Introduction to electrochemistry 2 by t. hara

Introduction to Electrochemistry 2
1
2. Electrochemistry, as a province of academic.
2.1. Overview
2.2. Thermodynamics
2.3. Interface
2.4. Kinetics
2.5. Experimental Methods
Recommended Text
Electrochemistry Principles, Methods, and Applications
C. M. A. Brett & A. M. O. Brett
OXFORD UNIVERSITY PRESS

2
2.1. Overview
(1) Electrochemistry is a study of redox reaction:
- Reduction [a reactant gains electron(s)]
- Oxidation [a reactant loses electron(s)]
(2) Reduction reactions take place heterogeneously at
Interfaces between electrodes and electrolyte(s):
- Anode at which oxidation reaction(s) take(s) place.
- Cathode at which reduction reaction(s) take(s) place.
(3) Chemical reactions including redox reactions are
thermodynamically and kinetically controlled/affected:
- Thermodynamics … Potential difference
- Kinetics … Charge and/or mass transfer

3
2.1. Overview
(1) Electrochemistry is a study of redox reaction:
- Reduction [a reactant gains electron(s)]
- Oxidation [a reactant loses electron(s)]
Zn
Cu
wire
Salt bridge
Cu(II) sulfateZn(II) sulfate
electrons
Reduction:
Cu2+(aq) + 2e- → Cu(s)
Oxidation:
Zn(s) → Zn2+(aq) + 2e-
Total Cell Reaction:
Zn(s)+Cu2+(aq) → Zn2+(aq)+Cu(s)

4
2.1. Overview
(2) Reduction reactions take place heterogeneously at
Interfaces between electrodes and electrolyte(s):
- Anode at which oxidation reaction(s) take(s) place.
- Cathode at which reduction reaction(s) take(s) place.
Zn2+(aq)
e-
Zn(s)
e-
Cu(s)
Cu2+(aq)

5
2.1. Overview
(3-a) Chemical reactions including redox reactions are
- Thermodynamics … Potential difference
Cu2+(aq) + 2e- → Cu(s) +0.34 V
Zn2+(aq) + 2e- → Zn(s) -0.76 V
Standard
Reduction
Potential
𝐄 𝐜𝐞𝐥𝐥 =1.10 V

6
2.1. Overview
(3-b) Chemical reactions including redox reactions are
- Kinetics … Charge and/or mass transfer
6
e-
Cu(s)
Cu2+(aq)
Electron
conduction is fast.Ion conduction is
slow.

7
2.2. Thermodynamics
“Why is it that some electrochemical reactions proceed
spontaneously in one direction?”
The cell potential tells us the maximum work (maximum
energy) that the cell can supply. This value is ΔG=-nF𝐄 𝐜𝐞𝐥𝐥.
For example, the cell notation, Zn|Zn2+(aq)||Cu2+(aq)|Cu, means
we consider the cell reaction as Zn+Cu2+Zn2++Cu. The half-
reactions are represented by Cu2++2e-Cu (E0=+0.34 V) and by
Zn2++2e-Zn (E0=-0.76 V). Then Ecell=+0.34-(-0.76)=+1.10 V.
The corresponding ΔG=-nF𝐄 𝐜𝐞𝐥𝐥 value is -212 kJ /mol which is
negative, showing that the reaction proceeds spontaneously as
written (n is the number of electrons involved in the reaction,
and F is Faraday constant, F≈96500 C/mole).

8
2.2. Thermodynamics
Nernst equation is given by
𝐄 𝐜𝐞𝐥𝐥 = 𝐄𝐜 𝐞𝐥𝐥, 𝟎 +
𝑹𝑻
𝒏𝑭
∏𝒂 𝒐𝒙
∏𝒂 𝒓𝒆𝒅
.
Ecell
0 is the value of the cell potential when the activities of
reactants equals the concentration of reactants.
The higher the concentration of i-th reactant (ci), the lower the
activity coefficient of it (γi). The activity, αi=γici.
When you need to the precise value of Ecell,0, you only have to
measure 𝐄 𝐜𝐞𝐥𝐥 s at different concentrations and extrapolate to
zero concentrations. That’s it!

9
2.2. Thermodynamics
The following is a list of
standard electrode potentials
of common half-reactions in
aqueous solution, that is
measured relative to the
standard hydrogen electrode at
25°C (298 K) with all species at
unit activity.
These ions are thermodynamically
unstable, eager for electrons, and
therefore quite strong oxidizing agents.
Reduced forms (metals) are stable;
therefore, those metals are valuable as
ornaments.
In contrast, this guy is eager to donate
electrons, and therefore a quite strong
reducing agent.

10
2.2. Thermodynamics
Why do these two guys have different
potentials?
The difference is caused from a
difference in electronic configurations:
the [Co(H2O)6]3+ has a (t2g)4(eg)2
electronic configuration; the [Co(NH3)6]3+
has a (t2g)6(eg)0.
(t2g)4(eg)2 (t2g)6(eg)0
PE

11
2.2. Thermodynamics
In electrochemistry, the
standard hydrogen
electrode (SHE)
potential is taken as a
reference point.

12
2.2. Thermodynamics
In alikaline solution, the
potential of hydrogen
evolution is going to be more
negative.

13
2.2. Thermodynamics
In acidic solution, the potential
of oxygen evolution is equal to
+1.23 V.

14
2.2. Thermodynamics
In alkaline solution, the
potential of oxygen evolution is
going to be less positive.

15
*Beyond Thermodynamics
“Why is it that electrochemical reactions are restricted
by the movement of ions?”
The movement of ions is slower than that of electrons.
Ions in electrolyte solutions are solvated. Solvated ions move at
different velocities, according to their size and charge.
- Diffusion is due to a concentration gradient. Diffusion occurs
for all species.
- Migration is due to electric field effects. Thus, migration
affects only charged species.

16
Diffusion is described by Fick's first law:
𝑱𝒊 = −𝑫𝒊
𝝏𝑪𝒊
𝝏𝒙
where Ji is the flux of species i of concentration ci in direction x,
and
𝝏𝑪𝒊
𝝏𝒙
is the concentration gradient. Di is the proportionality
factor between flux and concentration gradient, known as the
diffusion coefficient. The negative sign arises because the flux of
species tends to cancel the concentration gradient.

17
In the presence of an applied electric field of strength E=
𝝏ϕ
𝝏𝒙
(potential gradient),
𝑱𝒊=−𝑫𝒊
𝝏𝑪𝒊
𝝏𝒙
− 𝒛𝒊𝒄𝒊
𝑭
𝑹𝑻
𝑬
where the second term on the right-hand side represents
migration (z: charge, R: ideal gas constant). Opposing this
electric force there are three retarding forces: (1) a friction
force; (2) an asymmetric; and (3) an electrophoretic effect.

18
(1) a friction force:
The larger the size of solvated ion, the bigger the friction force.
We consider an isolated ion. The force due to the electric field is
F=zeE which is counterbalanced by a viscous force given by
Stokes' equation F=6πηrv where η is the solution viscosity, r the
radius of the solvated ion and v the velocity vector. The
maximum velocity is, therefore v=(zeE)/(6πηr)=μE where μ is
the ion mobility, and is the proportionality coefficient between
the velocity and electric field strength.

19
(2) an asymmetric effect: Because of ion movement the ionic
atmosphere becomes distorted such that it is compressed in
front of the ion in the direction of movement and extended
behind it.
(3) an electrophoretic effect: Ion movement causes motion of
solvent molecules associated with ions of the opposite sign. The
result is a net flux of solvent molecules in the direction contrary
to that of the ion considered.
+
E
+
δ-
δ+

20
2.3. Interface
The interfacial region in solution is the region where the value
of the electrostatic potential, ф, differs from that in bulk
solution. The basic concept was of an ordering of positive or
negative charges at the electrode surface and ordering of the
opposite charge and in equal quantity in solution to neutralize
the electrode charge (electric double layer is formed).
The models of electric double layer are listed as follows:
(1) Helmholtz model
(2) Stern model

21
2.3. Interface
(1) Helmholtz model: This model of the interface is comparable
to the classic problem of a parallel-plate capacitor. One plate
would be on the metal surface. The other, formed by the ions
of opposite charge from solution rigidly linked to the
electrode. So xH would be an ionic radius.

22
2.3. Interface
(2) Stern model: In the model, the compact layer of ions
(Helmholtz layer) is followed by a diffuse layer extending into
bulk solution. The distribution of species in the diffusion layer
with distance from the compact layer obeys Boltzmann’s low. So
the thickness of the diffuse layer would be (εrε0kBT/2ciz2e2)1/2
where εr is relative dielectric constant of the solution and ε0 is
vacuum permittivity.

23
2.4. Kinetics
In order for the consideration electrode reactions, the following
five steps must be listed up.
Step 1. Diffusion of the species to where the reaction occurs.
Step 2. Rearrangement of the ionic atmosphere.
Step 3. Reorientation of the solvent dipoles.
Step 4. Alterations in the distances between the central ion and
the ligands.
Step 5. Electron transfer.

24
2.4. Kinetics
The Arrhenius expression relates the activation enthalpy, ΔH*,
with the rate constant, k:
k=Aexp[-ΔH*/RT]
A is the pre-exponential factor. If we write the pre-exponential
factor, A, as
A=A'exp[ΔS*/R],
then
k=A'exp[-(ΔH*-TΔS*)/RT]=A'exp[-ΔG*/RT]
In an electron transfer reaction, the rearrangement of the ionic
atmosphere is a fundamental step, and thus it is useful to
include the activation entropy ΔS*.

25
2.4. Kinetics
By applying a potential to the electrode, we influence the highest
occupied electronic level in the electrode (the Fermi level, EF).
Electrons are always transferred to and from this level.

26
2.4. Kinetics
For a reduction we can write
ΔGc*=ΔGc*,o+αcnFE.
In a similar way, for an oxidation
ΔGa*=ΔGa*,o-αanFE
where E is the potential applied to the electrode and α is a
measure of the slope of the energy profiles in the transition state
zone and, therefore, of barrier symmetry. Values of αc and αa
can vary between 0 and 1, but for metals are around 0.5. A value
of 0.5 means that the activated complex is exactly halfway
between reagents and products on the reaction coordinate, its
structure reflecting reagent and product equally. In this simple
case of a one-step transfer of n electrons between an oxidant
and a reductant, it is easily deduced that (αa+αc) = 1.

27
2.4. Kinetics
We obtain for a reduction
kc=A'exp[-ΔGc*,o/RT]exp[-αcnFE/RT]
and for an oxidation
ka=A'exp[-ΔGa*,o/RT]exp[αanFE/RT].
These equations can be rewritten in the form
kc=kc,oexp[-αcnF(E-E0)/RT]
and
ka=ka,0exp[αanF(E-E0)/RT].
This is the formulation of electrode kinetics first derived by
Butler and Volmer. On changing the potential applied to the
electrode, we influence ka and kc in an exponential fashion.

28
2.4. Kinetics
The observed current for kinetic control of the electrode
reaction is proportional to the difference between the rate of the
oxidation and reduction reactions at the electrode surface and is
given by
I = nFA(ka[R]*-kc[O]*)
where A is the electrode area.

29
2.4. Kinetics
Note that kc[O]* and ka[R]* is going to be limited by the
transport of species to the electrode.
- When all the species that reach it are oxidized or reduced the
current cannot increase further.
- Diffusion limits the transport of electroactive species close to
the electrode; the maximum current is known as the
diffusion-limited current.

30
2.4. Kinetics
This is affected not only by the electrode reaction itself but also
by the transport of species to and from bulk solution. This
transport can occur by diffusion, convection, or migration.
Normally, conditions are chosen in which migration effects can
be neglected, this is the effects of the electrode's electric field are
limited to very small distances from the electrode. These
conditions correspond to the presence of a large quantity
(>0.1M) of an inert electrolyte (supporting electrolyte), which
does not interfere in the electrode reaction. Using a high
concentration of inert electrolyte, and concentrations of 10-3 м
or less of electroactive species, the electrolyte also transports
almost all the current in the cell, removing problems of solution
resistance and contributions to the total cell potential. In these
conditions we need to consider only diffusion and convection.

31
2.4. Kinetics
Diffusion is due to the thermal movement of charged and
neutral species in solution, without electric field effects.
Forced convection considerably increases the transport of
species. Natural convection, due to thermal gradients, also
exists, but conditions where this movement is negligible are
generally used.
We consider systems under conditions in which the kinetics of
the electrode reaction is sufficiently fast that the control of the
electrode process is totally by mass transport. This situation
can, in principle, always be achieved if the applied potential is
sufficiently positive (oxidation) or negative (reduction).

32
2.4. Kinetics
Diffusion is the natural movement of species in solution, without
the effects of the electric field. The rate of diffusion depends on
the concentration gradients. Fick's first law expresses this:
𝑱 = −𝑫
𝝏𝒄
𝝏𝒙
,
where J is the flux of species, 𝝏с/𝝏х the concentration gradient
in direction x (a plane surface is assumed) and D is the
proportionality constant known as the diffusion coefficient.

33
2.4. Kinetics
The D value in aqueous solution normally varies between 10-5
and 10-6 cm2/s, and can be determined through application of
the equations for the current-voltage profiles of the various
electrochemical methods.
Alternatively, the Nernst-Einstein (D=kTμ) or Stokes-Einstein
relations (D=kT/6πημ) may be used to estimate values of D.

34
2.4. Kinetics
The variation of concentration with time due to diffusion is
described by Fick's second law which, for a one-dimensional
system, is
𝝏𝒄
𝝏𝒕
= 𝑫
𝝏 𝟐
𝒄
𝝏𝒙 𝟐.
The solution of Fick's second law gives also the variation of flux,
and the diffusion-limited current, with time, it being important
to specify the conditions necessary to define the behavior of the
system (boundary conditions).

35
2.4. Kinetics
The experiment leading to the diffusion-limited current involves
application of a potential step at t=0 to an electrode, in a solution
containing either oxidized or reduced species, from a value where
there is no electrode reaction to the value where all electroactive
species that reach the electrode react. This gives rise to a diffusion-
limited current whose value varies with time. For a planar electrode,
which is uniformly accessible, this is called semi-infinite linear
diffusion, and the current is
𝑰 = 𝒏𝑭𝑨𝑫
𝝏𝒄
𝝏𝒙 0
where I=nFAJ, x is the distance from the electrode, and we consider,
for simplicity, an oxidation (anodic current) with c=[R]. If it were a
reduction, a minus sign would be introduced into the above equation.

36
2.4. Kinetics
𝒄 = 𝒄∞ 𝟏 − 𝒆𝒓𝒇𝒄
𝒙
𝒙 𝑫𝒕 𝟏
/
𝟐 𝑰 𝒕 = 𝑰𝒅 𝒕 =
𝒏𝑭𝑨𝑫 𝟏
/
𝟐
𝒄∞
𝝅𝒕 𝟏
/
𝟐

37
2.4. Kinetics
For small values of t there is a capacitive contribution to the
current, due to double layer charging, that has to be subtracted.
This contribution arises from the attraction between the
electrode and the charges and dipoles in solution, and differs
according to the applied potential (Q=CV); a rapid change in
applied potential causes a very fast change in the distribution of
species on the electrode surface and a large current during up to
about 0.1s.

38
2.4. Kinetics
The concentration gradient tends asymptotically to zero at large
distances from the electrode, and the concentration gradient is
not linear. However, for reasons of comparison it is useful to
speak of a diffusion layer defined in the following way:
𝑫
𝝏𝒄
𝝏𝒙 0=D
(𝒄∞
−𝒄 𝟎
)
𝜹
where 𝜹 is the diffusion
layer thickness.

39
2.4. Kinetics
The diffusion layer results from the extrapolation of the
concentration gradient at the electrode surface until the bulk
concentration value is attained. This approximation was
introduced by Nernst. 𝜹 is frequently related to the mass
transfer coefficient kd since when c0=0
𝒌d=D/𝜹,
kd has the dimensions of a heterogeneous rate constant.
The diffusion layer thickness is expressed as
𝜹=(𝝅𝑫𝒕) 𝟏/ 𝟐 .
The mass transfer coefficient is for c0=0
𝒌d=(𝐃/𝝅𝒕) 𝟏/ 𝟐.

40
2.4. Kinetics
So far, the kinetics of
(1) electrode processes
and
(2) mass transport to an electrode
have been discussed.
From now on, these two parts of the electrode process are
combined and we see how the relative rates of the kinetics and
transport cause the behavior of electrochemical systems to vary.

41
2.4. Kinetics
Mass transport to the electrode surface assumes that this occurs
solely and always by diffusion (except under forced convection).
The mass transfer coefficient kd describes the rate of diffusion
within the diffusion layer, and kc and ka are the rate constants
of the electrode
reaction for reduction
and oxidation,
respectively.
Thus for the simple
electrode reaction
O+ne-R,

42
2.4. Kinetics
kd,O and kd,R are the mass transfer coefficients of the species О
(oxidizing agent) and R (reducing agent). In general these
coefficients differ because the diffusion coefficients differ. We
already have the Butler-Volmer expressions for the kinetic rate
constants:
kc=kc,oexp[-αcnF(E-E0)/RT]
ka=ka,0exp[αanF(E-E0)/RT].
Assume that (𝝏c/ 𝝏t)=0, i.e. steady state, in other words the
rate of transport of electroactive species is equal to the rate of
their reaction on the electrode surface (Note that the rate of
mass transport is usually lower than that of reactions on the
electrode surface.). The steady state also means that the applied
potential has a fixed value.

43
2.4. Kinetics
The flux of electroactive species, J, is
𝑱 = −𝒌𝒄 𝑶
∗
+𝒌𝒂 𝑹
∗
= kd,O([O]*-[O]∞)
= kd,R([R]∞-[R]*)
When all О or R that reaches the electrode is reduced or
oxidized, we obtain the diffusion-limited cathodic or anodic
current densities Jl,C and Jl,a:
𝑱𝒍
,
𝒄
𝒏𝑭
= −𝐤𝐝, 𝐎[𝐎]∞,
𝑱𝒍
,
𝒂
𝒏𝑭
= −𝐤𝐝, 𝐑[𝐑]∞
Since kd=D/δ, we can write
kd,o/kd,R=p=(DO/DR)1/2,
𝑱 =
𝒌 𝒄 𝑱𝒍, 𝒄 + 𝒑𝒌𝒂𝒋𝒍, 𝒂
𝒌 𝒅, 𝑶 + 𝒌𝒄 + 𝒑𝒌𝒂

44
2.4. Kinetics
We can point out extreme cases for this expression:
Let us consider only О present in solution: Jl,a=0 and ka=0. Thus
𝑱 =
𝒌 𝒄 𝑱𝒍, 𝒄
𝒌 𝒅, 𝑶 + 𝒌𝒄
that is
𝟏
𝑱
=
𝒌 𝒅, 𝑶
𝒌 𝒄 𝑱𝒍, 𝒄
+
𝟏
𝒋𝒍, 𝒄
= −
𝟏
𝒏𝑭𝒌 𝒄 𝑶 ∞
−
𝟏
𝒏𝑭𝒌 𝒅, 𝑶[𝑶]∞
This result shows that the total flux is due to a transport and a
reaction term. When kc>>kd,o then
𝟏
𝑱
= −
𝟏
𝒏𝑭𝒌 𝒅, 𝑶[𝑶]∞
reaction transport

45
2.4. Kinetics
and the flux is determined by the transport. On the other hand,
when kc<<kd,o
𝟏
𝑱
= −
𝟏
𝒏𝑭𝒌 𝒄 𝑶 ∞
and the kinetics determines the flux.

46
2.4. Kinetics
We now consider the factors that affect the variation of kc, ka,
and kd. The kinetic rate constants depend on the applied
potential and on the value of the standard rate constant, k0.
When [O]*=[R]*, then kc=ka=k0.
At the moment we note that there are two extremes of
comparison between k0 and kd:
k0 >> kd … reversible system
k0 << kd … irreversible system
The word reversible signifies that the system is at equilibrium at
the electrode surface and it is possible to apply the Nernst
equation at any potential.

47
2.4. Kinetics
k0 >> kd … reversible system
k0 << kd … irreversible system
These are
the variation of current
with applied potential,
voltammograms.

48
2.4. Kinetics
Reversible reactions are those where ko>>kd and, at any
potential, there is always equilibrium at the electrode surface.
The current is determined only by the electronic energy
differences between the electrode and the donor or acceptor
species in solution and their rate of supply. Applying the Nernst
equation
𝑬 = 𝑬 𝟎 +
𝑹𝑻
𝒏𝑭
𝒍𝒏
𝑶
∗
𝑹
∗
and given that j/nF=kd,0([0]*-[O]∞) we have
𝑱
𝑱𝒍, 𝒄
=
𝑶
∗
−[𝑶]∞
[𝑶]∞
that is
𝑶
∗
=
𝑱𝒍
,
𝒄
−𝑱
𝑱𝒍
,
𝒄
[𝑶]∞.

49
2.4. Kinetics
Similarly,
𝑶
∗
=
𝑱𝒍
,
𝒂
−𝑱
𝑱𝒍
,
𝒂
[𝑶]∞.
Substituting above two equations in the Nernst equation,
assuming the electrode is uniformly accessible (I=AJ), we get
the steady-state expression
𝑬 = 𝑬 𝟎 +
𝑹𝑻
𝒏𝑭
𝒍𝒏
𝑰𝒍, 𝒄 − 𝑰
𝑰 − 𝑰𝒍, 𝒂
𝒌 𝒅, 𝑹
𝒌 𝒅, 𝑶
= 𝑬𝒓 𝒆𝒗
𝟏/ 𝟐 +
𝑹𝑻
𝒏𝑭
𝒍𝒏
𝑰 − 𝑰𝒍, 𝒂
where
𝑬 𝒓𝒆𝒗
𝟏/ 𝟐= 𝑬 𝟎 +
𝑹𝑻
𝒏𝑭
𝒍𝒏
𝒌 𝒅
,
𝑹
𝒌 𝒅
,
𝑶

50
2.4. Kinetics
E1/2 is called the half-wave potential and corresponds to the
potential when the current is equal to (Il,a+Il,c)/2.

51
2.4. Kinetics
For irreversible reactions, ko<<kd, kinetics has an important
role, especially for potentials close to Eeq. It is necessary to apply
a higher potential than for a reversible reaction in order to
overcome the activation barrier and allow reaction to occur.
This extra potential is called the overpotential, η. Because of the
overpotential only reduction or only oxidation occurs and the
voltammogram, or voltammetric curve, is divided into two
parts. At the same time it should be stressed that the retarding
effect of the kinetics causes a lower slope in the voltammograms
than for the reversible case.

52
2.4. Kinetics
reversible system irreversible system

53
2.4. Kinetics
The half-wave potential for reduction or oxidation varies with
kd, since there is not equilibrium on the electrode surface. For
cathodic and anodic processes respectively we may write
𝑬 = 𝑬𝒊 𝒓𝒓
𝟏/ 𝟐 −
𝑹𝑻
α 𝒄 𝒏𝑭
𝒍𝒏
𝑰
𝑬 = 𝑬𝒊𝒓𝒓
𝟏/ 𝟐 +
𝑹𝑻
α 𝒄
𝒏𝑭
𝒍𝒏
𝑰𝒍
,
𝒄
−𝑰
𝑰
where α is the charge transfer coefficient.

54
2.4. Kinetics
The electrolyte double layer affects
the kinetics of electrode reactions.
For charge transfer to occur,
electroactive species have to reach
at least to the outer Helmholtz
plane. Hence, the potential
difference available to cause
reaction is (фм-ф𝑰) and not (фм-фS).
Only when ф𝐈~фS we can say that
the double layer does not affect the
electrode kinetics. Additionally, the
concentration of electroactive
species will be, in general, less at
distance xH from the electrode than
outside the double layer in bulk
solution.
ф
фM
ф𝑰
ф 𝑺
x 𝑯

55
The electrochemical response to an AC perturbation is very
important in impedance techniques. This response cannot be
understood without a knowledge of the fundamental principles
of AC circuits. We consider the application of a sinusoidal
voltage
𝑽 = 𝑽 𝟎 sin 𝝎𝒕
where Vo is the maximum amplitude and ω the frequency (unit
is rad/s) to an electrical circuit that contains combinations of
resistances and capacitances which will adequately represent
the electrochemical cell. The response is a current, given by
𝑰 = 𝑰 𝟎 sin(𝝎𝒕 + ϕ)
where ϕ is the phase angle between perturbation and response.

56
Impedances consist of resistances, reactances (derived from
capacitive elements) and inductances. Inductances will not be
considered here, as for electrochemical cells, they only arise at
very high frequencies (>1 MHz).
In the case of a pure resistance, R, Ohm's law V=IR leads to
𝑰 =
𝑽 𝟎
𝑹
sin(𝝎𝒕 + ϕ)
and ϕ=0. There is no phase difference between potential and
current.

57
For a pure capacitor
𝑰 = 𝑪
𝒅𝑽
𝒅𝒕
=𝝎𝑪𝑽 𝟎 sin(𝝎𝒕 +
𝝅
𝟐
) =
𝑽 𝟎
𝑿 𝒄
sin(𝝎𝒕 +
𝝅
𝟐
)
We see that ф=π/2, that is the current lags behind the potential
by π/2. Хс=(ωC)-l is known as the reactance (measured in
ohms).

58
Given the different phase angles of resistances and reactances
described above, representation in two dimensions is useful.

59
On the x-axis the phase angle is zero; on rotating anticlockwise
about the origin the phase angle increases; pure reactances are
represented on the у -axis. The distance from the origin
corresponds to the amplitude. This is precisely what is done
with complex numbers as represented vectorially in the complex
plane: here the real axis is for resistances and the imaginary
axis for reactances. The current is always on the real axis. Thus
it becomes necessary to multiply reactances by -i.
-iX 𝐂
R
Z
ф

60
We exemplify the use of vectors in the complex plane with a
resistance and capacitance in series. The total potential
difference is the sum of the potential differences across the two
elements. From Kirchhoff's law the currents have to be equal,
that is I=IR=IC.
The differences in potential are proportional to R and Xc
respectively. Their representation as vectors in the complex
plane is …

61

62
The vectorial sum of - iXc and of R gives the impedance Z. As a
vector, the impedance is Z=R-iXc. The magnitude of the
impedance is |Z|=(R2+Xc2)1/2, and the phase angle is
ϕ = 𝒂𝒓𝒄𝒕𝒂𝒏
|𝑿𝒄|
|𝑹|
=
𝟏
𝝎𝑹𝑪
Often the in-phase component
of the impedance is referred to
as Z’ and the out-of-phase
component, i.e. at π/2, is called Z",
that is Z=Z'+iZ". Thus for this
case Z'=R, Z"=-Xc. This is a vertical
line in the complex plane impedance
plot, since Z' is constant but Z" varies with frequency.

63
For CR parallel circuit, the total current is the sum of the two
parts, the potential difference across the two components being
equal:
𝑰𝒕𝒐𝒕 =
𝑽 𝟎
𝑹
sin(𝝎𝒕) +
𝑽 𝟎
𝑿 𝒄
sin(𝝎𝒕 +
𝝅
𝟐
)
We need to calculate the vectorial sum of the currents. Thus
𝑰𝒕𝒐𝒕 = (𝑰𝑹 𝟐 + 𝑰𝑪 𝟐)1/2 = 𝑽(
𝟏
𝑹 𝟐 +
𝟏
𝑿 𝑪
𝟐)-1/2 .
The magnitude of the impedance is |𝒁| = 𝑽(
𝟏
𝑹 𝟐 +
𝟏
𝑿 𝑪
𝟐)-1/2
and the phase angle is ϕ = 𝒂𝒓𝒄𝒕𝒂𝒏
𝑰𝒄
𝑰 𝑹
=
𝟏
𝝎𝑹𝑪
, which is equal to
the CR series combination.

64
𝑰𝒕𝒐𝒕 =
𝑽 𝟎
𝑹
sin(𝝎𝒕) +
𝑽 𝟎
𝑿 𝒄
sin(𝝎𝒕 +
𝝅
𝟐
)
So, 1/Z=1/R+iωC, Z=R/(1+iωCR). This is easily separated into
real and imaginary parts via multiplication by (1-iωCR). Thus
𝒁 =
𝑹(1−iω𝑪𝑹)
𝟏+ ω𝑪𝑹 𝟐 , 𝒁′ =
𝑹
𝟏+ ω𝑪𝑹 𝟐, 𝒁" =
−𝑪𝑹 𝟐
𝟏+ ω𝑪𝑹 𝟐.
This is a semicircle in the complex plane of radius R/2 and
maximum value of |Z"| defined by ωCR=1.

65

66
2.5.1. Impedance methods
These methods involve the application of a small perturbation,
whereas in the methods based on linear sweep or potential
step the system is perturbed far from equilibrium. This small
imposed perturbation can be of applied potential, or of applied
current rate. The small perturbation brings advantages: it is
possible to use limiting forms of equations, which are normally
linear (e.g. the first term in the expansion of exponentials).

67
The response to the applied perturbation, which is generally
sinusoidal, can differ in phase and amplitude from the applied
signal. Measurement of the phase difference and the amplitude
(i.e. the impedance) permits analysis of the electrode process in
relation to contributions from diffusion, kinetics, double layer,
coupled homogeneous reactions, etc. There are important
applications in studies of corrosion, membranes, ionic solids,
solid electrolytes, conducting polymers, and liquid/liquid
interfaces.

68
Comparison is usually made between the electrochemical cell
and an equivalent electrical circuit that contains combinations
of resistances and capacitances (inductances are only important
for very high perturbation frequencies, > 1 MHz). There is a
component representing transport by diffusion, a component
representing kinetics (purely resistive), and another
representing the double layer capacity, this for a simple
electrode process. Another strategy is to choose a model for the
reaction mechanism and kinetic parameters, derive the
impedance expression, and compare with experiment. Given
that impedance measurements at different frequencies can, in
principle, furnish all the information about the electrochemical
system.

69
The impedance is the proportionality factor between potential
and current; if these have different phases then we can divide
the impedance into a resistive part, R where the voltage and
current are in phase, and a reactive part, Xc=l/ωC, where the
phase difference between current and voltage is 90°.

70
Any electrochemical cell can be represented in terms of an
equivalent electrical circuit that comprises a combination of
resistances and capacitances (inductances only for very high
frequencies). This circuit should contain at the very least
components to represent:
• the double layer: a pure capacitor of capacity Cd
• the impedance of the faradaic process Zf
• the un-compensated resistance, RΩ, which is, usually, the
solution resistance between working and reference electrodes.

71
• the double layer: a pure capacitor of capacity, Cd
• the impedance of the faradaic process, Zf
• the un-compensated resistance, RΩ, which is, usually, the
solution resistance between working and reference electrodes.

72
Impedance of the faradaic process, Zf
Resitance to charge transfer, Rct and,
Impedance that measures the difficulty of mass transport of
the electroactive species, Warburg impedance, Zw.

73
For kinetically favored reactions Rct0 and Zw predominates.
For difficult reactions Rct∞ and Rct predominates.

74
<Plot of the impedance in the complex plane>
The low-frequency limit
is a straight line, which
extrapolated to the real
axis gives an intercept.
The line corresponds to a
reaction controlled solely
by diffusion, and the
impedance is the
Warburg impedance, the
phase angle being π/4.

75
<Plot of the impedance in the complex plane>
At the high-frequency
limit the control is purely
kinetic, and RCT>>Zw.
The electrical analogy is
an CR parallel
combination..

76
2.5.1. Cyclic voltammetry and linear sweep technique
Cathodic current
Anodic current
Cyclic voltammogram
Linear sweep

77
These techniques are potential sweep methods. They consist in
the application of a continuously time-varying potential to the
working electrode. This results in the occurrence of oxidation or
reduction reactions of electroactive species in solution (faradaic
reactions) and a capacitive current due to double layer
charging. The total current is Itot=IF+IC=IF+Cd(dE/dt). Thus
IF∝ 𝒗 𝟏/ 𝟐 and I 𝐂 ∝ 𝒗: this means that the capacitive current must
be subtracted in order to obtain accurate values of rate
constants (usually IC decays to zero within <0.1 ms only when
an appropriate measuring system with a small CR time
constant is used).

78
These techniques are potential sweep methods. They consist in
the application of a continuously time-varying potential to the
working electrode. This results in the occurrence of oxidation or
reduction reactions of electroactive species in solution (faradaic
reactions) and a capacitive current due to double layer
charging. The total current is Itot=IF+IC=IF+Cd(dE/dt). Thus
IF∝ 𝒗 𝟏/ 𝟐 and I 𝐂 ∝ 𝒗: this means that the capacitive current must
be subtracted in order to obtain accurate values of rate
constants. Usually IC decays to zero within <0.1 ms (but only
when an appropriate measuring system with a small CR time
constant is used). Note that 𝑬 =
∆𝑬
𝑹
𝒆𝒙𝒑(−
𝒕
𝑪𝑹
) where R is the
solution resistance, RΩ, and C is the double layer capacitance,
Cd.

79
The observed current is different from that in the steady state
(dc/dt=0). Its principal use has been to diagnose mechanisms of
electrochemical reactions, for the identification of species
present in solution and for the semiquantitative analysis of
reaction rates. Although some improvements can be shown
recently, it is basically difficult to determine kinetic parameters
accurately from these experimental results.

Introduction to electrochemistry 2 by t. hara

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