ELECTROCHEMISTRY - Wits Structural Chemistry

ELECTROCHEMISTRY
Mrs Billing
GH840
Caren.Billing@wits.ac.za
Water and
effluent
treatment
Study and
control of
corrosion
Electroplating
and refining of
metals
Manufacture and
synthesis of
chemicals
Study of redox
reactions
Applications of
Electrochemistry
Manufacturing and
electromachining
Batteries and
fuel cells
Sensors
Bioelectrochemistry
EQUILIBRIUM
ELECTROCHEMISTRY
(POTENTIOMETRY)
NERNST EQUATION
THERMODYNAMICS
OUTLINE
TRANSPORT
DYNAMIC
ELECTROCHEMISTRY
(VOLTAMMETRY)
BUTLER-VOLMER EQN
KINETICS
EQUILIBRIUM ELECTROCHEMISTRY
REVISION
The Cell Potential
Nernst Equation
Types of electrodes
Standard Electrode Potentials
Thermodynamic Functions from the Cell Potential
Applications of Standard Potentials
MOLECULES and IONS
in MOTION
Part of Chapter 6
REVISION
Oxidation at anode
Reduction at cathode
Composition of an electrochemical cell:
2 electrodes: anode + cathode
Connected together
In contact with ionic conductor
(solution, liquid, solid)
Of the same composition
V
Galvanic cell: produces
electricity due to
spontaneous reaction.
Cell Notation:
Daniell Cell:
electrode
Separate
phases
in soln
Salt
bridge
in soln
Separate
phases
electrode
Zn(s) | Zn 2+ (aq,1 M) | | Cu 2+ (aq,1 M) | Cu(s)
Or different composition
Need a salt bridge to
complete the circuit
Electrolytic cell: the nonspontaneous
reaction is
driven by an external current
source.
Zn(s) looses e − ’s
Cu 2+ (aq) gains e − ’s
Overall reaction: Zn(s) + Cu 2+ (aq) → Zn 2+ (aq) + Cu(s)
Eventually equilibrium in reached: Zn(s) + Cu 2+ (aq) Zn 2+ (aq) + Cu(s)
1

w max = ∆G
Before Equilibrium
The Cell Potential
Consider a galvanic cell reaction:
The cell can do electrical work by driving
e − ’s through the external circuit.
The larger the potential difference between the
two electrodes, the more work can be done by
each e−.
Measure E cell by applying an opposing
potential s.t. the reaction occurs reversibly
and the composition is constant (no nett
current flows).
∆G = −nFE cell
for electrical work for a
reversible reaction
At Equilibrium
No more work can be done
E cell = 0 V
The composition in electrode compartment is expressed by:
the reaction quotient, Q
the equilibrium constant, K
Derive Nernst equation:
RT
E = E
o − ln Q
nF
Applies to:
∆G = −nFE
Half-reactions
RT
E
nF
and
Cells
RT
nF
o
o
= Ered
− lnQred
E = Ecell
− lnQrxn
Half-cell reaction written as a
reduction reaction.
Nernst Equation
o
∆G
= ∆G
+ RT ln Q
2.303 RT
E = E
o − log Q
nF
E
o
cell
= E
o
cathode
− E
o
anode
NOTE: E o used in the calculations are always for the reduction reactions.
loge
x
log x =
10
loge
10
2 .303 log x = ln x
0.05916
E = E
o − log Q at 25°C
n
E o = standard cell potential all reactants and products in their standard states,
thus all activities = 1
∆G = −nFE
Do Self-test 6.7
Standard Reduction Potentials in H 2 O at 25 o C
The cell reaction is spontaneous
if E cell > 0
At equilibrium:
∆G = 0 E cell = 0 and Q = K
Strong oxidising
agent
Gibbs free energy, G
∆G < 0
E > 0
∆G = 0
E = 0
∆G > 0
E < 0
RT
E = E
− lnQ
nF
RT
0 E − lnK
nF
= nF
ln K = E
RT
E > 0
spontaneous
reduction
reactants
Extent of reaction
products
The steeper the slope, the greater
the driving power of the cell.
Recall:
∆G°= –RT ln K
∆G°= –nFE°
E < 0 nonspontaneous
reduction
Strong reducing
agent
Example 1
Consider the Daniell cell:
Using the standard reduction potentials, calculate the equilibrium constant at 25 °C.
Standard reduction potentials at 25 °C:
Cu 2+ + 2e - → Cu(s)
Zn 2+ + 2e - → Zn(s)
Zn(s)
ZnSO 4 (aq) CuSO 4 (aq) Cu(s)
E o = +0.34 V
E o = −0.76 V
Do Self-test 6.8
Metal/metal ion:
e.g. Zn(s) in zinc ion solution
M(s) | M n+ (aq)
Metal-insoluble salt: M(s) | MX(s) | X - (aq)
e.g. silver-silver chloride electrode
−
Ag(s)
AgCl(s) Cl (aCl)
AgCl(s) + e - Ag(s) + Cl - (aq)
Inert electrode: Pt(s) | M n+ (aq), M p+ (aq)
e.g. Pt immersed in a solution containing a redox couple
4 + 3+
Pt(s) Ce (aq),Ce (aq)
Ce 4+ + e - Ce 3+
o RT a
E = E − ln
F a
Gas electrode:
Zn 2+ + 2e - Zn(s)
Pt(s) | X 2 (g) | X +/− (aq)
e.g. a standard hydrogen electrode
+
Pt(s)
H2(g)
H (aq)
2H + + 2e - H 2 (g)
Types of Electrodes
E
E
RT
2F
o
= E − ln
2+
RT
F
o
= E − ln a
−
Cl
RT
2F
1
a
p
3+
Ce
4+
Ce
o
H2
E = E − ln
2
a
Zn
+
H
2

Standard Electrode Potentials
The potential of a half-cell is measured relative to the standard
hydrogen electrode (SHE) under standard conditions.
Pt(s) | H 2 (g) | H + (aq)
activities = 1 mol kg -1
E o SHE = 0 V at all temperatures fugacities = 1 atm
Determining the standard potential of a silver-silver chloride electrode using
a Harned cell:
Pt⏐H 2 (g)⏐HCl(aq), a H+ , a Cl- ⏐AgCl(s)⏐Ag(s)
Half-reactions:
H + + e − → ½H 2 (g)
anode E o
1e - SHE = 0 V
AgCl(s) + e − → Ag(s) + Cl − (aq) cathode
Recall: E o is an intensive property!
E o cell = Eo indicator – Eo reference
SHE
Overall reaction:
AgCl(s) + ½H 2 (g) → H + (aq) + Ag(s) + Cl − (aq)
Nernst equation:
o RT a
+
a
−
H Cl
E = E − ln
cell cell
F PH 2
For SHE: P H2 = 1 atm
o RT
Ecell
= Ecell
− ln a + a −
F H Cl
Let a H+ = γ m ± H+ and a Cl- = γ ± m o RT 2 2
Cl- Ecell = Ecell
− ln (mHCl
γ ± )
F
γ ± = mean activity coefficient
m = molality (concentration in mol kg -1 )
o RT 2 2
Ecell = Ecell
− ln (mHCl
γ ± )
F
o RT 2 RT
Ecell = Ecell
− ln (mHCl
) − ln ( γ ±
F
F
2RT
o 2RT
Ecell
+ ln mHCl
= Ecell
− ln γ ±
F
F
Debye-Hückel law: log γ
for a 1:1 electrolyte:
±
= −
log γ ±
2 .303 log x = ln x
2
)
A z z
= −
+
−
ln = −A′
γ ±
I
A I = −A
mHCl
If E o cell is known, then using E cell
the activity coefficient γ ± can be
calculated for a solution of
molality m.
m HCl
1
2
I = m i z i
2 i
I
1
2
[(m
)( + 1) + (m
H +
2
)( −1)
]
=
2
− Cl
1
I = [ 2(m )]
HCl
2
Evaluating a standard potential from two others:
E o +
Cu 2 + =
/ Cu(s)
E o + = +
Cu / Cu(s)
0.340 V
0.522 V
E o =
Cu 2 + +
/ Cu
Cu 2+ (aq) + 2e - → Cu(s) E o = 0.340 V (A)
Cu + (aq) + e - → Cu(s) E o = 0.522 V (B)
(A) – (B): Cu 2+ (aq) + e - → Cu + (aq) E o (C) =
NOTE:
Thermodynamic Functions from the Cell Potential
This is NOT a cell reaction!!
→ the electrons do NOT cancel out
Redox couple Cu 2+ /Cu +
→ Reduction reaction
reduction
oxidation
2RT
o 2RT
E
cell
+ ln mHCl
= Ecell
+ A′
m
F
F
HCl
y = c + m x
Measure the cell potential for
various HCl concentrations.
Plot y vs x and extrapolate to the
y-intercept to determine E o cell
Applications of Standard Potentials
1) The determination of activity coefficients
2) The determination of equilibrium constants
3) The electrochemical series
4) The determination of thermodynamic functions
Example 2
The mean activity coefficients of HBr in 5.0 and 20.0 mmol kg –1 are 0.930 and 0.879,
respectively. Consider a hydrogen electrode in HBr(aq) solution at 25 °C operating at
1.15 atm.
Calculate the change in the electrode potential when the molality of the acid solution
is changed from 5.0 and 20.0 mmol kg –1 .
Thermodynamics has the striking ability of to relate apparently unrelated
relationships.
∆G = −nFE
E o cell used to calculate ∆Go f of ions in solution.
Using the relationship
dE ∆S
=
dT nF
∂G
= −S
∂T
p
The temperature coefficient of E o cell gives the standard entropy
of the cell reaction and hence entropies of ions in solution.
Combining both relationships:
dE
∆H
= ∆G
+ T∆S
= −nF
E − T
dT
Non-calorimetric measurement of ∆H°
and hence ∆H o f of ions in solution can
be determined.
3

Example 3
Devise a cell in which the cell reaction is:
Mn(s) + Cl 2 (g) → MnCl 2 (aq)
Give the half reactions at the electrodes and from the standard cell potential of 2.54 V
deduce the standard potential for the Mn 2+ /Mn(s) redox couple.
Given: E°(Cl 2 /Cl - ) = +1.36 V
Example 4
Estimate the cell potential at 25°C for
Ag(s)|AgBr(s)|KBr(aq, 0.050 mol kg –1 )||Cd(NO 3 ) 2 (aq,0.0034 mol kg –1 )|Cd(s)
E°(R-H) = –0.40 V E°(L-H) = +0.07 V (assume non-ideal solutions)
Write the spontaneous electrochemical reaction.
Example 5
The standard potential of the cell below at 25 °C is 0.95 V.
Ag(s) |AgI(s) | AgI(aq) | Ag(s)
Calculate: a) its solubility constant and b) the solubility of AgI .
MOLECULAR MOTION IN LIQUIDS
Mass Transport
Migration of ions - Conductivities of Electrolyte Solutions
Mobilities of Ions
Diffusion of Ions
Part of Chapter 20
Mass Transport
Three main ways to transport ions to the surface of an electrode
→ called mass transport
Conductivities of Electrolyte Solutions
Ions can be dragged through the solvent by applying a potential difference
between two electrodes in solution.
MIGRATION
To measure conductivity:
Incorporate conductivity cell into
one arm of resistance bridge and
search for balance point.
R 3
Convection
Movement due to
mechanical intervention.
e.g. stirring, pumping,
shaking
Affects molecules and
ions in solution.
Diffusion
Movement due to
concentration gradient.
e.g. movement across a
cell membrane
Affects molecules and
ions in solution.
Migration
Movement due to applied
electric field.
e.g. in an electrolysis cell
Affects only ions in
solution.
1
R =
κ A
1
κ =
R A
C
κ =
R
R = resistance of solution (Ω)
κ = conductivity (S cm -1 )
= distance between electrodes
A = area of electrodes
C = cell constant
κ depends on the number of ions in a solution.
R 4
2
R 1
R R =
1 2
R R
3
R 4
Resistance bridge R 1R
= R R3
4 2
Note: use ac current (ν~1kHz)
to avoid electrolysis and
polarisation of the electrodes
which would change the
composition of the solution at
the electrode surfaces.
4

Measuring the conductivity of tap water:
κ
=
c
Λ
Λ
m = molar conductivity (S cm 2 mol -1 )
m c = molarity
If κ ∝ c, the molar conductivity should be
independent of the concentration of an
electrolyte. Is this the case
Parallel
plates
293.3 µS / cm
1 S = 1 Siemen
= 1 Ω -1
NO!
Molar conductivity varies with
concentration because:
- no. of ions in solution may not be
proportional to concentration (especially
for weak electrolytes)
- ions interact strongly with one another
reducing effective charge
NB: Don’t confuse conductivity (κ) with conductance (G)
Strong electrolyte – Λ m
decreases slightly as
conc increases.
Weak electrolyte –
normal Λ m at low conc,
but Λ m decreases sharply
as conc increases.
1
G =
R
Kohlrausch’s law:
At low concentrations, molar conductivities vary with the square root of conc.
Λ = Λ − K c
m
Λ o m
= ν
o
m
+
λ
+
+ ν
−λ
−
STRONG ELECTROLYTES
Λ m o = limiting molar conductivity
(ions are infinitely far apart no interaction)
K → related to stoichiometry of electrolyte
Kohlrausch’s law of independent migration of ions:
Λ m o → expressed as sum of contributions from its individual ions.
λ = limiting molar conductivity of ions
ν = no. of ions per formula unit of electrolyte
H +
34.96 OH -
19.91
Na + 5.01 Cl -
7.63
K + 7.35 Br -
7.81
Zn 2+ 10.56 SO 2-
4 16.00
e.g. for MgCl 2 : ν + = 1 ν − = 2
Limiting ionic conductivies in water at 298 K, λ /mS m 2 mol -1
Weak electrolytes do NOT dissociate completely,
conductivity of weak electrolytes depends on the degree of ionization (α)
e.g. HA + H 2 O A - + H 3 O +
+ −
[H3O
][A ]
+
K
[ H3O
] = αc
a =
[HA]
−
[ A ] = αc
2
α c
Ka
=
[ HA] = ( 1−
α)c
1 − α
At infinite dilution the electrolyte is fully
ionised :
Λ o m
= ν
+
λ
+
+ ν
−λ
−
At very low conc’s:
Ostwald’s
dilution
law
WEAK ELECTROLYTES
Λ = α
o
m
Λ m
1 1 Λ
mc
= +
o
Λ
m Λ
m K
( Λ ) o 2
1 1 1
= + Λ
mc
o
o 2
Λ
m Λ
m K
a
( Λ
m
)
y c m x
a
m
(c = formal conc of HA)
Example 6
Calculate the degree of ionization and the acid dissociation constant at 298 K for a
0.010 M acetic acid solution that has a resistance of 2220 Ω. The resistance of a 0.100
M potassium chloride solution was also found to be 28.44 Ω.
Λ m (0.1 M KCl) = 129 S cm 2 mol -1
λ o (H + ) = 349.6 S cm 2 mol -1
λ o (CH 3 COO - ) = 40.9 S cm 2 mol -1
Example 7
The molar conductivity of a strong electrolyte in water at 25 °C was found to be 109.9
S cm 2 mol -1 for a concentration of 6.2 × 10 -3 mol L -1 and 106.1 S cm 2 mol -1 for a
concentration of 1.5 × 10 -3 mol L -1 .
Estimate the limiting molar conductivity of the electrolyte.
5

Example 8
The limiting molar conductivities of KCl, KNO 3 , and AgNO 3 are 149,9, 145.0, and 133.4
S cm 2 mol -1 , respectively (all at 25 °C).
What is the limiting molar conductivity of AgCl at this temperature
To understand conductivity measurements, we need to understand why
ions move at different rates
Motion of ions is largely random, but in the presence of an electric field the
motion is biased → migration
DC, uniform electric
field is applied
Mobilities of Ions
Electric field:
∆φ
ε =
∆φ = potential difference
= distance between electrodes
ε = electric field
V source
+ ∆φ
+
-
Force acting on the ion due to electric field:
ze∆φ
ze = charge of ion
F e
= zeε
=
e = 1.602×10 -19 C
causes ion to accelerate towards oppositely
charge electrode.
But ion undergoes frictional retardation
when moving through the solvent:
F Fric
viscosity drag
= fs = 6
πη
as
s = speed of ion
η = viscosity of solvent
a = radius of ion
The two forces (one due to the electric field force and the other due to the viscosity
drag, interactions between ions, etc.) act in the opposite directions.
Eventually, these forces are balanced
F e = F Fric
and the ions reach a terminal speed called the drift speed.
zeε
= 6
πη
as
Self-study:
What is the Grotthaus mechanism which is used to explain the very high mobility of
H + in water
zeε
Drift speed: s =
6πη
a
stronger electric field ε
faster ions drift
s = uε
u = ionic mobility =
ze
6
πη
a
The drift speed
governs the rate
at which charge
is transported.
theoretical
measurable
LINK between mobility and molar conductivity
∴ expect the conductivity to decrease with increasing solution viscosity and
increasing ion size.
Ionic mobilities in water at 298 K,
u /10 -8 m 2 s -1 V -1
H + 36.23 OH - 20.64
Na + 5.19 Cl - 7.91
K + 7.62 Br - 8.09
Zn 2+ 5.47 SO 2-
4 8.29
However, we see the larger ion (with the
same charge) has a larger mobility.
Need to consider:
a = hydrodynamic radius
i.e. includes hydration sphere
Small ions are more extensively
hydrated that larger ions (due to the
stronger electric field)
For cations and anions:
λ = zuF
And for very diluted solutions:
o
Λ m = ( ν+ z+
u+
+ ν−z−u
− )F
Faraday’s constant = F = N A e
Since Λ o m = ν+ λ+
+ν−λ−
Ion-ion interactions affecting mobilities:
In ionic solutions the ion-atmosphere also needs to be
considered when ions migrate.
When an electric field is introduced, two effects occur:
Relaxation effect:
When an electric field is introduced the ionic atmosphere is
distorted as it drags behind moving charge.
Central ion and atmosphere opposite in charge retardation
of moving ion.
Electrophoretic Effect:
The ion-atmosphere moves in the opposite direction to the
central ion which lead to the reduction in ion mobility.
Both effects results in lower mobilities!
Fick’s first law of diffusion:
The flux (J) of particles is proportional to the concentration gradient.
dc
J = −D
dx
D = diffusion coefficient
c = molar concentration
x = distance
Einstein equation:
Stokes-Einstein equation:
Nernst-Einstein equation:
Applies to ions and uncharged particles in a solution.
RT
D = u
zF
Diffusion of Ions
kT
D =
6
πη
a
c 1 > c 2
c 1
c 2
J
substitute expression for u
k = Boltzmann constant = 1.381×10 -23 J K -1
Λ =
F 2
o 2
( ν+ z+
D+
+ ν−z
2 Or for a
m −D−
)
λ+ =
F 2
o
( ν + z
2
+ D+
)
RT
single ion: RT
c
Do Self-test 20.4
dc
dx
x
6

Fick’s second law of diffusion or the
diffusion equation:
2
∂c
∂ c
= D
2
∂t
∂x
→ describes time
dependence of the
diffusion process
Diffusion coefficients at 298 K, 10 -9 m 2 s -1 Example 9
H + in water
9.31
The mobility of the NO 3- ion in aqueous solution at 25 °C is 7.40×10 -8 m 2 s -1 V -1 .
Na + in water
1.33
(Viscosity of water is 0.89110 -3 kg m -1 s -1 ).
sucrose in water 0.522
Calculate its diffusion coefficient and the effective radius at this temperature.
It shows that the rate of change of conc is proportional to the
curvature (the second derivative) of the concentration with
respect to the distance.
This equation shows that:
1) If the concentration changes sharply from point to
point, then the concentration changes rapidly in time.
c
vs
c
x
x
2) If the curvature = 0, then the concentration does not
change in time.
3) If the concentration decreases linearly with distance,
then the concentration is constant at any point,
because the inflow is exactly balanced by the outflow
of particles or ions.
c
x
Remember: in the calculations you have to show the work on units.
Without that, the work might be considered as not done at all.
DYNAMIC ELECTROCHEMISTRY
Double Layer at the Electrode-Solution Interface
Dynamic Equilibrium
Butler-Volmer Equation
No Mass Transport Effects
Mass Transport Effects
Tafel Plot
Electrolysis and Galvanic Cells
Double Layer at the Electrode-Solution Interface
Electrode reaction = heterogeneous chemical process
e - ’s move from electrode surface to chemical species in solution (at the surface), or
vice versa
Diffusion layer – region
dominated by unequal
(1 µm − 1 mm) charge distribution
Region of charge neutrality
Part of Chapter 22
(1 − 10 nm)
Double layer generated due
to electrostatic forces
Faradaic process:
Helmholtz Model
Gouy-Chapman Model
Electrons transferred across the electrode-solution interface (reduction/oxidation)
Governed by Faraday’s Law: current ∝ charge
Non-Faradaic process:
Under some conditions an electrodesolution
interface shows a range of
potentials where no charge transfer
reaction occurs.
E.g. capacitive or charging currents
i
Charging currents decay rapidly
after applying a potential.
t
φ M
+
+
+
+
+
+
+
Electrode
-
-
-
-
-
-
-
Solution
φ s
Distance
Double layer is like a capacitor
20-40 µF/cm 2
∆φ = φ M - φ S = Galvani potential difference
The potential difference between points
in the bulk metal and the bulk solution.
φ M
Electrode
OHP
Solvated
ions
φ S
φ M
+
+
+
+
+
+
+
Electrode
Solution
Distance
Double layer has no thickness and is
diffused by thermal motion of ions
Stern Model
φ M
+
+
+
+
+
+
+
Electrode
-
-
-
-
-
-
-
OHP
rigid Helmholz plane
Diffusion of
charges caused
by thermal
motion of ions
Distance
7

Exchange Current Density
Under zero current conditions:
Dynamic Equilibrium
Ox + ne - Red
o RT a
Eeq
= E − ln
nF a
Re d
Ox
EQUILIBRIUM!
Overpotential (η):
the difference between the applied potential and the
equilibrium potential.
η = E – E eq
Equilibrium potential: potential where i = 0
By definition, no net chemical change occurs i = 0
However, this is a dynamic equilibrium! i = ia + ic
= 0
Increasing rate of reduction
η = E – E eq
E eq
Increasing rate of oxidation
η = E – E eq
i.e. there is still current flowing:
i a = −ic
= io
E eq
exchange current
η > 0
Recall: current is the variation of charge with time:
current ∝ rate of electron transfer
dq
i =
dt
η < 0
i o is a measure of the electron transfer activity in both cathodic and anodic directions
at equilibrium.
It is a very useful kinetic parameter in dynamic electrochemistry.
Rate of e - transfer depends on the electrode potential which drives the
transfer of e - ’s.
Current is defined as:
i = nFAν
A = surface area of electrode
ν = reaction rate
Forward reaction:
Reverse reaction:
k k ox
Ox + ne - red
Red
Red Ox + ne-
ν
red
= k
red
C
i
ν c
red
=
nFA
Butler-Volmer
Equation
ox
1 st order
Current at electrode:
ν = k C
ox
ox
i
ν a
ox
=
nFA
Overall rate:
1
ν = νox
− νred
= koxCred
− kredCox
= (ia
− ic
)
nFA
i = ia
− ic
= nFA(k oxCred
− kredCox
)
red
Note: c red or c ox refers to the
conc’s near the electrode surface
Forward reaction:
Reverse reaction:
k
Ox + ne - red
k ox
Red
Red Ox + ne-
Current at electrode:
The rate constant varies with applied potential (in an exponential way):
(1−α
)nF
−αnF
η
η
o
o
RT
RT
k = kred
= k e
ox
k e
k o = standard rate constant (k when potential applied is E eq )
α = transfer coefficient (0 < α < 1) → symmetry factor
(1− α)nF
o
i = nFAk C RT
rede
i = ia
− ic
= nFA(k oxCred
− kredCox
)
η
− Coxe
− αnF
η
RT
(1− α)nF
− αnF
η
η
i = i RT − RT
o e e
Butler-Volmer equation
and
o
io = nFAk C
Current (rate of reaction) depends on:
• Electrode area, A
• Concentration of reactant, C
• Temperature
• The kinetic parameters i o and α
• Overpotential, η
Current density, j:
i
j =
A
(1− α)nF
− αnF
η
η
j = j RT − RT
o e e
(1−α)nF
− αnF
η
η
i = i RT − RT
o e e
i = nFAν
j is independent of electrode area
Butler-Volmer equation in
terms of current density
For small overpotentials (η < ~10 mV):
nF
j = joη
RT
In most practical situations, only the forward or the reverse reaction is significant and
a simplified equation is sufficient.
Large, positive overpotential predominantly oxidation:
In practise, η ≥ ~ +0.12/n V
j
(1−α
)nF
η
RT
= j e o
(1− α)nF
ln j = ln jo
+ η
RT
Large, negative overpotential predominantly reduction:
In practise, η ≤ ~ -0.12/n V
j
−αnF
η
RT
= j e o
αnF
ln j = ln jo
− η
RT
(1− α)nF
j = j RT
o e
η
− e
Anodic current
− αnF
η
RT
Cathodic current
i.e. η nF < 1
RT
Low field region
High field region
Tafel equations
8

(1− α)nF
ln j = ln jo
+ η
RT
Tafel Plot
Tafel Plot
Anodic current
Butler-Volmer: No Mass Transport Effects
j
j a
High field
Oxidation: Red → Ox + ne -
For oxidation reaction
anodic current:
nF
slope = (1−
α)
RT
∴ find α
j = j a + j c
Low field
i ∝ E
E
ln j o
Deviation from linearity at low
η as the reverse reaction can
no longer be ignored
High field
j c
Reduction: Ox + ne - → Red
j = jo
e
Anodic
current
(1−α
)nF
η
RT
− e
Cathodic
current
−αnF
RT
η
Focusing on low field region
j
Oxidation
j a
Effect of transfer coefficient (α) on j vs η relationship:
j
α = 0.25 α = 0.5
η0
Oxidation favoured
α = 0.75
j a = j o
-j c = j o
E = E eq potential at
zero current flow
E
Reduction:
Ox + ne - → Red
Oxidation:
Red → Ox + ne -
η
j c
Reduction
Reduction favoured
(1− α)nF
− αnF
η
η
j = j RT − RT
o e e
α = the fraction of applied potential that
influences the rate of electrochemical rxn
Butler-Volmer: Mass Transport Effects
Assumption in derivation of Butler-Volmer equation:
negligible conversion of electroactive species at low current densities
uniformity of concentration near the electrode
j
Predicted by the
Butler-Volmer equation
The Butler-Volmer equation fails at high current densities → current is limited
by transport of ions towards the electrode.
Larger η is needed to produce given current.
η
Current
controlled only
by mass
transport
This effect is called concentration polarisation
and its contribution to the total overpotential is called the polarisation
overpotential, η c .
Current
controlled only
by electron
transfer and j 0
Exponential
increase in
current with η
(B-V equation)
Current
controlled by η
and transport
9

Example 10
The electron transfer coefficient of a certain electrode in contact with the redox couple
M 3+ / M 4+ in aqueous solution at 25 °C is 0.39. The current density is found to be 55.0
mA cm –2 when the overvoltage is 125 mV.
What is the overvoltage required for a current density 75.0 mA cm –2
What is the exchange current density
Example 11
The exchange current density and the electron transfer coefficient for the reaction
2H + + 2e H 2 (g) on nickel at 25 °C are 6.3 × 10 –6 A cm –2 and 0.58, respectively.
Determine what current density would be required to obtain an overpotential of 0.20 V
as calculated from the Butler-Volmer equation and the Tafel equation.
Electrolysis Cells:
Electrolysis and Galvanic Cells
To induce current to flow through an electrolytic cell, have to apply a
potential difference greater than that required to reach equilibrium:
E applied > E eq
In theory:
Have to also overcome the cell overpotential:
η cell = |η c | + |η a | + | iR s |
There is an activation or charge
transfer overpotential (η a and η c )
at the anode and cathode.
E eq = -E cell
Additionally, the rate of e - transfer depends on the
electrode potential which drives the transfer of e - ’s.
E applied > E eq + |η cell |
∆G > 0
Ohmic drop due to the solution E cell < 0
resistance, ∴ apply an overpotential to
drive current through the solution. V = IR
j
η
Many other factors also need to be taken into account
E.g.: The magnitude of the exchange current density (j o ) also determines the
overpotential required for a significant current to flow:
j
Decreasing j o
For high j o :
observable current at low η
η
Galvanic Cells:
Conversely, in a galvanic cell a smaller potential difference is
generated than expected due to these overpotentials.
V
j o /A m -2 : 10 10 2 10 3
Self-test 6.7
Express the formation of H 2 O from H 2 and O 2 in acidic solution (a redox reaction)
as the difference of two reduction reactions.
(You may need to consult the full table of standard reduction potentials in the Appendix)
Self-test 6.8
Write the half-reaction and the reaction quotient for a chlorine gas electrode.
Self-test 20.4
Use the experimental value of the mobility to evaluate the diffusion coefficient, the
limiting molar conductivity and the hydrodynamic radius of the NH 4+ ion in
aqueous solution. (u = 7.63 × 10 -8 m 2 s -1 V -1 )
10