Electrochemistry

Electrochemistry
By: Getachew Y.
18/10/2016

Presentation- II-
 Redox Reactions and basics of electrode-solution interface
 Nonfaradaic process and charge transfer
 Faradaic Process and types Electrochemical cell.
Faradaic Processes and Factors Affecting Rates of Electrode Reactions
Potentials and Thermodynamics of Cells
Kinetics of Electrode Reactions
Mass Transfer by Migration And Diffusion

Gibbs free energy
 • from the First Law of Thermodynamics and some standard
thermodynamic relations. We find
"Useful" work is that which can be extracted from the cell by electrical means
to operate a lamp or some other external device.
dU  dqdw dq = T dS
dw  PdV  dwelectrical
dHP  dUP  PdVdU  T dS  PdV  dwelectrical
dGT  dHT  T dS
 dUT,P  PdV  T dS
 T dS  PdV  dwelectrical  PdV  T dS
dGT,P  dwelectrical
the electrical work at constant
pressure and temperature,
under reversible conditions,
is the free energy change of
the reaction

• By convention, we identify work which is negative with work which is being
done by the system on the surroundings. And negative free energy
change is identified as defining a spontaneous process.
GT,P  welectrical  n F E
• Note how a measurement of a cell potential directly calculates the Gibbs
free energy change for the process.
welectrical  V Q
since Q  n F
 nF E
Electrical work is just the amount of charge Q and the potential V through
which we move it.
Cont.…

The propensity for a given material to contribute to a reaction
is measured by activity, a.
How “active” is this substance in this reaction compared to how it
would behave if it were present in its standard state?
• activity scales with concentration or partial pressure.
a  C/C˚ OR a  P/P˚
Definition of activity a  
C
C
a  
P
P
Activity coefficients close to 1 for dilute solutions and low partial pressures.
• it changes with concentration, temperature, other species, etc.
Cont.…
In order to analyze a chemical process mathematically, we form
this reaction quotient. wAxB  yC zD
Q 
aC
y
aD
z
aA
w
aB
x

When all participants have unit activity (a=1), then Q=1 and ln Q = 0.
This special Q* (the only one for which we achieve this balance) is
renamed Keq, the equilibrium constant.
G  G  RT lnQ
G  G
Reaction proceeds, Q changes, until finally G=0. The reaction stops.
This is equilibrium.
0  G  RT lnQ*
 G  RT lnQ*
Q*
 Keq
The free energy of the system changes as temperature and concentration
of the species are changed from the standard state using :
Cont.…
dynamic equilibrium

Nernst Equation
from the expression of the Gibbs dependence
on activity and turn this around for an
expression in terms of the cell potential.
G  G  RT lnQ
The relation between cell potential E and free energy gives
n F E  n F E  RT lnQ
Rearrange and obtain the Nernst Equation.
E E 
RT
nF
lnQ At T = 25 °C
E E 
0.0257
n
lnQ
E E 
0.0592
n
logQ
“n” – the number of moles of electrons transferred in the process
according to the stoichiometry chosen.
Walther Nernst
1864-1941

Nernst and Half-Cells
The Nernst equation can be accurately applied to the half cell
reactions. The same rules of “products over reactants” applies to
forming the activity ratio in the logarithm. The number of electrons is
as specified by the stoichiometry.
Example : Cd2+(aq) + 2e–  Cd(Hg)
ECd2
/Cd
 E
Cd2/Cd

RT
2F
ln
aCd
a
Cd
2







ΔG°
∆𝑯 𝟎
, ∆𝑺 𝟎
Electrochemical
celldata𝑬𝟎
𝒄𝒆𝒍𝒍
Equilibrium
constantan
ΔG° = -nFE°cell
Reaction Parameters at the Standard State
ΔG° Q E°cell
Reaction at standard-state
conditions
< 0 > 1 > 0 spontaneous
= 0 =1 = 0 at equilibrium
> 0 < 1 < 0 nonspontaneous
• When Q < 1, [reactant] > [product], ln Q < 0, so Ecell > E°cell
• When Q = 1, [reactant] = [product], ln Q = 0, so Ecell = E°cell
• When Q > 1, [reactant] < [product], ln Q > 0, so Ecell < E°cell
Ecell = E°cell - ln Q
RT
nF
Summery of interrelationship of G°, E°cell, and Q.
Cont.…

v 
i
nFA
kinetically
controlled
mass-transfer
controlled
Do,
no kinetic info
Transition-state
theory
variation of activation
energy by electrode
potential
Butler-Volmer
equation
Tafel Plot
ko, io, 
chemical
reactions
Faraday’s Law

Bulk concentration of electroactive
species (Co, CR) Concentrations of
other species (electrolyte, pH,...) Solvent
Material Surface
area (A)
Mode (diffusion,
convection,...) Surface
concentrations
Adsorption
Temperature (T)
Pressure {P)
Time (t)
Potential (E)
Current (i)
Quantity of
electricity (Q)
VElectrode variables
Mass transfer variables
Solution variables
Electrical variables
External variables
An investigation of electrochemical behavior consists of holding certain variables of an
electrochemical cell constant and observing how other variables (usually current, potential,
or concentration) vary with changes in the controlled variables. parameters of importance in
electrochemical cells are shown
The Electrochemical Experiment and Variables in Electrochemical Cells
we will focus
Mode (diffusion,
convection,...)
Surface concentrations
Adsorption
Concentration of species
Potential (E)
Current (i)
Quantity of
electricity (Q)
Rate and current at the interface

Potential Drop Across The Electrochemical Cell
 When we apply a potential in any electrochemical cell, there will be a voltage drop across
the interface, solution, and reference electrode .
 𝑽𝒂𝒑𝒑 = 𝑽𝒆𝒍𝒆𝒄 + 𝑽𝒔𝒐𝒍𝒖 + 𝑽𝒓𝒆𝒇,
the voltage drop across the the solution = 𝐢 𝐑𝐬
 In constructing an ideal electrochemical cell we must reduce/minimize Vref, Vslou,
and we need the potential drop across the solution and electrode interface;
♦ Vref is must be designed to have zero voltage drop , we can be made it small by
making ideally polarized electrode.
♦ Vsolu can be reduced by reducing the solution resistance ,adding high concentration of acid..

  uCHg2Cl2HgsatKCl1M3NO,TlTlCu 

𝑇𝑙+
+ ҧ𝑒 ⇌ Tl , 𝐸0
= 0.336 V
SCE , 𝐸0
= 0.242
 What would happen if we apply the same
potential using power supply ?
• Example , if 0.572 V applied , then i=0
because there is no driving force for the
reactions .
 If V < 0.572𝑉 𝑖𝑠 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 we will get
Oxidation current (anodic current )
 If V > 0.572V reductive current or cathodic
current.
we know that current flow in the cell is directly related to the electron transfer at the
interface.
So that 𝑖 =
𝑟𝑎𝑡𝑒 𝑜𝑓 𝑓𝑙𝑜𝑤 𝑜𝑓 𝑐ℎ𝑎𝑟𝑔𝑒
𝑡𝑖𝑚𝑒
=
𝑑𝑞
𝑑𝑡
and

𝑄
𝑛𝐹
= N where N is moles of materials electrolyzed. Where F is Farday
 The amount of electricity flowing in the system related to the amount of chemical
change happing in the system.
 The rate 𝑣
𝑚𝑜𝑙
𝑠𝑒𝑐
=
𝑑𝑁
𝑑𝑡
=
𝑑
𝑑𝑡
𝑄
𝑛𝐹
=
𝑖
𝑛𝐹
(for homogeneous rxn )
Cont..
Let an arbitrary V=0.65 V is
applied, so that what current will
flow?

R (reduced
species)
O (oxidized
species)
- e
O + ne- ⇄ R
(electrode reaction)
electrode solution Interface
nF
j
AnF
i
AnFt
q
cmmolv 
1
d
d
).sec/( 2
Heterogeneous reactions occur at the electrode-
solution interface, and they are characteristic of
electrochemistry, the rate depends on area of the
phase boundary where the reaction occurs:
Since electrode reactions are heterogeneous, their reaction rates are
usually described in units of mol/s per unit area; that is,
E
i
E
i
E
i 𝜂
a)
c)
a) Ideal polarized, b) ideal non polarized, c) over potential=E-Eeq
Electrode Reaction and Factors Affecting Electrode Reaction
Electrode

Consider an overall electrode reaction, O + ne- ⇄ R (electrode reaction) the current (or electrode
reaction rate) is governed by the rates of processes such as a series of steps that cause the
conversion of the dissolved oxidized species, O to a reduced form R in solution
To make electron transfer occur, all of these transfer system play an important role.
Electrode Reactions can be controlled by, Mass transfer control and kinetic control
In/out of electron
𝒏ത𝒆
Molecules that stuck on the
electrode surface can under
go redox by transferring
electrons
non adsorbed species
Interface Bulk solutionElectrode
Cont..
The magnitude of the current is limited by the slowest process: rate-determining step

So that Over potential is a couple of three things for the case above process
Over potential can be due to:
 Mass transfer = 𝛈 𝐦𝐭 or we call it concentration polarization
 Electron transfer = 𝛈 𝐞𝐭 or charge transfer over potential (activation polarization )
 Chemical reaction = 𝛈 𝐫𝐱𝐧 chemical reaction polarization
(all of these process will shift the equilibrium potential of a certain chemical
reaction)
All the above system will have an over all rxn that needs some energy to over come the
barrier for electron transfer. So that we need extra amount of energy to overcome these
effect, and we call this potential over potential.
For this reaction to proceed, O is required to move from the bulk solution near the
electrode surface. The mechanism is related to mass transfer and is governed by
equations such as Fick’s laws of diffusion and Nernst–Planck. Mass transfer from
the bulk solution towards the electrode surface could limit the rate of the reaction if all
of the processes leading to the reaction are fast, this leaves the electron transfer reaction
as the limiting factor.
Cont..

Simple case of electron transfer
Let consider the reaction limited by mass transfer only. Which means the rates of
charge transfer is limited by the rate of mass transfer.
𝑡 = 0
CO
CR
𝑡 = 𝑡1
𝑡1 > 𝑡 = 0
Draw of concentration vs distance from
the electrode to the bulk solution. (Only
to show the concentration change at
the Electrode interface )
XO
MASS-TRANSFER-CONTROLLED REACTIONS
 The rate of all chemical reactions is very rapid compared to the rate of mass transfer
processes.

When we apply potential to the system, the initial concentration changes, and species
start to move from the bulk solution to the electrode surface. After Redox reaction occur
at the electrode, the product species will be maximum and the reactant species will
reduce in concentration like shown in above concertation figure.
The amount of species that moves towards to a certain area (A) per unit time is called
flux(J) 𝑱 𝒙, 𝒕 =
𝒎𝒐𝒍/𝒔𝒆𝒄
𝒄𝒎 𝟐 which is also called the rate 𝒗 =
𝒊
𝒏𝑭𝑨
(
𝒎𝒐𝒍/𝒔𝒆𝒄
𝒄𝒎 𝟐 )
The simplest electrode reactions are those in which the rates of all associated chemical
reactions are very rapid compared to those of the mass-transfer processes. Under
these conditions, the chemical reactions can usually be treated in a particularly simple
way, and the surface concentrations of species involved in the faradaic process are
related to the electrode potential by an equation of the Nernst form.
So that , the net rate of the electrode reaction is then governed totally by the rate at
which the electroactive species is brought to the surface by mass transfer, vmt hence:
nFA
i
vv mtrxn 
(Such electrode reactions are often called reversible or nernstian, because the principal
species obey thermodynamic relationships at the electrode surface)
Cont..

Mass transfer to an electrode is governed by the Nernst-Planck equation, written for one-
dimensional mass transfer along the x-axis as
𝑱𝒊 𝒙 = −𝑫𝒊
𝝏𝑪𝒊 𝒙
𝝏 𝒙
−
𝒛𝒊 𝑭
𝑹𝑻
𝑫𝒊 𝑪𝒊
𝝏∅ 𝒙
𝝏𝒙
+ 𝑪𝒊 𝒗 𝒙
where Ji(x) is the flux of species i (mol /s/cm2) at distance x from the surface, Di is
the diffusion coefficient (cm2/s),
𝝏𝑪 𝒊 𝒙
𝝏 𝒙
is the concentration gradient at distance x,
𝝏∅ 𝒙
𝝏𝒙
is the potential gradient, zi and Ci are the charge (dimensionless) and
concentration (mol cm-3) of species i, respectively, and v(x) is the velocity (cm/s) with
which a volume element in solution moves along the axis.
diffusion, migration, and convection,
If the mass transfer is the slowest step of the electrode reaction, then the electrode reaction is
termed as being “electrochemically reversible”. At each potential difference (E) of the
interface, the electrode reaction is in redox equilibrium, which is described by the Nernst equation:
0x
0x'0
[R]
[O]
ln



nF
RT
EE
  )(equlibrumat1
),(
,
E
0

txC
txC
R
O
Cont..

Semiempirical Treatment of Steady-State Mass Transfer

OC
Electrode
0
o
)
d
dC
(  xo
x
Dv





 


)0(*
xcc
Dv oo
o








o
oooo
D
mxccm
nFA
i
));0(( *
O + ne- ⇄ R





 


)0(
D
*
0
xcc
nFA
i oo
The rate of mass transfer is
proportional to the concentration
gradient at the electrode surface,
For linear concentration
Gradient (1D)
mo, called the mass-transfer coefficient has units
of cm/s (we call it heterogeneous rate constant)
𝑖 𝑖𝑠 𝑝𝑜𝑠𝑡𝑖𝑣𝑒 𝑏𝑐 𝐶 𝑜
∗
> 𝐶 𝑜 (x=0)
Lets consider the species O only
0
O
)
d
dC
(  x
x
v
x
)(*
R bulkC
)(*
O bulkC)0( xCO
)0(R xC
The flux at the electrode (i.e., the rate of the electrode reaction, thus the current),
depends on the diffusion rate only (i.e., depends on the mass transfer only). According
to the First Fick law, the rate of diffusion depends on the diffusion coefficient (D) and
the concentration gradient (dc/dx); (D – diffusion coefficient (it is the rate constant of
the diffusion (cm 2 s-1)).









O
RRRR
D
mcxcm
nFA
i
);)0(( *
)0(  xCm
nFA
i
RR
*
oo
l
cm
nFA
i

0.)concbulk(*
R C
The maximal flux of O will be if CO(x =
0) = 0. Thus, the corresponding
current is termed limiting current, il
Under the conditions of a net cathodic reaction, R is produced at the electrode surface,
so that 𝐶 𝑅(𝑥 = 0) > 𝐶 𝑅
∗
(where 𝐶 𝑅
∗
is the bulk concentration of R). Therefore,
If the potential is high enough, the values of 𝐶𝑜(𝑥 = 0) and 𝐶 𝑅(𝑥 = 0) are
functions of electrode potential, E. The largest rate of mass transfer of О occurs when
𝐶𝑜(𝑥 = 0) = 0 (or more precisely, when 𝐶𝑜(𝑥 = 0) < < 𝐶𝑜∗
, so that 𝐶 𝑜
∗
− 𝐶𝑜(𝑥 =
0) ~ 𝐶 𝑜
∗
). The value of the current under these conditions is called the limiting
current, , where
no R in the bulk solution
li
*
ool cnFAmi 
When the limiting current flows, the electrode process is occurring at the maximum rate possible
for a given set of mass-transfer conditions, because О is being reduced as fast as it can be
brought to the electrode surface.
R
R
nFAm
i
xC  )0(
Cont..

))0(( *
 xccm
nFA
i
ooo
*
ool cnFAmi  o
l
o
nFAm
ii
xc

 )0(
Thus, the concentration of species О at the
electrode surface is linearly related to the
current and varies from 𝐶 𝑜
∗
, when i = 0, to a
negligible value, when i = 𝑖𝑙.
If the kinetics of electron transfer are rapid, the concentrations of О and R at the electrode
surface can be assumed to be at equilibrium with the electrode potential, as governed
by the Nernst equation for the half-reaction
0x
0x'0
[R]
[O]
ln



nF
RT
EE Such a process is called a nernstian reaction
We can derive the steady-state i-E curves for nernstian reactions under several different conditions.
I. R Initially Absent
When 𝐶 𝑅
∗
=0, can be obtained
R
R
nFAm
i
xc  )0(
o
l
nFAm
ii
xc

 )0(0









)0(
)0(
ln
R
O'0
xc
xc
nF
RT
EE a)(..........lnln'0





 

i
ii
nF
RT
m
m
nF
RT
EE l
O
R
)0( xCR
Using this, and
Cont..

II. Both О and R Initially Present
When both members of the redox couple exist in the bulk, we must distinguish between a
cathodic limiting current, 𝑖𝑙,𝑐, when 𝐶 𝑜(𝑥 = 0) ≈ 0, and an anodic limiting current, 𝑖𝑙,𝑎
when CR(x= 0)= 0. We still have 𝐶 𝑜 (x = 0) given by;
The limiting anodic current naturally reflects the maximum rate at which R can be brought to the
electrode surface for conversion to O.
cl,l ibyreplacedibut with)0(
o
l
o
nFAm
ii
xc


*
, RRal cnFAmi 
The negative sign arises because of our convention that
cathodic currents are taken as positive and anodic ones
as negative
Thus CR(X = 0) is given by
R
cl
R
nFAm
ii
xc
,
)0(


alR
R
i
i
C
xc
,
*
1
)0(


)...(..........lnln
)0(
)0(
ln'
,
,0'0
R
O0
b
ii
ii
nF
RT
m
m
nF
RT
EE
xc
xc
nF
RT
EE
al
cl
R














o
Cl
o
nFAm
ii
xc

 ,
)0(
Cont..

A plot of two of these equations are shown below
Current-potential curve for a nernstian
system involving two soluble species
with both forms initially present.
Current-potential curve for a nernstian
reaction involving two soluble species
with only oxidant present initially











al
cl
R ii
ii
nF
RT
m
m
nF
RT
EE
,
,0'0
lnln








ii
i
nF
RT
m
m
nF
RT
EE
lO
R
lnln'0
When i = 0, E = Eeq and the system is at equilibrium. Surface concentrations are then equal to
the bulk values. When current flows, the potential deviates from Eeq, and the extent of this
deviation is the concentration over potential.
Cont..

Semi empirical Treatment of the Transient Response
The steady state case employed in an approximate way to time dependent (transient)
phenomena, for example, the buildup of the diffusion layer, either in a stirred solution (before
steady state is attained) or in an unstirred solution where the diffusion layer continues to grow
with time.





 


)0(
D
*
0
xcc
nFA
i oo
0)(  t
This approximate treatment predicts a
diffusion layer that grows with 𝑡1/2
and
a current that decays with 𝑡−1/2
x
𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑛𝑔 𝑓𝑜𝑟 𝑖 𝑟𝑒𝑠𝑢𝑙𝑡𝑠
In the absence of convection, the current continues to decay, but in a convective system, it
ultimately approaches the steady-state value characterized by





 

)(
)0(
2
D *
1/2
1/2
o
t
xcc
tnFA
i oo

Moles of О electrolyzed =
in diffusion layer
  
t
nF
idttA
xCC
0
00
2
)(
)0(


 Chemically reversibility consider 𝑃𝑡|𝐻2|𝐻+, 𝐶𝑙−
|𝐴𝑔𝐶𝑙|𝐴𝑔, 𝐸 = 0.222 𝑉
Overall reaction 𝐻2 + 2𝐴𝑔𝐶𝑙 ⇋ 2𝐴𝑔 + 2𝐶𝑙 − + 2𝐻 + may reverse the reaction
upon the application of an outside voltage of 0.222 V/
(Reversing the cell current merely reverses the cell reaction. No new reactions appear, thus the
cell is termed chemically reversible. But if the net process upon current reversal give new or
different reaction hence this cell is said to be chemically irreversible)
 Thermodynamic Reversibility (fast ET kinetics)
• Achieve thermodynamic equilibrium, Can be readily reversed with an
infinitesimal driving force
• Concentration profiles follow Nernstian equation
Reversibility







RC
Co
nF
RT
EE ln0
POTENTIALS AND THERMODYNAMICS OF CELLS
If a system follows the Nernst equation or an equation derived from it, the electrode reaction is
often said to be thermodynamically or electrochemically reversible (or nernstian).
Thermodynamic parameters ∆𝑆 = −
𝜕∆𝐺
𝜕𝑇 𝑃
= 𝑛𝐹
𝜕𝐸
𝜕𝑇
∆𝐻 = ∆𝐺 + 𝑇∆𝑆
∆𝐻 = 𝑛𝐹 𝑇
𝜕𝐸
𝜕𝑇
− 𝐸
∆𝐺 = −𝑛𝐹𝐸 , ∆𝐺0
= −𝑛𝐹𝐸0
= −RTln𝐾𝑒𝑞
useful for predicting electrochemical
properties from thermochemical data
Rxn thermodynamics
determines the
electromotive force of
the cell

Formal Potentials
It is usually inconvenient to deal with activities in evaluations of half-cell potentials,
because activity coefficients are almost always unknown. A device for avoiding them is
the formal potential, This quantity is the measured potential of the half-cell (vs. NHE)
when
(a) the species О and R are present at concentrations such that the ratio 𝐶 𝑂
𝑉0
/𝐶 𝑅
𝑉𝑅
is
unity and
(b) other specified substances, for example, miscellaneous components of the
medium, are present at designated concentrations.
𝑭𝒆 𝟑+
+ ത𝒆 ⇋ 𝑭𝒆 𝟐+
)
Fe
Fe
ln( 3
3
0


 

nF
RT
EE
 
 





 

22
33
0
ln
FeFe
FeFe
nF
RT
EE


 
 





 

2
3
'0
ln
Fe
Fe
nF
RT
EE 





 

2
3
0'0
ln
Fe
Fe
nF
RT
EE


So that the formal potential incorporates the standard potential and some activity coefficients 𝛾𝑖,
Cont..
formal potential

At equilibrium (null current), all conducting phases exhibit an equipotential surface;
that means, the potential difference only occurs at the interface.
Inner potential (𝜙) is the potential that we measured inside the phase. The charge that
provides the inner potential resides on the surface like shown on the sphere. In our case
if we consider an electrode immersed in solution then charges will surrounded the
electrode and will create a potential differences.
Cont..Interfacial Distribution of Potential
{The ultimate sources of inner potential is the presence of excess charges }

So that charges for the inner potential can be due to:
Charges that arises due to the electronic properties of materials(electrons and holes)
 Ions at interfaces for example a Pt electrode in contact with chloride ion, the chloride
ions will be absorbed at the platinum surfaces
 Electrostatic filed
- +
- +
- +
Pt H2O molecules
Cont..
 Dipole a the interfaces. for example a Pt electrode in
contact with water molecules the water molecules will
orient themselves in the direction of the platinum electrode
and form dipole
Interfacial Distribution of Potential
Zn Cu
 If we have two conductors with different fermi
level, electron will transfer to balance the fermi
level. And the electron can flow from high fermi
level to low fermi level.
+
+
+
+
-
-
-
-
And we will get interfacial potential
difference at the interface
If we have
• M/semiconductor
• M/solution
• Solution/solution
We will have
interfacial potential
For example 1M HCl 0.01M HCl
Because of concentration gradient we will have
potential and we call this Liquid Junction potential

Cu
Zn
Electrolyte
Ag
Cu’
Distance across the cell
𝝓
Potential profile across a whole cell at equilibrium.
 The difference in the inner potentials, Δ𝝓 of two phases in contact is a factor of
primary importance to electrochemical processes occurring at the interface
 It comes from the local electric fields reflecting the large changes in potential in the
boundary region .These fields can reach values as high as 107
𝑉/𝑐𝑚. They are large
enough to distort electro-reactants and to alter reactivity, and they can affect the
kinetics of charge transport across the interface.
 Another aspect of Δ𝝓 is its direct influence over the relative energies of charged
species on either side of the interface.
Cu|Zn|Zn2+, Cl-|AgCl|Ag|Cu’
The measured cell potential is a sum
of several interfacial differences,
none of which we can evaluate
independently Vetter's representation
still contain contributions from two separate interfacial potential differences.
( In this way, 𝜟𝝓 controls the relative electron
affinities of the two phases; hence it controls
the direction of reaction)

Electrochemical Potentials
From Cu|Zn|Zn2+, Cl-|AgCl|Ag|Cu’ , In solution, zinc ion is hydrated and may interact
with Cl-. The potential arising from such interaction is called chemical potential(𝚫𝝁). The
sum of chemical potential in a electrochemical cell is also gives free energy 𝚫𝑮.
Let 𝜇𝑖
𝛼
(the chemical potential of species i in phase 𝛼 𝝁𝒊
𝜶
= 𝝁𝒊
𝒐𝜶
− 𝐑𝐓𝐥𝐧 𝒂𝒊
𝜶
In addition, there is the energy required simply to bring the +2 charge, disregarding the
chemical effects, to some location . This second energy is clearly proportional to the potential
𝝓 at the location; hence it depends on the electrical properties of an environment very much
larger than the ion itself.
So that if we consider large scale interaction the total chemical potential will be arise due on
the ion-ion interaction and ions-electric filed interaction. To include both effect we use
Electrochemical potential ҧ𝜇𝑖
𝛼
=𝝁𝒊
𝜶
+𝑍𝑖 𝐹𝜙 𝛼
𝜇𝑖
𝛼
=
𝜕𝐺
𝜕𝑛𝑖 𝑇,𝑃.𝑛𝑗≠𝑖 𝐺 = ෍
𝑖=1
𝑛
𝜇𝑖
activities
Chemical potential at standard state
ҧ𝜇𝑖
𝛼
=𝝁𝒊
𝟎𝜶
+𝑅𝑇𝑙𝑛𝑎𝑖
𝛼
+𝑍𝑖 𝐹𝜙 𝛼
Where 𝑛𝑖 is the number of moles of 𝑖 in phase 𝛼.

Properties of the Electrochemical Potential
1. For an uncharged species: ҧ𝜇𝑖
𝛼
=𝝁𝒊
𝜶
2. For any substance:𝝁𝒊
𝜶
=𝝁𝒊
𝟎𝜶
+𝑅𝑇𝑙𝑛𝑎𝑖
𝛼
3. For a pure phase at unit activity ҧ𝜇𝑖
𝛼
=𝝁𝒊
𝟎𝜶
4. For electrons in a metal (z =1): ҧ𝜇𝑖
𝛼
=𝝁𝒊
𝟎𝜶
− 𝐹𝜙 𝛼
5. For equilibrium of species i between phases a and𝛽: ҧ𝜇𝑖
𝛼
=𝝁𝒊
−𝜷
Formulation of Cell Potential
 Zn + 2AgCl + 2e (Cu’)  Zn2+ + 2Ag + 2Cl- + 2e (Cu)
ҧ𝜇 𝑧𝑛
𝑧𝑛+2 ҧ𝜇 𝐴𝑔𝐶𝑙
𝐴𝑔𝐶𝑙
+2 ҧ𝜇 𝑒
𝑐𝑢′ = ҧ𝜇 𝑧𝑛2+
𝑠
+2 ҧ𝜇 𝐴𝑔
𝐴𝑔
+2 ҧ𝜇 𝐶𝑙−
𝑠
+2 ҧ𝜇 𝑒
𝑐𝑢
2( ҧ𝜇 𝑒
𝑐𝑢′ − ҧ𝜇 𝑒
𝑐𝑢)= ҧ𝜇 𝑧𝑛2+
𝑠
+2 ҧ𝜇 𝐴𝑔
𝐴𝑔
+2 ҧ𝜇 𝐶𝑙−
𝑠
− ҧ𝜇 𝑧𝑛
𝑧𝑛 −2 ҧ𝜇 𝐴𝑔𝐶𝑙
𝐴𝑔𝐶𝑙
But , 2( ഥ𝝁 𝒆
𝒄𝒖′
− ഥ𝝁 𝒆
𝒄𝒖
) = -2F(𝝓 𝑪𝒖′
− 𝝓 𝑪𝒖
) = -2FE
-2FE=𝜇 𝑧𝑛2+
0𝑠
+ RTln 𝑎 𝑍𝑛2+
𝑠
+ 2𝐹𝜙 𝑠
+ 2𝜇 𝐴𝑔
0𝐴𝑔
+ 2𝜇 𝐶𝑙−
0𝑠
+ 2𝑅𝑇𝑙𝑛 𝑎 𝐶𝑙−
𝑠
− 2𝐹𝜙 𝑠
− 𝜇 𝑧𝑛
0𝑧𝑛
− 2𝜇 𝐴𝑔𝐶𝑙
0𝐴𝑔𝐶𝑙
−2𝐹𝐸 = Δ𝐺0
+ 𝑅𝑇 ln 𝑎 𝑍𝑛2+
𝑠
(𝑎 𝐶𝑙−
𝑠
)2
𝑤ℎ𝑒𝑟𝑒, Δ𝐺0
= 𝜇 𝑧𝑛2+
0𝑠
+ 2𝜇 𝐴𝑔
0𝐴𝑔
+ 2𝜇 𝐶𝑙−
0𝑠
− 𝜇 𝑧𝑛
0𝑧𝑛
− 2𝜇 𝐴𝑔𝐶𝑙
0𝐴𝑔𝐶𝑙
= −2𝐹𝐸0
At equilibrium,
Expanding
  2
2
0
ln
2
S
Cl
S
Zn aa
F
RT
EE  which is the Nernst equation for the cell.

Liquid Junction Potential
many real cells are never at equilibrium, because they feature different electrolytes around the
two electrodes. There is somewhere an interface between the two solutions, and at that point,
mass transport processes work to mix the solutes. Unless the solutions are the same initially, the
liquid junction will not be at equilibrium, because net flows of mass occur continuously across it
 Potential differences at the electrolyte-electrolyte interface
 Cu|Zn|Zn2+|Cu2+|Cu’
E = (Cu’ – Cu2+) – (Cu – Zn2) + (Cu2+ – Zn2+)
Obviously, the first two components of E are the expected interfacial potential
differences at the copper and zinc electrodes. The third term shows that the measured
cell potential depends also on the potential difference between the electrolytes, that is, on
the liquid junction potential.

1. Two solutions of the same electrolyte at different concentrations, as in Figure
2. Two solutions at the same concentration with different electrolytes having an ion in
common, as in Figure b.
3. Two solutions not satisfying conditions 1 or 2, as in Figure c.
 Three major cases Liquid Junction Potential
Types of liquid junctions. Arrows show the direction of net transfer for each ion, and
their lengths indicate relative mobility's.
𝜶 phase 𝜶 phase
𝛂 phase
𝜷 phase𝜷 phase
𝜷 phase

Conductance, Transference Numbers, and Mobility
When an electric current flows in an electrochemical cell, the current is carried in solution
by the movement of ions. For example
Where 𝑎2 > 𝑎1When the cell operates galvanically, an oxidation occurs at the left electrode
𝐇 𝟐 → 𝟐𝐇(𝛂)
+
+ 𝟐𝐞(𝐏𝐭) and a reduction happens on the right, 𝟐𝐇(𝛃)
+
+ 𝟐𝐞(𝐏𝐭′
) → 𝐇 𝟐
Therefore, there is a tendency to build up a positive charge in the 𝜶 phase and a negative charge
in 𝜷. This tendency is overcome by the movement of ions: 𝐇+
to the right and 𝐂𝐥−
to the left.
For each mole of electrons passed, 1 mole of 𝐇+
is produced in 𝜶, and 1 mole of 𝐇+
is consumed
in β. The total amount of 𝐇+
and 𝐂𝐥−
migrating across the boundary between 𝛼 and 𝛽 must equal
1 mole.
The fractions of the current carried by 𝐇+
and 𝐂𝐥−
are called their transference numbers (or
transport numbers). If we let 𝑡+ be the transference number for H+ and 𝑡− be that for Cl−
, then
clearly, 𝑡+ + 𝑡− = 1
In general, for an electrolyte containing many ions, i,
෍
𝒊
𝒕𝒊 = 𝟏
ΤΤ⊖ 𝐏𝐭 𝐇 𝟐 𝟏 𝐚𝐭𝐦 𝐇+
, Τ𝐂𝐥−
𝐇+
, 𝐂𝐥−
∕ 𝐇 𝟐(𝟏 𝐚𝐭𝐦) ∕ 𝐏𝐭′
⊕
𝜶 𝜷𝑎1
𝑎1

Transference numbers are determined by the details of ionic conduction, which are understood
mainly through measurements of either the resistance to current flow in solution
or its reciprocal, the conductance, L..
L= 𝜿
𝑨
𝒍
The conductance, L, is given in units of Siemens. and к is
expressed in S cm-1
Since the passage of current through the solution is accomplished by the independent movement
of different species, к is the sum of contributions from all ionic species, i.
• The magnitude of the force exerted by the field is 𝑧𝑖 𝒆𝝃 where e is the electronic charge. The
frictional drag can be approximated from the Stokes law as 6𝝅𝜼𝒓𝒗 where 𝜂 is the viscosity of
the medium, r is the radius of the ion, and v is the velocity.
𝜿 = 𝑭 ෍
𝒊
𝒛𝒊 𝒖𝒊 𝑪𝒊
Direction of movement
Drag force Electric force
When a field of strength 𝝃 is applied to an ion, it will accelerate under the force imposed by the
field until the frictional drag exactly counterbalances the electric force. Then, the ion continues
its motion at that terminal velocity. The forces balance at the terminal velocity.
𝒖𝒊 =
𝒛𝒊 𝒆
𝟔𝝅𝜼𝒓
The transference number for species i is merely the
contribution to conductivity made by that species
divided by the total conductivity:
𝒕𝒊 =
𝒛𝒊 𝒖𝒊 𝑪𝒊
σ𝒋 𝒛𝒋 𝒖𝒋 𝑪𝒋

Essentials of Electrode Reactions
Reactions can be visualized in terms of progress along a reaction coordinate connecting a
reactant configuration to a product configuration on an energy surface.
Standardfreeenergy
Reaction coordinate
product
Reactant
 Simple representation of potential energy changes
during a reaction.
Decreasing rate cons.
K is decreasing
But over potential is
increasing
𝑖 ∝ 𝑒−𝑏𝐸
Most of the electrode transfer process are
not reversible at time scale,
Kinetic controlled reactions
Arrhenius was first to recognize
the generality of this behavior,
and he proposed that rate
constants be expressed in the
form 𝒌 = 𝑨𝒆
−𝑬 𝑨
𝑹𝑻
EA the activation energy, A is constant

 The idea of activation energy has
led to pictures of reaction paths in
terms of potential energy along a
reaction coordinate.
For homogeneous reaction
This idea applies to electrode
reactions too, but the shape of
the surface turns out to be a
function of electrode potential.
Cont..
we can understand EA as the
change in standard internal
energy in going from one of the
minima to the maximum,
PE of EAPE Reactant PE of Product

Ox
Red
Ox
Red
StandardFreeEnergy
Reaction coordinate
Progress of reaction along the rxn coordinates, 𝑭𝒆 𝟑+
+ ത𝒆 ⇋ 𝑭𝒆 𝟐+
When the rates are equal, and the system is at
equilibrium, and the potential is Eeq
Now suppose the potential is changed to a more
positive value. The main effect is to lower the
energy of the "reactant" electron; hence the curve
corresponding to 𝑭𝒆 𝟑+
drops with respect to the
corresponding 𝑭𝒆 𝟐+
as shown
Setting the potential to a value more negative than
Eeq, raises the energy of the electron and shifts the
curve for 𝑭𝒆 𝟑+
+𝒆 to higher energies
Since the reduction barrier drops and the
oxidation barrier rises, relative to the condition
at Eeq, a net cathodic current flows.
𝑭𝒆 𝟑+
𝑭𝒆 𝟐+
𝑭𝒆 𝟐+
𝑭𝒆 𝟐+
𝑭𝒆 𝟑+
𝑭𝒆 𝟑+
𝑤𝑒 𝑐𝑎𝑙𝑙 𝑡ℎ𝑖𝑠 𝑒𝑛𝑒𝑟𝑔𝑦 𝑎𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛 𝑒𝑛𝑒𝑟𝑔𝑦
Cont..
These arguments show qualitatively the way in which the potential affects the net rates and
directions of electrode reactions

at 𝐄 𝟎
′
at 𝑬
RO + ne-
Cont..
∆𝐆 𝐚
‡
= ∆𝐆 𝟎,𝐚
‡
− 𝟏 − 𝛂 𝐅(𝐄 − 𝐄 𝟎′
)
∆𝐆 𝐂
‡
= ∆𝐆 𝟎,𝒄
‡
+ 𝛂𝐅(𝐄 − 𝐄 𝟎′
)
𝐅(𝐄 − 𝐄 𝟎′
)
∆𝐆 𝟎,𝐚
‡∆𝐆 𝐚
‡
∆𝐆 𝟎,𝒄
‡
∆𝐆 𝐂
‡
Suppose the upper curve on the О +nе side of applies when the electrode potential is equal to E°'.
The cathodic and anodic activation energies are then ∆𝐆 𝟎,𝒄
≢
and ∆𝐆 𝟎,𝐚
≢
respectively. If the
potential is changed by ∆𝑬 to a new value E, the relative energy of the electron resident on the
electrode changes by −𝐹∆𝐸 = −𝐹(𝐸 − 𝐸°’); hence the О +nе curve moves up or down by that
amount.

 

x
FαE𝟏−𝛂𝐅𝐄
R
E = 𝑬
E =0
The transfer coefficient, 𝛼, is a measure of the symmetry of
the energy barrier. This idea can be amplified by considering a
in terms of the geometry of the intersection region, as shown





tantan
tan
/)1(tan
/tan




xFE
xFE
If the intersection is symmetrical, 𝜙 = 𝜃,
𝛼 = 1/2 Otherwise 0 ≤ α <
1
2
or
1
2
< 𝛼 ≤ 0
Cont..
free energy profiles are not likely to be linear over large
ranges of the reaction coordinate; thus the angles 𝜃 𝑎𝑛𝑑 𝜙
can be expected to change as the intersection between reactant
and product curves shifts with potential.Relationship of
the transfer
coefficient to the
angles of
intersection of the
free energy curves.

O + ne- ⇌ R
Kf
Kb
…. Are called the heterogeneous rate constants
off ck Rbb ckf
Most of the electrode transfer process are not reversible at time scale,
Cont..
𝐤 𝐟 = 𝐀 𝐟 𝐞𝐱𝐩 −
𝚫𝐆 𝐜
‡
𝐑𝐓
𝐤 𝐛 = 𝐀 𝐛 𝐞𝐱𝐩 −
𝚫𝐆 𝐚
‡
𝐑𝐓
Now let us assume that the rate constants Kf and Kb have an
Arrhenius form that can be expressed as
∆𝐆 𝐚
‡
= ∆𝐆 𝟎,𝐚
‡
− 𝟏 − 𝛂 𝐅(𝐄 − 𝐄 𝟎′
)
The forward component proceeds at a rate, which must be proportional to the
surface concentration of CO
∆𝐆 𝐂
‡
= ∆𝐆 𝟎,𝒄
‡
+ 𝛂𝐅(𝐄 − 𝐄 𝟎′
)activation
energies,
𝐤 𝐟 = 𝑲 𝟎
𝐞𝐱𝐩 −𝛂
𝐅
𝐑𝐓
(𝐄 − 𝐄 𝟎′
)
𝐤 𝐛 = 𝐀 𝐟 𝐞𝐱𝐩
−𝜟𝑮 𝒐,𝒂
‡
𝑹𝑻
𝒆𝒙𝒑 (𝟏 − 𝜶)
𝑭
𝑹𝑻
(𝑬 − 𝑬 𝟎′
)
The first two factors in each of
these expressions form a
product that is independent of
potential and equal to the rate
constant at E = E°'.
The rate constants at other potentials can then be expressed simply in terms of k°:
𝐤 𝐟 = 𝐀 𝐟 𝐞𝐱𝐩 −
𝚫𝐆 𝐨,𝐜
‡
𝐑𝐓
𝐞𝐱𝐩 −𝛂
𝐅
𝐑𝐓
(𝐄 − 𝐄 𝟎′
)
𝐤 𝐛 = 𝑲 𝟎
𝒆𝒙𝒑 (𝟏 − 𝜶)
𝑭
𝑹𝑻
(𝑬 − 𝑬 𝟎′
)

nFA
i
ck c
Off 
nFA
i
ck a
Rbb 
nFA
i
ckck Rbfbfnet  0
][ RbOfac ckcknFAiii 
)](exp[ '00
EE
RT
nF
kkf   )]()1exp[( '00
EE
RT
nF
kkb  
Rate constants depend on the potential! The unique feature of electrochemical rate constants.
Thus, the rate of the electrode reaction can be controlled by the potential!
overall current, i[A], can be viewed as the
difference of the cathodic (reduction)
current, ic [A], and the anodic (oxidation)
current, ia [A]:
ac iii 
Each of the currents is proportional to their
corresponding heterogeneous rate
constant
 
 t)(0,CFAk
t)(0,CFAk
Rba
ofC


i
i
Net current:
][
][
O
R
K
k
k
b
f

BUTLER-VOLMER MODEL FOR THE ONE-STEP, ONE-ELECTRON PROCESS
O
Rai
ci
O + ne- ⇌ R
Kf
Kb

At equilibrium, the net current is zero, and the electrode is known to adopt a potential based on
the bulk concentrations of О and R as dictated by the Nernst equation
ac iii 







 )()1()(
0
'0'0
),0(),0(
EE
RT
nF
R
EE
RT
nF
o etcetcFAki

basis for all accounts of heterogeneous electrode kinetics, including the
expression for the Butler–Volmer equation.
)()1(
0
)(
0
'0'0
),0(),0(
EE
RT
nF
O
EE
RT
nF
o etcFAketcFAk



 0
exp0
EE
RT
nF
RC
C 



which is simply an exponential
form of the Nernst relation: 


RC
C
nF
RT
EE 00
ln
N.B Nernst equation is
always based on bulk
concentration
Cont..

Even though the net current is zero at equilibrium, we still envision balanced faradaic
activity that can be expressed in terms of the exchange current, io, which is equal
in magnitude to either component current, ic or ia. That is,
   ),0(or,),0( tCFAkitCFAki RbaofC 
)](exp[ '00
0 EE
RT
nF
kk  With ,
 ),0(FAk0
tCi Oo 
)(
0
'0
EE
RT
nF
oo eCFAki



raised to the -a power,  0
exp0
EE
RT
nF
RC
C 


 












 



 


 0
0
EE
RT
nF
e
C
C
R


 
 )()( )1(
Ro
o
o CCFAki
At equilibrium, the net current is zero, and the electrode is known to adopt a
potential based on the bulk concentrations of О and R as dictated by the Nernst
equation.
Cont..

The exchange current is therefore proportional to k° and can often be substituted for k°
in kinetic equations. For the particular case where Co = CR = C, io = FAk°C
the exchange current density Jo = Fk°C
The Current-Overpotential Equation can found from :







 )E(E
RT
nF
α)(1
R
)E(E
RT
nF
α
o
0
0'0'
t)e(0,ct)e(0,cFAki
 
 )()( )1(
Ro
o
o CCFAki
and,












)()(
),0(),0(
)1(
)()1()(
0
'0'0
Ro
o
EE
RT
nF
R
EE
RT
nF
o
o CCFAk
etcetcFAk
i
i Simplifying










 



R
RT
nF
R
o
RT
nF
o
o
c
etc
c
etc
ii
 )1(
),0(),0(
This equation, known as the
current-overpotential equn.
For any type of heterogeneous rxn
Note that the first term describes the cathodic component current at any
potential, and the second gives the anodic contribution.
Cont..











 

o
RT
nF
o
oc
c
etc
ii

),0(
Current-overpotential
curves for the system
The current increases
exponentially with the
potential as predicted by
the dependence of the rate
constants on the potential!










 

R
RT
nF
R
oa
c
etc
ii
 )1(
),0(










 



R
RT
nF
R
o
RT
nF
o
o
c
etc
c
etc
ii
 )1(
),0(),0(

Approximate Forms of the 𝒊 − 𝜼 Equation Cont..
I. No Mass-Transfer Effects
known as the Butler-Volmer
equation







 
RT
nF
RT
nF
eeii
)1(
0
If the solution is well stirred, or
currents are kept so low that the
surface concentrations do not
differ appreciably from the bulk
values, then
Fig. Effect of exchange
current density on the
activation overpotential
required to deliver
net current densities

2. Linear Characteristic at Small 𝜼







 
RT
nF
RT
nF
eeii
)1(
0
For small values of 𝜼, the exponential 𝑒 𝜂
can be approximated as 1 + 𝜼 ; hence the following
equation can be simplified as :




 
RT
nF
ii 0
Using 𝑒 𝑥
=1+x
which shows that the net current is linearly related to
overpotential in a narrow potential

i i  The ratio has units of resistance and is
often called the charge-transfer
resistance, Rct:
0
nF
RT
i i

 
ct
0
nF
R
RTi

This parameter is the negative reciprocal slope of the 𝒊 − 𝜼 curve where that curve passes through
the origin (𝒊 = 𝟎 , 𝜼 = 0). It can be evaluated directly in some experiments, and it serves as a
convenient index of kinetic facility. For very large k°, it approaches zero. Bc of large k means ….

3. Tafel behavior large 𝜼
For large values of 𝜼 (either negative or positive), one of the bracketed terms shown below
becomes negligible. For example, at large negative overpotentials







 
RT
nF
RT
nF
eeii
)1(
0
Solving for
overpotential term
i
RT
i
RT
lnln 0

 
Tafel equation can be written in a very simple form as
iba logwhere a and b are constants from above terms
2.3RT
b
nF
 0log
3.2
i
nF
RT
a


At 25 oC, when n = 1,  = 0.5
118 mVb 
The typical Tafel slope ……

0 -100 -200 -300300 200 100
/ mV
lgi
0lgi
A plot of logi vs 𝜼 known as a Tafel plot, is a useful device for evaluating kinetic parameters
In general, there is an anodic branch with slope and a cathodic branch with slope
RT
nF
3.2
RT
nF
3.2
)1( 
Tafel plots for anodic and cathodic branches of the current-overpotential curve for
O + ne- ⇄ R
At 25 oC,
when n = 1,
 = 0.5
𝑖0 = 10−6
ൗ𝐴
𝑐𝑚2

Exchange Current Plots
 
 )()( )1(
Ro
o
o CCFAki
From our derivation above that the exchange current can be restated
 Plot of log(io) vs Eeq
 Slope  
 Intercept  ko
Eeq
log(io)EeqEo'  log io  log FAko  log CO 
 2.3RT 
F F
 2.3RT 
  *
Very Facile Reactions
Nernstian process: kinetically facile reactions
io   i
 0
io
)
)0(
)0(
ln'
R
O0



xc
xc
nF
RT
EE










 



R
RT
nF
R
o
RT
nF
o
o
c
etc
c
etc
ii
 )1(
),0(),0(

RT
nF
R
RT
nF
o etcetc
)1(
),0(),0(


io becomes very large
compared to any current of
interest

Electrodes solutions Electrode reaction i0 / Acm-2
Hg 0.5 M sulfuric acid H++2e– = H2 510-13
Cu 1.0 M CuSO4 Cu2++2e– = Cu 210-5
Pt 0.1 M sulfuric acid H++2e– = H2 110-3
Hg 110-3 M Hg2(NO3)2 +
2.0M HClO4
Hg2
2++2e– = 2Hg 510-1
 The exchange current of different electrodes differs a lot
Dependence of exchange currents on electrolyte concentration
Electrode reaction c (ZnSO4) i0 / Acm-2
Zn2++2e– = Zn
1.0 80.0
0.1 27.6
0.05 14.0
0.025 7.0
High electrolyte concentration
is need for electrode to achieve
high exchange current.
Use of Ag/AgCl electrode.

Effects of Mass Transfer
Rct « Rmt: mt controlled Rct » Rmt: ct controlled
  iRct  Rmt,c  Rmt,a 
A more complete 𝑖 − 𝜂 relation can be obtained by substituting for
alR
R
i
i
C
xc
,
*
1
)0(












 



R
RT
nF
R
o
RT
nF
o
o
c
etc
c
etc
ii
 )1(
),0(),0(
η
F
α)(1
al,
η
F
α
cl,0
exp
i
i
1exp
i
i
1
i
i RTRT














at small  the complete Taylor expansion give the following
N.B
𝛼𝜂𝐹
𝑅𝑇
≪ 1


RT 1 1 1
    


  i
l,ao l,ci inF i
cl
o
i
i
xC
xc
,0
1
)0(
)0(





MASS TRANSFER BY MIGRATION AND DIFFUSION
In 1D(x)
 C v(x)J j (x)  Dj
C j(x)

z j F
D C
xx RT
jj j
In a three-dimension system,
z j F
C (r)  C v(r)(r)  D(r)  D C
RT
J jj jj jj
    
diffusion migration convection
diffusion
current
migration
current
convection
current
Diffusion and migration
result from a gradient in
electrochemical potential, .
Convection results from an
imbalance of forces on the
solution.
In solution mass transfer can be done by diffusion. migration, and
convection. The first two can be caused by electrochemical potential.
Consider species j moves from one potential to other as shown
ҧ𝜇 𝑗(x)
ҧ𝜇 𝑗(x + ∆𝑥)If ҧ𝜇 𝑗 x ≠ ҧ𝜇 𝑗 x + ∆𝑥 then flux will be developed
𝐽𝑗 x ∝
𝜕ഥ𝜇 𝑗
𝜕𝑥
gradient of chemical potential 𝐽𝑗 x = −
𝐶𝑗 𝐷𝑗
𝑅𝑇
𝜕 ҧ𝜇 𝑗
𝜕𝑥

Under quiescent conditions, that is, in an unstirred or stagnant solution with no density
gradients, the solution velocity, v, is zero, and the general flux equation
(x)
J j (x)  Dj
C j(x)

z j F
D C
xx RT
j j
If species J is charged, then the flux, J
j is equivalent to a current density
Let us consider a linear system with a cross-sectional area, A, normal to the axis of mass
flow. Then, 𝑖𝑗 (𝑚𝑜𝑙/𝑠 𝑐𝑚2
) is equal to
𝑖 𝑗
𝑍 𝑗 𝐹𝐴
, where 𝑖𝑗is the current component at any
value of x arising from a flow of species j . So that the above equation can be written as
J j (x) =
𝒊 𝒋
𝒁 𝒋 𝑭𝑨
=
𝒊 𝒅,𝒋
𝒁 𝒋 𝑭𝑨
+
𝒊 𝒎,𝒋
𝒁 𝒋 𝑭𝑨
with 𝒊 𝒅,𝒋
𝒁 𝒋 𝑭𝑨
= -𝐷𝑗
𝜕 ҧ𝐶 𝑗
𝜕𝑥
𝒊 𝒎,𝒋
𝒁𝒋 𝑭𝑨
=
𝒁𝒋 𝑭𝑫𝒋
𝑹𝑻
𝑪𝒋
𝝏𝝓
𝝏𝒙
where 𝒊 𝒅,𝒋and 𝒊 𝒎,𝒋are diffusion and migration currents of species , respectively.
At any location in solution during electrolysis, the total current, i, is made up of contributions
from all species; that is
𝒊 = ෍
𝒋
𝒊𝒋 𝒐𝒓 𝒊 =
𝑭 𝟐
𝑨
𝑹𝑻
.
𝝏𝝓
𝝏𝒙
෍
𝒋
𝒁𝒋
𝟐
𝑫𝒋 𝑪𝒋 + 𝑭𝑨 ෍
𝒋
𝒁𝒋
𝟐
𝑫𝒋
𝝏𝑪𝒋
𝝏𝒙
where the current for each species at that location is made up of a migration component
(first term) and a diffusional component (second term).
 In the bulk solution (away from the electrode),
concentration gradients are generally small, and the
total current is carried mainly by migration. 𝒊𝒋 =
𝒁𝒋
𝟐
𝑭 𝟐
𝑨𝑫𝒋 𝑪𝒋
𝑹𝑻
.
𝝏𝝓
𝝏𝒙

 Mixed Migration And Diffusion Near An Active Electrode
Near the electrode, an electroactive substance is, in general, transported by both processes and the
associated current can be separated into diffusion and migration currents : 𝒊 = 𝒊 𝒅 + 𝒊 𝒎
Note that 𝒊 𝒎and 𝒊 𝒅 may be in the same or opposite directions, depending on the direction of the
electric field and the charge on the electroactive species. Examples of three reductions of a
positively charged (a), a negatively charged (b), and an uncharged substance(c)are shown in
-
Cu2+
id
im -
Cu(CN)4
2-
id
im - Cu(CN)2
id
a) b) c)
𝐂𝐮 𝟐+
+ 𝟐𝐞 → 𝐜𝐮 𝐂𝐮 𝐂𝐍 𝟒
−𝟐
+ 𝟐𝐞 → 𝐂𝐮 + 𝟒𝐂𝑵− 𝑪𝒖 + 𝐂𝐍 𝟐 + 𝟐𝐞 → 𝐂𝐮 + 𝟐𝐂𝑵−
The migrational component is always in the same direction as id for cationic species reacting at
cathodes and for anionic species reacting at anodes. It opposes id when anions are reduced at
cathodes and when cations are oxidized at anodes.
𝒊 = 𝒊 𝒅 + 𝒊 𝒎 𝒊 = 𝒊 𝒅 − 𝒊 𝒎 𝒊 = 𝒊 𝒅

DIFFUSION
It is possible to restrict mass transfer of an electroactive species near the electrode to the diffusive
mode by using a supporting electrolyte and operating in a quiescent solution. Most
electrochemical methods are built on the assumption that such conditions prevail; thus diffusion
is a process of central importance.
Fick's laws are differential equations describing the flux of a substance and its concentration
as functions of time and position
-𝐽𝑗 x = 𝐷0
𝜕𝐶0(𝑥,𝑡)
𝜕𝑥
Fick's second law pertains to the change in concentration of О with time:
𝜕𝐶0(𝑥,𝑡)
𝜕𝑡
= 𝐷0
𝜕2 𝐶0(𝑥,𝑡)
𝜕2 𝑥
for 1D
𝜕𝐶0(𝑟,𝑡)
𝜕𝑡
= 𝐷0
𝜕2 𝐶0(𝑟,𝑡)
𝜕2 𝑥
+
2
𝑟
𝜕𝐶0(𝑟,𝑡)
𝜕2 𝑟
spherical equations
Consider the situation where О is an electroactive species transported purely by diffusion to an
electrode, where it undergoes the electrode reaction If no other electrode
reactions occur, then the current is related to the flux of О at the electrode surface (x = 0), Jo(0,
t), by the equation
O + ne- ⇌ R.
-𝐽0 0, t =
𝑖
𝑛𝐹𝐴
= 𝐷0
𝜕𝐶0(𝑥,𝑡)
𝜕𝑥 𝑥=0
If several electroactive species exist in the
solution, the current is related to the sum
of their fluxes at the electrode surface.
Thus, for q reducible species,
𝑖
𝑛𝐹𝐴
= ෍
𝑘=1
𝑞
𝑛 𝑘 𝑗 𝑘(0, 𝑡) = ෍
𝑘=1
𝑞
𝑛 𝑘 𝐷 𝑘
𝜕𝐶 𝑘(𝑥, 𝑡)
𝜕𝑥
𝑥=0

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