Electrochemical problems
Equilibrium: no electric current, no spatial nor temporal changes in concentrations.
• Nernst equation applies
• Transport equations not needed, c ≠ c(x,y,z,t),
Stationary state: electric current is constant, no temporal changes in
concentrations.
• 2c = 0, I ≠ I(t), c ≠ c(t), c = c(x,y,z)
• At the electrode, either Nernst equation or a kinetic equation (Butler-Volmer) as
the boundary condition of the transport equation.
Transient: concentrations change as a function of both time and position.
• Fick’s 2.law, I = I(t), c = c(x,y,z,t), c  D 2c
t
x 2
• In addition to boundary conditions, also initial conditions, c(x,y,z,t=0) = c0.
Obs.: Electric current is continuous everywhere, I ≠ I(x,y,z)
Transport equations

zF

Jk   Dk ck  k Dk ck   v ck
RT
diffusion + migration + convection
H
~k
Jk  Lk 
Nernst-Planck equation

v  velocity of the solution
Phenomenological equation, Lk = phenomenological coefficient
The driving force of transport is the gradient of the electrochemical potential.
~k  
~ 0k  RT ln  k ck  zk F  Vk p

H
Jk  Lk RT  ln ck  RT  ln  k  zk F   Vk p 
X
Lk 
Dk ck
RT
H
  ln  k  zk F
 
Jk  Dk ck  1 
Dk ck 

ln
c
RT

k
non-physical!
Convection term in Nernst-Planck equation means a change from Hittorf reference

system to a fixed laboratory coordinates; velocity v is due to mechanical forces and
 
has to be calculated using Navier-Stokes equation. v  v ~ k .




Hittorf: JkH  ck vk  v1  ; v1  solvent (water) velocity
Laboratory coordinates:

 H
 H

Jk  ckvk  Jk  ckv1  Jk  ckv
All the transport quantities (diffusion coefficient, transport number, conductivity)
are defined only in Hittorf reference, but experiments are done in fixed laboratory
coordinates. Strictly speaking, phenomenological equations should be written in
the center-of-mass coordinates (barycentric reference), but in practice it is the
same as the Hittorf reference.
Solution flow is due to recycling, viz pumping, (electro)osmosis or stirring.
Calculating it can be very difficult, because Navier-Stokes equation can be solved
in closed form only in some special cases.
Do activity coefficients matter?
Figure: Mean acitivity coefficient of
NaCl in the concentration range of
5 mM - 1 M. A single ionic diffusion
coefficient cannot be measured,
but it must be estimated with a
non-thermodynamic procedure.
As can be seen from the figure above, in a binary system an error can be as high as 10 %.
Since in multicomponent systems the estimation of activity coefficients is, however, very
difficult, they are usually ignored. Yet, in the derivation of Nernst-Planck equation a number
of simplifying assumptions have been made, such as ignoring the mutual coupling of ionic
fluxes; hence, ignoring the activity coefficient term is not that serious. Also, Nernst-Planck
equation has proved to be very useful and workable. More rigorous approaches are
hopelessly complicated and include several parameters the values of which are unknown.
Analysis of transport equations
Each species has its own Nernst-Planck equation. Constraints:
• electroneutrality
 z k ck  0
• electric current density
i  F  zk Jk
k
k
Binary system = salt + water:
  J   c 

 D

 J
    c  
 D
zF
v
c   
c
RT
D
z F
v
c   
c
RT
D
+
u
J
J
u 
     u  u c      vc
D D
 D D 
c+ = u+c ja c = uc  z+u+ + zu = 0
z
u
   
z
u
J 
i
z F
 1
u
u 
i
u 
 J     
 u  u c      vc
 D uD  zDF
 D D 

z
i
u
J 
  J
z
z F u
D
i
 J   u  u DD c 
 vc

 
uD  uD
zD  zD F

u  u DD
D
i
 J   
c  
 vc

uD  uD
zD  zD F
 J  D c  t  i  vc



 
zF

t i
 J  Dc    vc

z F
D 
u  u DD
uD  uD

D±, t+ ja t− are concentration independent constants,
which apples only in binary systems! They can be
measured, contrary to ionic diff. coefficients.
u  u u u


D
D D
D± = salt diffusion coefficient
Nernst-Hartley equation
Transport number:
z2 D c
z2 D u
z  z uD
zD
t  2



zD ck  z2 Dc z2 D u  z2 D u z u zD  zD  zD  zD
t+ + t− = 1 or in general
 tk  1.
k
Q.E.D.
1:1 electrolyte (e.g. NaCl, c+ = c− = c):
 J   2DD c  D i  vc   D c  t  i  vc

 
D  D
D  D F
F

2D D
D i
t i
 J      c 
 vc   Dc    vc

D  D
D  D F
F
Ionic diffusion coefficient can be calculated by measuring the salt diffusion coefficient
and the transport number in the binary system:
D

t
u  u DD 
uD   u 
   1  D
1


uD   u 
 u D 
uD  1     
 u D 
1:1 electrolyte
D
D
D 
ja D  
21  t  
2t 
ja
2:1 electrolyte (CaCl2)
D 
D
2D
ja D  
31  t  
3t 
D  u 
  1  D
t   u 
1:2-electrolyte (Na2SO4)
D 
2D
D
ja D  
31  t  
3t 
Potential gradient in the solution in a general case:
Nernst-Planck equation:
dc
d
F
 Ji  Di  i  zi fci  ; f 
dx 
RT
 dx
ci d ln ci  dci :
Multiply with zi and sum:
i
dc
d
  zi Ji     zi Di i  f  zi2Di ci
F i
dx
dx i
i
d
RT
 2 i

dx
F
F
2
z
D
c
 i ii
RT i
d
i RT
 
dx
 F

i
 ziDi ci
ti d ln ci
zi dx
i
  ziDi ci d ln ci  f d  zi2Di ci
i
dx
dx
i
d ln ci
dx
 zi2Di ci
i
F2
  conductivity 
zi2Di ci

RT i
zi2Di ci
concentration dependent
ti 

 zi2Di ci transport number
i
d
i
RT
 

dx

F

i
ti d ln ci
zi dx
diffusion potential, reversible, usually very
low, as all ionic mobilities are of the same
order of magnitude (except H+ and OH−)
ohmic loss, irreversible,
dissipates into heat
In electrochemical experiments, an inert electrolyte that does not react on the electrode is
usually added in excess for two reasons:
• conductivity increases, making ohmic loss insignificant
• transport number of the electroactive ion reduces practically to zero, reducing N-P
equation to Fick’s law, see below.
A transport number can formally be written also as (no convection)
Jk   Dk ck 
tk i
zk F
if, again, ionic coupling is ignored. But since the transport number depends on concentrations
and changes, therefore, as a function of position, the equation above cannot be solved. Since
for a trace-ion tk ≈ 0, its N-P equation becomes Fick’s law: Jk   Dk c
Time dependent transport equation:


}=0
}=0

ck
zF
   Jk  Dk 2ck  k Dk ck    ck 2  ck   v  v  ck 
t
RT
Poisson equation: 2 = −re/e; re = charge density, e = permittivity = ere0. In an electroneutral system re = 0.
In an incompressible fluid ∙v = 0.
ck
zF
 Dk 2ck  k Dk ck    v  ck
t
RT
If it is arranged so that  ≈ 0 ja v = 0
ck
 Dk 2ck
t
Fick’s 2. law
For a trace-ion, the above equation can be solved using the boudary condition
 Jk 0 
i
zk F
 Dk ck 0
Additionally, on the electrode, either Nernst equation (reversible reaction) or a kinetic
equation is needed.