Transcript Document 7755487

 Gc   Gc (0)  F
 Gc   Gc (0)  F
 Ga   Ga (0)  (1   ) F
The parameter α is called the transient coefficient and lies in the range 0 to 1.
Based on the above new expressions, the net current density can be expressed
as:


j  FBa [RED ]e   Ga ( 0) / RT e (1 ) F / RT  FBc [Ox]e   Gc ( 0 ) / RT e F / RT
assuming f 
then
F
RT
ja  FBa [RED ]e

  Ga ( 0 ) / RT (1 ) f
e
jc  FBc [Ox]e   Gc ( 0) / RT e f
j  ja  jc
• Example 25.1 Calculate the change in
cathodic current density at an electrode
when the potential difference changes
from ΔФ’ to ΔФ
• Self-test 25.5 calculate the change in
anodic current density when the potential
difference is increased by 1 V.
Overpotential
• When the cell is balanced against an external source, the
Galvani potential difference, ∆Φ , can be identified as the
electrode zero-current potential, E.
• When the cell is producing current, the electrode potential
changes from its zero-current value, E, to a new value, E’.
• The difference between E and E’ is the electrode’s
overpotential, η.
η = E’ – E
• The ∆Φ = η + E,
• Expressing current density in terms of η
ja = j0e(1-a)fη
and
jc = j0e-afη
where jo is called the exchange current density, which is
when there is no net current flow, i.e. ja = jc
• The Butler-Volmer equation:
j = j0(e(1-a)fη - e-afη)
• The low overpotential limit
η < 0.01 V
• The high overpotential limit
η ≥ 0.12 V
The low overpotential limit
• The overpotential η is very small, i.e. fη <<1
• When x is small, ex = 1 + x + …
• Therefore ja = j0[1 + (1-a) fη]
jc = j0[1 + (-a fη)]
• Then j = ja - jc = j0[1 + (1-a) fη] - j0[1 + (-a fη)]
= j0fη
• The above equation illustrates that at low overpotential limit, the
current density is proportional to the overpotential.
• It is important to know how the overpotential determines the property
of the current.
Calculations under low overpotential
conditions
• Example 25.2: The exchange current density of a Pt(s)|H2(g)|H+(aq)
electrode at 298K is 0.79 mAcm-2. Calculate the current density
when the over potential is +5.0mV.
Solution: j0 = 0.79 mAcm-2
η = 5.0mV
f = F/RT =
j = j0fη
• Self-test 25.6: What would be the current at pH = 2.0, the other
conditions being the same?
The high overpotential limit
• The overpotential η is large, but could be positive or
negative !
• When η is large and positive
jc = j0e-afη = j0/eafη becomes very small in comparison to ja
Therefore j ≈ ja = j0e(1-a)fη
ln(j) = ln(j0e(1-a)fη ) = ln(j0) + (1-a)fη
• When η is large but negative
ja is much smaller than jc
then j ≈ - jc = - j0e-afη
ln(-j) = ln(j0e-afη ) = ln(j0) – afη
• Tafel plot: the plot of logarithm of the current density
against the over potential.
Applications of a Tafel plot
• The following data are the anodic current through a platinum
electrode of area 2.0 cm2 in contact with an Fe3+, Fe2+ aqueous
solution at 298K. Calculate the exchange current density and the
transfer coefficient for the process.
η/mV 50
100
150
200
250
I/mA
8.8
25
58
131
298
Solution: calculate j0 and α
Note that I needs to be converted to J
• Self-test 25.7: Repeat the analysis using
the following cathodic current data:
η/mV
-50 -100 -150 -200 -250
I/mA
-0.3 -1.5 -6.4 -27.61 -118.6
• In general exchange currents are large when the redox
process involves no bond breaking or if only weak bonds
are broken.
• Exchange currents are generally small when more than
one electron needs to be transferred, or multiple or
strong bonds are broken.
The general arrangement for
electrochemical rate measurement
25.10 Voltammetry
• Voltammetry: the current is monitored as the potential of the
electrode is changed.
• Chronopotentiometry: the potential is monitored as the current
density is changed.
• Voltammetry may also be used to identify species and determine
their concentration in solution.
• Non-polarizable electrode: their potential only slightly changes when
a current passes through them. Such as calomel and H2/Pt
electrodes
• Polarizable electrodes: those with strongly current-dependent
potentials.
• A criterion for low polarizability is high exchange current density
due to j = j0fη
Concentration polarization
• At low current density, the conversion of the electroactive
species is negligible.
• At high current density the consumption of electroactive
species close to the electrode results in a concentration
gradient.
• Concentration polarization: The consumption of electroactive
species close to the electrode results in a concentration gradient
and diffusion of the species towards the electrode from the bulk may
become rate-determining. Therefore, a large overpotential is needed
to produce a given current.
• Polarization overpotential: ηc
• Consider the reduction half reaction:
Mz+ + ze → M
• The Nernst equation is
E = Eө + (RT/zF) ln(a)
• When using a large excess of support electrolyte, the mean activity
coefficients stays approximately constant,
E = Eө + (RT/zF)ln(γ) + (RT/zF)ln(c)
E = Eo + (RT/zF)ln(c)
• The ion concentration at OHP decreases to
c’ due to the reaction, resulting
E’ = Eo + (RT/zF)ln(c’)
• The concentration overpotential is
ηc = E’ – E = (RT/zF)ln(c’/c)
(typo in the 8th edition)
• The thickness of the Nernst diffusion layer (illustrated in
previous slide) is typically 0.1 mm, and depends strongly
on the condition of hydrodynamic flow due to such as
stirring or convective effects.
• The Nernst diffusion layer is different from the electric
double layer, which is typically less than 1 nm.
• The concentration gradient through the Nernst diffusion
layer is dc/dx = (c’ – c)/δ
• This concentration gradient gives rise to a flux of ions
towards the electrode
J = - D(dc/dx)
• Therefore, the particle flux toward the electrode is
J = D (c – c’)/ δ
• The cathodic current density towards the electrode is the
product of the particle flux and the charge transferred per
mole of ions (zF)
j = zFJ = zFD(c – c’)/ δ
• The maximum rate of diffusion across the Nernst layer is
when c’ = 0 at which the concentration gradient is the
steepest.
jlim = zFJ = zFDc/ δ
• Using the Nerst-Einstein equation (D = RTλ/(zF)2),
jlim = cRTλ/(zFδ)
where λ is ionic conductivity
• Example 25.3: Estimate the limiting current density at
298K for an electrode in a 0.10M Cu2+(aq) unstirred
solution in which the thickness of the diffusion layer is
about 0.3mm.
• Solution: one needs to know the following information
molar conductivity of Cu2+: λ = 107 S cm2 mol-1
δ = 0.3 mm
employing the following equation:
jlim = cRTλ/(zFδ)
• Self-test 25.8 Evaluate the limiting current density for an
Ag(s)/Ag+(aq) electrode in 0.010 mol dm-3 Ag+(aq) at
298K. Take δ = 0.03mm.
• From j = zFD(c – c’)/ δ,
one gets c’ = c - jδ/zFD
• ηc = (RT/zF)ln(c’/c) = (RT/zF)ln(1 - jδ/(zFDc))
• Or
j = (zFDc)(1 – ezfη)/δ