Download Electrochemistry Extended Notes

Some Concepts of Electrochemistry
(Ref. Bard and Faulkner II ed.)
1. Background:
Electrochemistry is broadly defined as the branch of chemistry that examines the
phenomena resulting from combined chemical and electrical effects. This field covers
reactions in which chemical changes occur on the passage of an electrical current
(electrolytic processes), and chemical reactions that result in the production of
electrical energy (galvanic or voltaic processes). This chapter will attempt to provide
a brief review of electrochemistry concepts, and will limit itself to the later category
of electrochemical phenomena.
2. A Brief Review:
a.
Electrochemical cell, Oxidation and Reduction: An electrochemical cell
typically consists of two electronic conductors (also called electrodes) and an ionic
conductor (called an electrolyte). Charge transport in the electrodes occurs via the
motion of electrons (or holes), while in the electrolyte, charge is transported by ions.
At each electrode, an electrochemical reaction (called a half cell reaction) occurs. The
overall chemical reaction of the cell is given by combining the two individual half
cell reactions. One of the half cell reactions involves the loss of an electron, and is
called an oxidation reaction, typically represented by the equation:
R = O + ne (1)
Where R represents the reduced form, O the oxidized form, e represents a single
electron, and n is an integer, and represents the number of electrons transferred. The
other half cell reaction is the reverse of reaction 1, and involves the gain of an
electron. This reaction is also called the reduction reaction:
O + ne = R (2)
Oxidation and reduction reactions therefore involve the transfer of electrons from the
compound to the electrode and from the electrode to the compound respectively.
They are energy driven processes. Reduction occurs when the energy of the electrode
is higher than that of the lowest vacant molecular orbital of the compound, thereby
facilitating the transfer of the electron from the electrode to the vacant orbital.
Similarly oxidation occurs when the energy of the electrode dips below the highest
occupied molecular orbital of the compound, thereby facilitating the transfer of an
electron from the orbital to the electrode. This is illustrated in Fig. 1.
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Fig. 1
An example of an electrochemical cell is one with zinc and copper metals (Fig. 2 http://hyperphysics.phy-astr.gsu.edu) placed in a solution of their respective sulfates,
and separated by a semi permeable membrane. In this case, the zinc metal gets
oxidized, and goes into solution:
Zn = Zn 2+ + 2 e (3)
At the other electrode, the copper ions in solution get reduced, and copper metal is
deposited on the copper electrode. The electrons for this process are obtained from
the zinc electrode through an external wire:
Cu2+ + 2e = Cu (4)
The sulfate ions that are created due to reaction (4) migrate through the membrane,
and react with the zinc ions formed in (3) to produce more zinc sulfate.
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Fig. 2
Conventionally, the electrode at which the oxidation reaction occurs is called the
anode, and the one at which the reduction reaction occurs is called the cathode.
Generally, the electrode at which the reaction of interest occurs (for example the
electrode at which copper is deposited) is called the working electrode, and the
electrode at which the other (coupled) reaction occurs is called the counter electrode.
A third electrode, called the reference electrode may also be used (discussed later)
b. What gets oxidized? – half cell potential and the electrochemical series: In the
previous example, we had the zinc get oxidized, and the copper reduced. However,
this was not an arbitrary assignation. For a given set of two reversible redox reactions,
one can predict (based on thermodynamics) which reaction proceeds as an oxidation
and which one proceeds as a reduction. This prediction is made based upon the
potential maintained at an electrode (that behaves as a half cell) at which the redox
couple occurs. This potential is typically measured against a reference electrode (the
other half cell) such as a standard hydrogen electrode (SHE) at standard conditions of
1 atm. Pressure, unit hydrogen activity and 298K, whose potential is arbitrarily
assigned to be 0. The potential for a given reaction is derived directly from the free
energy change for that reaction (section on thermodynamics). Also, it is a trivial
statement that the potential for the oxidation reaction (standard oxidation potential) is
equal in magnitude, but opposite in sigh to the potential for the reduction reaction
(std. reduction potential). For a set of two competing reactions (such as zinc and
copper in the above examples), the reaction with the lower standard reduction
potential gets oxidized. The other reaction occurs as a reduction reaction. Tables that
list the standard reduction potentials of all conceivable half cell reactions are
available in standard references such as the Lange’s Handbook of chemistry.
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c. Faraday’s law, faradaic and non faradaic processes: Faraday’s law provides a
relationship between the quantity of current (charge) passed through a system, and the
quantity of (electro) chemical change that occurs due to the passage of the current.
The law may be writte in equation form as:
m = M I t /n F (5)
where m is the mass of substance, M is the molecular weight of the substance, I is the
current passed in amperes, t is the time for which the current is passed in seconds, n is
the number of electrons transferred, and F is the Faraday constant (96475 C / eqv).
Thus, the amount of change is proportional to the amount of current passed. The
above is Faraday’s first law. Faraday’s second law is a restatement of the first for a
fixed quantity of charge passing through the system.
All processes that obey Faraday’s law are termed faradaic processes. All these
processes involve electron transfer at an electrode / electrolyte interface. These
reactions are also called electron (charge) transfer reactions (due to the transfer of
electrons), and electrodes at which these processes occur are called charge transfer
electrodes.
There can be a change in the electrode / electrolyte interface without charge
transfer taking place. These changes are due to processes such as adsorption and
desorption . Though no electrons flow through the interface, external currents
(transient in nature) can be generated by these “nonfaradaic” processes. Typically,
both faradaic and non faradaic processes occur when an electrochemical reaction
takes place. Since the faradaic processes are the ones that interest us the most, care
must be taken to ensure that the effects of the nonfaradaic processes are understood,
and accounted for during data interpretation. More information about nonfaradaic
processes is provided below.
An electrode at which there is no charge transfer across the electrode / electrolyte
interface over all potential ranges is called an ideally polarized electrode (IPE).
There are certain electrodes that behave as an IPE over a given potential range
(though none exists that can cover the entire potential range in solution). Examples
include a mercury electrode in contact with deaerated KCl (potential range of ~ 2V).
Other electrodes are ideally polarized over much smaller ranges. Since no charge
transfer is possible across an ideally polarized electrode / electrolyte interface, the
behaviour of such an interface when the potential across the electrode is changed
resembles that of a plain capacitor (2 metallic sheets separated by a dielectric
material). The behaviour of such an electrode follows the standard capacitor equation:
Q = CE (6)
Where Q is the charge stored in coloumbs, C is the capacitance in farads and E is the
potential across the capacitor (IPE) in volts. Therefore, when voltage is applied across
an IPE, the “electrical double layer” is charged until eqn. 6 is obeyed. During this
charging process, a current (known as a charging current) will flow. Typically, it is
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not appropriate to ignore this charging current, especially when dealing with low
faradaic currents (dilute solutions). In this case, the charging currents may even
exceed the faradaic current. In order to better understand the effect of charging, we
examine mathematically the responses of an IPE to various electrochemical stimuli.
Before doing that we assume that the IPE system can be represented as a capacitance
(C)(as discussed above) in series with the electrolyte resistance (R) (Fig. 3)
Fig. 3
If a potential step is applied (fig. 4), or in other words, the voltage is suddenly
Fig. 4
raised to a value E, the double layer is charged according to eqn. 6:
Q = C E (6)
Since (from fig. 3) the applied voltage must equal the sum of the voltage drops across
the resistor (Er) and capacitor (Ec), we have:
E = Er + Ec (7)
Now:
Er = I R (8) [ohms law]
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And :
Ec = q/C (9) [from (6)]
Therefore,
E = IR + q/C (10)
Note that current is the rate of flow of charge. Therefore:
I = dQ/dt (11)
Thus, equation 10 can be rewritten as:
I = dQ/dt = -Q/RC + E/R (12)
If we assume (as an initial condition) that the capacitor is initially uncharged (namely
Q = 0 at t = 0), we have the solution of 12 to be:
Q = E C [1 – e (-t/RC)] (13)
Differentiating 13 wrt time, we get:
I = (E/R) * e (-t/RC) (14)
Equation 14 describes the time dependence of the charging current in response to a
potential step. For typical values of R and C, the charging current dies down to an
insignificant level is a few hundred microseconds. Any faradaic current must be
measured after this time period to avoid the influence of the charging current.
d. Factors to be considered in faradaic processes: Electrochemical behaviour
investigation involves controlling certain variables constant while observing the
trends in others. Several possible variables are shown in Fig. 5.
Fig. 5
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The main aspects that interest us from a fuel cell operation point of view are the
electrochemical kinetics, ionic and electronic resistances (of electrolyte and
electrodes) and mass transport through the electrodes. This is illustrated in Fig. 6. A
brief overview of these aspects is provided below. The issues will be discussed in
greater detail in following chapters.
Fig. 6
e. Current and reaction rate: In general, current may be written as the rate of change
of charge:
I = dQ/dt (15)
Also, the number of moles (N) consumed or generated by an electrochemical reaction
consuming (or liberating) n electrons is given by:
N = Q/nF (16)
Therefore, one can write an expression (similar to chemical kinetics) for the
electrochemical reaction rate as:
Rate (mol/s) = dN/dt = (1/nF) dQ/dt = I/nF (17)
This can be written as a flux by dividing the rate by the active area of the electrode:
Flux (J, mol/cm2.s) = I/nFA = i/nF (18)
Where i = current density = I/A.
f. Polarization and overpotential: For any faradaic reaction, there exists an
equilibrium potential based upon the reaction free energy. Upon application of a
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faradaic current, the potential shifts from this equilibrium potential. This shift is
termed polarization, and the extent of this shift is measured by overpotential (ή)
ή = E – Eeq, (19)
Information about the faradaic reaction is gained determining current as a function of
potential or vice versa. The resulting curves are called polarization curves (or V-I
curves or E-I curves). Obtaining such curves is a critical part of fuel cell research.
g. Resistance in electrochemical cells: Resistance plays an important role in
electrochemical reactions. During operation of an electrochemical cell, two different
currents may be considered. One is the electronic current flowing due to redox
reactions at the electrodes. This current is typically directed through an external
circuit. Additionally, there is an ionic current due to the transport of ions through the
electrolyte. Evidently, then there are two different sources for cell resistance, namely
electronic and ionic resistances. The electronic resistance can be minimized
(especially in fuel cells) by using electron conducting materials at the electrodes.
However, the ionic resistance needs to be considered very seriously.
Current is carried in solution (in the electrolyte) by the movement of ions. The
positive ions are in excess at the anode (due to the oxidation process that occurs at the
anode), and lead to a build up of positive charge. Similarly, there is a buildup of
negative charge at the cathode (due to the reduction process that occurs at the
cathode). This buildup of charge is released by the movement of ions, with positive
ions moving from anode to cathode and vice versa (keep in mind that this scenario is
only when the reactants are in solution. – In a fuel cell, only positive ions are
transported from anode to cathode). The fractions of currents carried by the positive
and negative ions are given by their transport numbers (also called transference
numbers) t+ and t- respectively. Since each mole of current passed corresponds to one
mol of electrochemical change at each electrode, the amount of ions transported in the
electrolyte also equals 1 mol. Therefore:
t+ + t- = 1 (20).
More generally,
Σ ti = 1 (21)
The transport numbers of ions are determined by the conductance (L) of the
electrolyte. The conductance of an electrolyte is given by:
L = κ A / l (22)
Where A is the active area and l is the length (thickness). The constant of
proportionality (κ) is termed the conductivity, and is an intrinsic property of the
electrolyte. The conductance has units of Seimens (S), and conductivity of S/cm. The
conductivity of the electrolyte has a contribution from every ion n solution, and is
proportional to the concentration of the ion (C), its charge (z) and a property that
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determines its migration velocity. The latter property is the mobility of the ion which
may be defined as the limiting velocity of the ion in an electric field of unit strength.
Sine the force exerted by an electric field of strength E is given by:
F = eE*z (23)
Where e is the electronic charge and z is the ionic charge. AN opposing force exists
due to frictional drag. This is represented by the Stokes equation:
Fopp = 6Πηrv (24)
Where η is the viscosity of the solution, r is the ionic radius and v is the velocity of
the ion in solution. When the forces exactly counterbalance each other, the ion attains
a steady state velocity called the terminal velocity. This velocity is termed the
mobility (u) when the electric field strength is unity:
u = z e / 6Πηr (25)
With this expression for mobility, we can proceed to define the conductivity as
follows:
κ = F Σ zi ui Ci (26)
Since we have defined the transport number as the contribution made by each
individual ion to the total current carried in solution, it may be represented as the ratio
of the contribution to conductivity made by that ion to the total conductivity of the
solution:
ti = zi ui Ci / Σ zi ui Ci (27)
The resistance (R) of an electrolyte may be defined as follows:
R = L/ κ A (28)
Clearly, the ionic resistance is an inherent property of the ion in motion and the
electrolyte. The effect of resistance in electrochemical cells is understood by
considering Ohm’s law:
V = I R (29)
Therefore, even if we ignore electronic resistances, the ionic resistance will introduce
a potential drop V which will increase with current. Clearly, it is in our interest to
minimize the resistance to avoid this ohmic potential drop. This may be accomplished
by minimizing the distance between the electrodes of interest. This lowers resistance
because resistance is directly proportional to this distance (Eqn. 29). One might also
think that increasing the area of contact will reduce resistance. While this is true, one
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must keep in mind that this area is dictated by numerous other considerations (such as
the amount of current required), and therefore cannot be altered easily. In any case, a
portion of the resistance is always extant, and therefore, any voltage measured across
two electrodes does not accurately represent the potential difference across the
electrodes as it also incorporates the potential drop in solution:
Emeasured = Ecathode – E anode – IR (30)
This ohmic loss can be compensated if the resistance of the electrolyte is known. The
resistance can be determined using techniques such as electrochemical impedance
spectroscopy. Once the resistance is know, the real voltage across the two electrodes
can be estimated:
Eactual = Emeasured + IR (31)
h. Mass transport in electrochemical cells: There are 3 fundamental modes of mass
transport in solution. They are:
1. Migration – this is the motion of a charge body (such as an ion) under the
influence of an electrical potential gradient
2. Diffusion – this is the motion of a species under the influence of a chemical
potential gradient
3. Convection – this is hydrodynamic transport either due to density gradients
(natural convection) or due to external means such as stirring (forced convection)
The governing equation for mass transfer is the Nernst –Plank equation, which can be
written for one (x) dimension (and one species) as follows:
J (x) = -D [δC(x)/δx] – [z F/RT][D*C δφ(x)/δx] + C*v(x) (32)
Where J is the flux in mol / cm2.s, D is the diffusion coefficient (cm2/s), δC(x)/δx is
the concentration gradient, δφ(x)/δx is the potential gradient and v is the velocity with
which an element of solution moves in the x dimension. From Faraday’s law (5), it is
clear that the current produced by a cell is directly proportional to the number of
moles of reactant present in solution. Thus, there is a direct relationship between the
current and the flux of a reacting species. This may be represented as follows:
I / nFA = m/M.t.A = N/t.A = J (33)
Where m/M = N = no. of moles. The first half of equation 33 follows from Faraday’s
law. In the case of a reaction where the kinetics are very fast, the reaction is
essentially limited by the rate of mass transfer of the reactant to the electrode. Thus
J = I /nFA (34)
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Now, if we assume that there is no convective mass transport at the electrode surface,
and that the transport due to migration is negligible (often the case if there is an
excess of supporting electrolyte – i.e. an electrolyte present just to carry charge), then
based on equations 32 and 34, the flux at the electrode may be written as follows
J = D(dC/dx)x=0 (35)
Where D is the diffusion coefficient of the reactant. If we assume that the
concentration gradient is linear very close to the electrode (in a layer called the Nernst
diffusion layer, which bounds the electrodes and the bulk), then eqn. 35 can be
written as:
J = [D/δ][C* -C(x=0)] (36)
Where δ is the thickness of the Nernst diffusion layer, and C* is the bulk
concentration of the reactant. It is not easy to precisely determine the value of δ, and
therefore it is grouped along with D to yield a new constant:
m = D/ δ (37)
where m is called the mass transfer coefficient (units: cm/s). Writing flux in terms of
current (eqn 34), and combining equations 36 and 37, we have:
I/nFA = m[C* -C(x=0)] (38)
As the reaction proceeds at a faster rate, a point is reached when all the reactant that
reaches the electrode surface is immediately consumed. In other words, the
concentration of the reactant at the electrode surface [C(x=0)] is zero (or, more
realistically, C(x=0)<< C*). Under these conditions, the current levels off, and is
called the limiting current (Il). Therefore, from eqn. 38, we have:
Il = nFAmC* (39)
Under these conditions, the electrode process is occurring at the maximum possible
rate for a given set of system conditions. This concept is employed to significant
advantage in fuel cell development research. From equations 38 and 39, we can write
the following relationship between surface and bulk concentrations:
C(x=0)/C* = [1 – (I/Il)] (40)
Substituting for C* from equation 39, we have:
C(x=0) = [(Il – I)/(nFAm)] (41)
3. Some Important Electrochemical Experiments
With the above background information, it is possible to go ahead and discuss
certain simple (but extremely useful) electrochemical experiments that are very
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pertinent (directly or indirectly) to fuel cell research. In doing so, a few new concepts
may be introduced where necessary. Further information about these techniques will
be provided as the course progresses, and as part of laboratory experiments.
a. Linear Sweep Voltammetry: In this experiment, the electrode of interest (working
electrode) is subject to a potential sweep (at a given sweep rate in V/s, see Fig. 7)
within a potential range (say 0 – 800 mV). Initially, when the potential is below from
the standard reduction potential of the cell, there are no faradaic processes occurring.
However, when the potential reaches the standard reduction potential, faradaic
reaction begins (either oxidation or reduction, depending upon the direction of the
sweep). As the potential moves farther away from the standard reduction potential,
the “overpotential” and hence the electrochemical driving force for faradaic reaction
is increasing, and therefore the reaction proceeds faster. Above a given potential, the
reaction becomes mass transport limited. If the concentration of the reactant in the
bulk is limited, then a dip in the current is seen (Fig. 7). This is due to a reduced flux
to the electrode due to the reduction of the concentration gradient. If the concentration
gradient is constant (well stirred system or flow system), the familiar plateau limiting
current behaviour is seen. In the case of limiting current behaviour, the current can
be converted into a reactant flux using Faraday’s law. The experiment is typically
done at a low sweep rate to enhance the thickness of the Nernst diffusion layer and
obtain a “true” limiting current behaviour, while maintaining Nernstian behaviour.
Fig. 7 (http://www.bath.ac.uk/~chsacf)
b. Cyclic Voltammetry: This is also a potential sweep experiment (similar to LSV),
with the difference being an additional reverse sweep (i.e. start from E = E1, sweep
up to E = E2 and sweep down back to E = E1 – See Fig. 8). Typically, the same
sweep rate is employed for both the forward and reverse sweeps. The resultant current
on the forward sweep is the same as discussed for LSV. The reverse process occurs
on the backward sweep (i.e. if the forward sweep is oxidation , reduction occurs
during the reverse sweep). The sweep rates (v) employed are typically higher than
those in LSV (~ 20-30mV/s as opposed to 2-4 mV/s). This is because the peak height
scales as v 0.5. Therefore, larger, more discernable peaks are seen. Care is however
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taken to ensure that the sweep rate is not so high that the system tends to become
quasi-reversible or irreversible. CV is very useful in studying the reversibility of
reactions and redox behaviour of compounds. Additionally, it can be used to obtain
quantitative information about the extent of reaction occurring on a surface, and from
that information, to compute certain properties of the electrode surface.
Fig. 8 (http://www.bath.ac.uk/~chsacf)
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