffusion and migration, is not adequate to meet the conditions of actual operation.
Nernst (2, 10) and Brunner (11) developed the qualitatively correct concept of ionic mass transfer by the combined
mechanisms of diffusion and convection. As a result of an
elaborate experimental study of heterogeneous reaction rates
Brunner developed the following argument.
During operation of an electrolytic cell, due to natural
and forced convection the concentration in the bulk of the
solution can be considered to be uniform. I t may be assumed
that a stagnant layer of electrolyte of thickness 8 exists between the bulk of the solution and the electrode surface,



Brunner (11) found that the thickness of the diffusion

layer in some geometrical arrangements varied with the rate
of stirring, and suggested the relation

8 = R~/--%


where: R = rate of stirring (rotation), rpm; and a = a constant for a given solvent, geometry, temperature.
In moderately stirred solutions in a number of experiments
8 was found to be 0.02-0.05 mm (11, 12). Glasstone (13)
suggested that a = 0.05 gives a fair agreement with experimental findings?
Brunner realized that the assumption of a stagnant layer of
electrolyte of 8 thickness at the interface is physically incorrect. He also found that 5 depends on the diffusion coefficient. His mass transfer experiments were carried out in a
system of impractical geometry, and, therefore, only qualitative deductions can be made for different experimental
Methods of hydrodynamics were applied first by Eucken
(14) to predict the effect of forced convection on mass transfer
rates to electrode surfaces (Fig. 2). The limiting current in
case of laminar flow in the y direction past an electrode of
height h and width yl was evaluated for cases when
the velocity of flow increases linearly with the distance from
the electrode plane.
a In equation [19] 8 is expressed in cm if used with a = 0.05.
Conditions for application of this equation, such as rotor size,
solution properties, etc., are not stated in reference.





Under steady state conditions the increase in concentration of a volume element dx.dy.dz due to convection will be
balanced by a decrease of concentration due to diffusion in a
direction (x) perpendicular to the electrode.
By solving the differential equation for diffusion and


~I l X ~


= D &--~


with the boundary conditions: at x = 0, c = 0; at x = * ,

c = c~; where W, denotes the constant fluid velocity unit
distance away from the electrode plane, the following expression was obtained for the total linfiting current:
iL = 77,800 hcb "~/D"g{IV~.

[21 ]

Dividing both sides of equation [19] by the surface area

A = h-y~ yields the average limiting current density along
the electrode. The result indicates that I~.=~g decreases with
the .~i power of the length of the electrode in the direction of
a flow, and increases proportional]y to D 2ja. This conclusion
is also supported by the expression obtained for the average
thickness of the diffusion layer:
8,,~ = 1.24 ,a/DYe-.


~- q- (~'~grad) c* --- DV2c *


December t 9 5 2

which upon integration yields:


I~;;]-' (~: D1 -- n2D2) c* grad ~b

q- F(D1 -- D2) grad c* ~- 3
I l l 7l 2

wherej represents the vectors of current density and R = gas

The solution of equation [25] allows the determination of the
exact relation betwcen the potential across the electrolytic
cell and the current. This import'mr separation of the convective diffusion from the potential distribution has been shown
to be possible also for a tervalent system when the concentration of one kind of ions is low compared to that of the other
two kinds.
A number of solutions for specific cases of definite geometric
systems and hydrodynamic conditions were given as reviewed
A fiat infinite disk rotating with a constant angular velocity.Using yon Karman's solution for the hydrodynamics of fluid
motion about such a rotating disk the following result was
obtained for steady state conditions in the discharge of one
ion of a binary system:


To obtain the simplified diffusion convection equation [20],

Eucken made the assumption that no velocity component
exists in the x direction. The results derived therefore nmy be
of questionable value for practical applications (1). Eucken
himself met with considerable difficulties trying to achieve
such conditions in experiments designed to test equation [21]
A fundamental approach to the solution of mass transfer
problems based upon modern theories of fluid mechanics was
initiated in the excellent series of papers by B. Levich (1, 15).
The method involves application of the general equation for
convection, migration, and diffusion (equation I2]), coupled
with the Navier-Stokes equations for fluid motion and the
Poisson equation for charge distribution in the solutions.
Levich succeeded in simplifying these nonlinear partial
differential equations for the case of a bina W electrolyte to
the form

where c* -



where: 6
Thus for

= diffusion layer thickness (equation [6]); D =

diffusion coefficient fro" the salt; v = kinematic
of the fluid; and Z = number of revolutions per
", system 4 with D = 10-s cm2/sec and v = 10-=


0 00647

Using the above value for 6, equations [8] and [11] may be
utilized to give limiting currents or concentr'~tion polarization. Equivalent expressions are presented by Levich.
Flow past a fiat electrode set into a landnar flow of liquid.-Levich developed a general theory of the laminar boundary
layer by considering the analogy of the convective diffusion
Ox q- V O!j = D \ a x e q- ay'- ]



to the Navier-Stokes equations for the velocity distribution.

The thickness of the diffusion boundary layer, 8, for the
case of "~ flat electrode inserted into a laminar flow of fluid
(16) is given for an)" distance x along the electrode by:

and the effective diffusion coefficient

D = (nl -- n.2) DID.~
1/1D1 - - n2 D~
D1D~. = diffusion coefficients of ions 1 and 2, respectively;
*hn,~ = number of valence charges per ions of species
1 and 2, respectively;
c~, c2 = concentrations of ions 1 and 2, respectively.
Equation [23] from which the potential has been eliminated
applies to the case of convective diffusion and determines for
a known ~ the concentration at any time and point in the
solution. Then the distribution of the potential is obtained
from the distribution of the concentration by the equation:
RT(D1 -- De) V2c.
div (c* grad 4~) = -- F(nlD1 -- n2D2)


8 = 3 D


where U = velocity of the fluid and x = distance along the

electrode from the leading edge.
For the case of the flat electrode equation [29] shows that
for a given fluid velocity U the diffusion layer thickness increases as xl, the square root o f distance from the leading edge
m the direction of fluid flow. This in turn affects the current
4 In the original paper this equation is given as ~ = ~
which appears to be in error.

Vol. 99, No. 12



distribution so that the local limiting current density (IL)x

varies in the x-direction according to the equation:
(I~)~ =


!~ *FDcb




This is valid if the flow past the electrode and the electrolysis
begin at the same point. Upon integration for an electrode of
width b and length L the total limiting current is found to be:


,~.~ nFDcb D






I t should be noted that the development of the Prandtl

hydrodynamic boundary layer at the plate cannot go on
indefinitely in an apparatus of finite size. How far in the x-(tirection its formation will extend, until a steady state boundary
layer of constant thickness is formed, depends not only upon
the geometry of the entire cell (presence of any close parallel
wall, for instance) but also upon the length of the electrode.
Levich's equation [29] is acceptable from the hydrodynamical
viewpoint only for a short electrode at a relatively low fluid
velocity and in the absence of any close parallel boundary.
The problem of current distribution is also complicated.
First, because near an electrode edge submerged into an dectrolyte the current distribution will be nonuniform and the
equipotential surfaces by no means parallel to the electrode
surface. Second, due to depletion of the ions, as the fluid
moves along the electrode the electrode-potential changes
continuously in the x-direction. This nonuniform potential
is, in turn, responsible for a nonuniform current distribution
throughout the electrode. These possible limitations should
be considered in the general application of equations [30]
and [31].
Turbulent flow over a fiat plate electrode.--Levich considered
the boundary layer (region of disturbed velocity) developed
over the plate to be divided into three regions with respect
to transfer of mass and momentmn. In region 1, immediately
adjacent to the surface, turbulent pulsations are sufficiently
damped out so that transfer of momentum occurs by viscous
shear and transfer of mass occurs by molecular diffusion. In
region 2, adjacent to region l, turbulent pulsations are large
enough to cause the transfer of mass to be predominantly by
eddy diffusion or turbulent pulsations, but because of the
relatively high kinematic viscosity of the liquid, momentum is
transferred predominantly by viscous shear. In region 3, the
magnitude of turbulent pulsations is sufficient to cause transfer of both momentum and mass to occur by turbulent pulsations. Within these assumptions by application of the nmss
and momentum transfer analogy an equation was derived
which may be presented in the following expression for limiting current density:



In general Cz depends upon the shape of the surface, roughness, and Reynolds number so that no complete generalization is possible.
As a more general relation for turbulent flow Levich

IL = nFClcouor


where: ~o(Sc, Re) is a function which must be determined

experimentally; Sc = v/D, the Schmidt number.
Alternate expressions could be based upon the treatment of
yon Karman (6) for heat transfer. For low Schmidt numbers
equation [32] agrees well with a formula obtained by Prandtl
08) for the corresponding case in heat transfer.
At high turbulence the thickness of the diffusion layer will
be practically constant over most of the electrode, and the
current and potential distribution will be correspondingly
In a subsequent paper (15) Levich extended the treatment
of turbulent flow to nonstreamlined electrodes (cylinder
placed perpendicularly to the electrolytic flow) and to electrodes of rough surfaces.
Along the lines suggested by Agar (19), Putnam, et al.
(20) correlated limiting current densities for the inner electro(te of an annular cell, with the mass transfer heat transfer
an'dogy. Mass transfer correlations were obtained in terms
of equivalent film resistances using experimental data obtained from measurements with four different electrode reactions. The electrolytes were passed through an annular electrode system at various constant rates of flow. The experiments covered viscous and turbulent regions (Re =
From the measured limiting currents and the inlet concentrations to the cell, the mass transfer coefficient was obtained
as follows:

= ( (d.~ - - d~)'~ in - Co
\ 4dlL ]
c, -- 4i/nF(d~ -- d~)V


where: kL = mass transfer coefficient; d,d2 = diameter of

inner and outer electrode, em; L = length of electrode, cm;
Co = concentration of inlet stream gram mole/cc; I = limiting current, amp; n = number of electrons per molecule or
ion reacted at the electrode; F = Faraday; and V = velocity

For the systems investigated in the laminar flow region the
Leveque ai)proximatiou (21) can be applied in the following


\pDL] \ L ]


where: CI = coefficient of resistance, or friction factor;

where: & = equivalent diameter, d2 - d~ cm; D = diffusivity,

cmUsec; G = nmss velocity, gram/cm2/sec; and p = density
of electrolyte, gram/cc.
The agreement of experimental data with the Leveque heat
transfer equation was found to be satisfactory (within about

uo = velocity of flow past the plate; Co = bulk concentration


of transferred species; and A, B, a = constants to be evaluated experimentally. For tm'bulent flow over a flat plate
at ordinary Reynolds numbers the coefficient of resistance is
given by:

The mass transfer under turbulent flow conditions was

correlated in terms of the jv parameter, defined by Chilton
and Colburn (22) as:

nFCf7lo Co
IL = 2[A q-(aSca/4--A)B%//(~fl


CI = 0.07 Re -c 2
where Re = Reynolds nmnber for flow.
5 This equation is essentially equivalent to (12'), p. 287 in
original reference.

where V = velocity, cm/sec, and t~ = viscosity, gram/cin/sec.

A satisfactory correlation of jD as a function of Reynolds
number was obtained. The function was found to agree



rather closely with the friction factor data of Carpenter,

Colburn, and Wurster (23), i.e.,
j o = f = 0,023


where f = friction factor.

However, it is doubtful that great significance can be
attached to this apparent agreement between momentum
transfer and mass transfer in view of the high values
of Schmidt numbers prevailing in liquid systems, and possible
differences in surface roughness.
The correlation should be ve~T useful in predicting mass
transfer behavior in electrolytic cells, under flow conditions,
in presence of a large excess of indifferent electrolyte.

Natural Convection
Natural convection or "spontaneous stirring" results when
density changes accompany concentration changes in the
region of the electrode. Fluid motion thus induced aids the
mass transfer process by supplying fresh electrolyte from the
bulk solution.
For a vertical plate electrode Levich 05) derived the following relation for the local limiting current density at any
height X on the electrode surface:

(IL)x = 0.3S2 nFD


where: a = density coefficient (defined by equation [38]);

g = acceleration of gravity; and c0 = bulk concentration of
species reacting.
The density coefficient, a, relates density changes to concentration changes:

po Oc



po(cb - cD

where Pb and p~ are fluid densities in bulk solution and at the

electrode surface, respectively.
Equation [37] indicates that the limiting current density
varies as the ~ power of bulk concentration and inversely
as the ~ power of height on the electrode.
An alternate theoretical treatment has been presented by
Wagner (24) who treated the electrolysis of a salt with a large
excess of a "neutral" electrolyte (CuS04 with an excess of
H~SO4). Wagner's derivation indicated the concentration of
H + ions to increase at the cathode surface by ~/~ of the decrease of that of the Cu++-ion, and the S0%-ion concentration
to decrease by the same factor. Using von Karman's (25)
analysis of the boundary layer formed under free convection
he derived for the limiting current density at the height X
from the lower edge of the electrode

(IL)x = 4FD(cb)~ X-I


126-D,vOcc.so, --

1.422 01up ~I


Natural convection has also been treated by Agar (19)

who utilized a theoretical solution for the heat loss from a
vertical plate derived by Schmidt, et al. (26) and obtained a
result which may be expressed for limiting current density
as follows:

(IL)x = 0.394 nFD

\ re] X -~ Cb~.


A further expression was derived by Keulegan (17), which

yields for limiting current density:

(It.)x = 0,471 nFD


X -~ cb~.


On the basis of experimental studies for deposition of

CuSO4 from sulfuric acid solutions the authors (27) have
found the following equation to be applicable:

(IL)x = 0.505

[g(p0 --e,)r X-"

\1)] L








r.+ LbH~j


where ac~so~ and an2so ~ = density coefficients for CuS04 and

H2S04 respectively; and VH+ = transference number for
hydrogen ion.
Thus it may be seen that all of the theories agree in essential form, differing primarily by a numerical constant. It
should be noted that only equations [39] and [42] include the
effects of migration of nonreacting species on the density
difference established between the bulk solution and the
The total limiting current density (IL)~g, at an electrode
of height Xo is obtained by the relation:

(IL)~ = Xoo

(I~.)x dz =



Some authors (13) have attempted to treat free convection

by assuming a constant diffusion layer thickness ~ over the
electrode of 0.05 em. The foregoing theoretical analyses which
have been substantiated by experimental evidence indicate,
however, that ~ depends on the electrolyte composition,
viseosib , diffusion constant, density coefficient, and nature
of the electrode reaction in regard to initiation of density
changes, electrode height, shape, and orientation of the electrode in the solution. Unfortunately use of 6 = 0.05 em, which
can lead to serious error in prediction of limiting currents,
has been furthered in much of the subsequent literature on
polarization phenomena and textbooks of electrochemistry
(28, 29). The concept of diffusion layer with this numerical
value is frequently presented without adequate qualification.


or in a general form

(IL)x = 2nFD(cb) ~X -I [ 1-f~'g

December 1952


~ Ki OciO
In p Il ]


where: D = diffusion coefficient of the species participating

in the electrode reaction; c0 = bulk concentration of transferred component; p = density of the solution; and Ki =
dimensionless constants calculated by equations analogous to
[33] to [40] of the original reference.
Wagner obtained a fairly good agreement of his equation
with experiments on a 0.1M H2S04 system.

As described above, the early theoretical developments

have at best provided treatment for ideal cases of diffusional
supply to the electrode. The Nernst concept of a stationary
diffusion layer was fruitful in this early period for a qualitative understanding of the mechanism of ionic transfer in the
vicinity of the electrode. However, the diffusion layer concept
is merely a convenient form for describing the mass transfer
to the electrodes, and its thickness represents no real physical
constant. Misunderstanding of this limitation has often led
to erroneous interpretation of experimental work.

Vol. 99, No. 12



I t is unfortunate that the theory of Prandtl's boundai~

layer, patterns of velocity distribution, and other aspects of
hydrodynamic theory have not been introduced into the field
of electrochemistry until recently. Mass, heat and momentum
transfer analogies, and dimensional analysis are valuable
tools in the development of modern theories of electrolysis.
The use of concentration instead of activity in the electrochemical equations may lead to serious errors due to large
changes of the activity coefficients with concentration. The
form of some equations, e.g., those accounting for diffusion
only, may still be acceptable if one merely considers the
symbols for concentration as representing activity. However,
when mass transfer analogies are carried over to electrolysis,
the distribution of concentration is obtained. Therefore, the
prediction of the concentration polarization (which is a function of the activities) is not possible unless the activity coefficients are known for the entire range of concentrations.
A further major difficulty arises from the fact that the methods available at present for activity measurements yield
the mean activity coefficients and not the activity coefficient
of the potential-determining ion. The appreciation of these
facts is essential for future theoretical and experimental
developments in this field.
The variation of the diffusion coefficient with concentration
becomes a serious matter when high currents are applied. For
this reason the validity of the now available relations for
limiting currents, derived with the diffusion coefficient assumed constant, is subject to question. For a given system
the functional dependence of D on concentration can be
established in general only by experiment. Such diffusivity
measurements must be performed under conditions where
only the potential-determining ion diffuses under its own
concentration gradient.
In conclusion it may be stated that the recent developments
which have been reviewed in this paper represent a major
step toward establishment of quantitative theoretical methods
for the prediction of mass transfer effects. This in turn should
facilitate a more complete understanding of other aspects of
1. B. LEvicn, Acta Physicochim. U.R.S.S., 17,257 (1942).
2. W. NERNST, Z. physik. Chem., 47, 52 (1904).
3. H. F. WEBER, Wied. Ann., 7,539 (1879).



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