Download 2. Electrodics

1. Introduction
1.1. Basic concepts
The origins of electrochemistry can be traced back 200 years ago (1791) and is due to Luigi
Galvani who first performed an "electrochemical" experiment while dissecting a frog. Nine years later,
Volta discovered the first electrochemical cell, having salt water between two plates, made of silver
and zinc. In the following years, pioneering work of Nicholson (1800), Davy (1807 – 1808), Faraday
(1833), Kohlrausch, Hittorf, Arrhenius, Nernst and Leblanc in the XIXth century lead to the
development of electrochemistry as an important branch of science.
We can say that now electrochemistry deals with two major issues: the physical chemistry of
ionically conducting solutions or pure substances (such as molten salts) –the ionics – and the physical
chemistry of electrically charged interfaces – the electrodics. The ionics describes mainly ions and
solvents, as well as the interaction between them. The electrodics is concerned with the interface
between an electrode (metal or semiconductor) and an electrolyte and all the phenomena that happen
when such interfaces are brought together. In the following we need to define some basic concepts,
which will be encountered throughout the course.
An electrolyte is a substance, either dissolved in a solution or in a molten salt, that forms
charged species (ions). An electrode consists of a second phase (usually solid, e.g. a metal) which is
immersed in an electrolyte. The electrode charged positively, i.e. having a deficit of electrons, is called
the anode, while the electrode charged negatively, i.e. having an excess of electrons, is called the
cathode. The charged species in solution move towards the electrode having opposite charges and are
called cations (positively charged – they move towards the cathode) and anions (negatively charged –
they move towards the anode). The terms ion, anion and cation were introduced by Michael Faraday
in 1834.
The process of adding electrons to either an ion or a neutral species is called reduction, while
the reverse process (i.e., removal of electrons) is called oxidation.
1.2. Solvents and ion solvation
For many years, electrochemistry dealt mostly with aqueous solutions, but in time, with the
development of electrochemistry, non-aqueous solvents became important as well. The aluminum
industry for example is entirely based on electrolysis in a molten salt system (fused cryolite). There are
three types of solvents used in electrochemistry, outlined below.
1. Molecular solvents – which consist of molecules. The forces between solvent molecules range from
–2–
hydrogen-bond type (water) and other type of "bridges" (oxygen, halogen) – these are highly polar
solvents – to dipole-dipole interactions (moderately polar liquids, e.g. acetone) and van der Waals
interactions (non-polar liquids, such as hydrocarbons). The latter solvents are dielectrics and do not
conduct appreciably; in some of them the autoionization phenomenon occurs, conducting electricity
to some extent (very little however):
2H2O
H3O+ + OH; 2HgBr2
HgBr+ + HgBr3; 2NO2
NO+ + NO3;
2. Ionic solvents – which consist of ions, and are mostly molten salts. Not all salts yield ions when
fused, some form instead molecular liquids (like HgBr2). Usually, molten salts exist at high
temperatures (at standard pressure, NaCl is liquid between 800 and ca. 1450 oC), but in the past years
"room-temperature" molten salts were discovered, which have low melting points (ethylpiridinium
bromide, -114 oC, tetramethylammonium thiocianate, -50.5 oC). In some cases, mixtures of salts
(called eutectics) have also low melting points, such as the AlCl3 + KCl + NaCl in the ratio 60:14:26
(mol %) which melts at 94 oC. The ions in these melts can be monoatomic (like Na+ and Cl) or
polyatomic (molten cryolite, Na3AlF6, contains Na+, AlF63, AlF4 and F ions).
3. Polymer solvents – which contain polymeric chains capable of dissolving salts. These are (almost)
solid electrolytes and they are very important in the manufacturing of solid-state batteries and any
other practical device that needs a solid electrolyte. The most important solvents of this type are
polyethylene oxide (PEO) and polypropylene oxide (PPO). Ions are dissolved by coordination of the
cation by electronegative heteroatoms (such as oxygen), the anions surrounding the polymer chain
which adopts a helical structure (Figure 1).
C C O
H2 H2
H
C C O
H
CH3 2
n
PEO
PPO
O
O
O
O
O
OO
+
_
ClO4
Li
Li
O
O
+
Figure 1. Schematic structure of a PEO – LiClO4 "complex".
2
_
ClO4
n
–3–
In a fluid medium, most commonly used in electrochemistry, the dissolved ions interact
strongly with the solvent molecules: the higher the dielectric constant of the solvent, the stronger the
interaction. The solvent-solute interaction is called solvation (or hydration, if the solvent is water).
The energy changes accompanying this interaction are very large for ions (~ 400 kJ/mol for single
charged ions), and much smaller for non-polar species (~10 – 15 kJ/mol). Transport parameters, such
as ionic mobilities and diffusion coefficients, are influenced by the solvation: the ion does not move
alone, as a single entity, but carries some solvent molecules (in some cases quite many of them) with
it.
Normal water
Primary hydration
shell
Secondary
hydration shell
Disorganized
water
Figure 2. Schematic of a hydrated cation, showing the different water layers surrounding the
cation.
1.3. Electrolysis, Faraday's law and electrode types.
The electrolysis is an (electro)chemical process which occurs due to the passage of electric
current through an electrolyte by applying a large enough voltage between two electrodes.
According to Faraday's law, the amount of substance transformed during the passage of current
is related to the charge:


m = KQ = K I (t )dt = KIt (at constant current)
0
where Q is the charge passed, I is the current, t is the electrolysis time and K is the equivalent of the
3
–4–
substance:
K
A
M
A

or K 
nF nF
nF
where M is the molar mass of the substance (atomic mass, A, if we deal with an element), F is the
Faraday constant (96487 C/mole) and n is the number of transferred electrons.
e
ELECTRODE
Solution
ELECTRODE
Fe3+
Fe3+
Cu
e
Cu
Cu2+
Cu2+
Cu
Cu
Solution
Cu
Cu
Fe2+
Fe2+
Cu
Cu
(A)
(B)
Cl2
e
PbO2
Cl
e
Cl
H+
H+
Cl
Solution
Cl
Cl
ELECTRODE
ELECTRODE
PbSO4
Cl
SO42
SO42
Solution
e
(C)
(D)
Figure 3. Common electrode processes. (A) – simple electron transfer; (B) – metal deposition;
(C) – gas evolution; (D) – surface film transformation.
4
–5–
Some examples of common electrode processes are shown in Figure 3.
Fe
Solution
ELECTRODE
e
Fe2+
Fe2+
(E)
(E)
Figure 3. Common electrode processes. (E) – anodic dissolution.
2. Ionics
2.1. Ion migration and transference numbers
Although positive and negative ions are discharged in equivalent amounts at the electrodes, the
anions and cations do not necessarily move with the same velocity in an electric field. The total
amount of ions, and hence the corresponding quantity of electricity, carried through the solution is
proportional to the sum of the anion and cation velocities.
If u+ is the absolute migration velocity (or mobility) of the cation and u- for the anions (in the
same solution), the total amount of electricity passed will be proportional to the sum u+ + u-. The
amount of electricity carried by each ionic species, Qi, is proportional to its own mobility. The fraction
of current carried by each ionic species is called transference (or transport) number, and for a 1:1
electrolyte it is given by the simple equation:
t 
u
u
and t  
; t+ + t- = 1
u  u
u  u
(1)
In general, for a z+:z- electrolyte, one can write:
5
–6–
t 
z  c u 
z  c u 
and t  
z  c u   z  c u 
z  c u   z  c u 
(2)
If z+ = z- = 1 (1:1 electrolyte), then c+ = c- as well, and we recover eq. (1).
Obviously, the faster the ion, the greater its contribution to the total current. If, and only if, the
mobilities of anions and cations are exactly the same, the current will be transported in the same
proportion (50%) by each species. To calculate the transference number one does not need the absolute
mobility of an ion, but only the ratio between the two mobilities. The transference number is not
constant with concentration, because the mobilities change with changing the concentration (due to
ionic interactions – see ). As a rule, if the transference number is close to 0.5, it changes only slightly
with concentration. Also, if the transference number for the cation is less than 0.5, then it decreases
with increasing concentration , while if t- > 0.5, it increases with increasing concentration.
The mobilities u represent the migration rate of an univalent ion under a potential gradient of 1
V/m and can be calculated through a force balance: the electric force must balance the frictional force
of movement in the fluid medium. The electrical force can be written as:
Fe = zeE
(3)
where E is the electric field (dV/dx)
The frictional force is assumed to be given by Stokes law for spherical particles:
Ff = 6rv
(4)
where  is the solution viscosity (for dilute solutions it can be taken equal to the solvent's viscosity), r
is the radius of the ion and v is its speed (in m/s). From the balance of the two forces (i.e., equality of
eqs. (3) and (4)) one obtains:
zeE = 6rv, or u 
v
ze

E 6 πη r
(5)
Eq. (5) holds well for large ions, but large deviations are seen for small ions, as Stokes' law is
not appropriate to describe the movement of very small particles. One can define also an effective
hydrodynamic radius if the mobility is known:
ri 
zi e
(6)
6 πη u i
As with eq. 5, the hydrodynamic radius is close to the real radius (including the solvent
molecules in the solvation shell!) for large ions, but it is usually larger for small ions.
2.2. Measurement of transference numbers
In metallic conductors the current is carried by electrons only, and for such conductors one can
6
–7–
write t- = 1 and t+ = 0. For electrolyte solutions it is often difficult to guess a priori what fraction of the
current is carried by positive and negative ions. The simplest method for measuring transference
numbers is due to Hittorf, and it is called actually the "Hittorf's method". In general, the number of
equivalents removed from any compartment during the passage of current (or electrolysis) is
proportional to the speed of the ion moving away from it:
Equivalents lost from anode compartment
speed of cation u 


Equivalents lost from cathode compartment speed of anion u 
(7)
The total number of equivalents lost from both compartments, which is proportional to u+ + u-,
is seen to be equal to the number of equivalents deposited on each electrode; hence:
u
Equivalents lost from anode compartment
 t 
 Faradaic loss
u  u
Equivalents lost from both electrodes
(8)
u
Equivalents lost from cathde compartment
 t 
 Faradaic loss
u  u
Equivalents lost from both electrodes
(9)
and
Figure 4. Hittorf's apparatus for determining transference numbers.
The two expressions provide a basis for experimental determination of transference number by
the Hittorf method (1853). A schematic diagram of a Hittorf cell is shown in Figure 4. Stirring is
performed only near the anode and cathode, in order to enhance the mass transfer, while the central
part is not stirred. Consider such a cell which is filled with a e.g. HCl solution and let as assume that
we pass 1 Faraday charge. The current is carried across the cell by the flow ions, and in view of the
definition of the two transference numbers, the passage of 1 Faraday of charge means that t+
equivalents of H+ move towards the cathode and t- equivalents of Cl move towards the anode. The net
flow across the cell's section is t+ + t- = 1 equivalents of ions, which corresponds to 1 Faraday of
charge. Obviously, the number of equivalents in the middle of the cell is not changed by the passage of
7
–8–
current. Let us consider now the changes that occur in the cathode region. The change in equivalent of
H+ and Cl due to ion migration is given by the transfer across the cross section line. In addition to
migration, there is a removal of 1 equivalent of H+ through the electrode reaction (H+ + e  ½H2).
The net change in the cathode compartment is:
change in equivalents of H+ = electrode reaction + migration = –1 + t+ = t+ – 1 = –t-
(10)
change in equivalents of Cl = electrode reaction + migration = 0 – t- = –t-
(11)
The passage of 1 Faraday results thus in the removal of t- equivalents of HCl from the cathode
compartment. In a similar manner, the change in the anode compartment is:
change in equivalents of H+ = electrode reaction + migration = 0 – t+ = = –t+
(12)
change in equivalents of Cl = electrode reaction + migration = –1 + t- = t- – 1 = -t+
(13)
+ 
1 Faraday
Cl
t+ H+
t+ H+
t- Cl
t- Cl
H+ + e  ½H2
 ½Cl2
Figure 5. Schematic of the Hittorf's cell showing the changes that occur in each compartment.
The net effect at the anode is the loss of t+ equivalents of HCl; the faradaic loss of material can
be easily measured using a coulometer. Thus, the experimental procedure for measuring the
transference numbers consists in filling the Hittorf cell with the desired solution (e.g., HCl) previously
measuring accurately its concentration. Then electrolysis is performed and the charge passed is
accurately measured. The anode and cathode compartments are drained and analyzed to give the
concentration after passing the current. The concentration change is related to the number of
equivalents lost during electrolysis. If the charge passed is not too large and if no mixing occurs in the
central compartment, then it is found that the concentration in the central compartment is unchanged.
8
–9–
The changes in concentration in the anodic and cathodic compartments will give the transference
numbers for the anions and cations; Table 1 shows some measured values for various electrolytes at
different concentrations.
Table 1. Transference numbers of cations at various concentrations in water solution.
Electrolyte
HCl
CH3COONa
CH3COOK
KNO3
NH4Cl
KCl
KI
KBr
AgNO3
NaCl
LiCl
CaCl2
1/2Na2SO4
1/2K2SO4
1/3LaCl3
1/4K4Fe(CN)6
1/3K3Fe(CN)6
0
0.8209
0.5507
0.6427
0.5072
0.4909
0.4906
0.4892
0.4849
0.4643
0.3963
0.3364
0.4380
0.3860
0.4790
0.4770
---
c (mol/L)
0.02
0.05
0.8266
0.8292
0.5550
0.5573
0.6523
0.6569
0.5087
0.5093
0.4906
0.4905
0.4901
0.4899
0.4883
0.4882
0.4832
0.4831
0.4652
0.4664
0.3902
0.3876
0.3261
0.3211
0.4220
0.4140
0.3836
0.3829
0.4848
0.4870
0.4576
0.4482
0.555
0.604
-0.475
0.01
0.8251
0.5537
0.6498
0.5084
0.4907
0.4902
0.4884
0.4833
0.4648
0.3918
0.3289
0.4264
0.3848
0.4829
0.4625
0.515
--
0.1
0.8314
0.5594
0.6609
0.5103
0.4907
0.4898
0.4883
0.4833
0.4682
0.3854
0.3168
0.4060
0.3828
0.4890
0.4375
0.647
0.491
0.2
0.8337
0.5610
-0.5120
0.4911
0.4894
0.4887
0.4841
-0.3821
0.3112
0.3953
0.3828
0.4910
0.4233
---
2.3. Electrical conductivity of ionic solutions
Ionic solutions, just like metallic conductors, obey the Ohm's law (provided that the applied
voltage is not too large and no electrode reaction takes place), which relates the applied voltage to the
current flowing through the electrolyte solution:
I
V
R
(14)
where V is the applied voltage. The resistance of any uniform conductor is proportional to its length, l,
and inversely proportional to its cross section area, A, so that:
R
l
A
(15)
The proportionality factor, , is called the specific resistance (or resistivity); in
electrochemistry the inverse of the specific resistance,  = 1/, is more often used, and it is called
specific conductance, its units being -1cm-1, or Scm-1. In the same way, one can define the
conductance of the electrolyte solution, as the inverse of the resistance:
9
–10–
1
A

R
l
(16)
which is measured in -1 (also called Siemens, S, or mho, as the word "mho" is just the reverse of
"ohm").
Practical measurement of conductance require a cell with known values of interelectrode
distance (l) and electrode area (A), and therefore, since these values are constant for the same cell, their
ratio is a constant called cell's constant. Thus, when measuring the conductance of a solution, we can
write that:

l 1
1
= (cell constant)
AR
R
(17)
The cell constant is either known from the manufacturer, or it can be determined (as a
calibration procedure) by measuring the conductance of a standard solution for which the conductance
is known very accurately (e.g. a solution of KCl 0.02 M at 25 oC, having  = 2.76810-3 -1cm-1).
As the conductance of an electrolytic solution depends on the concentration (because the
number of charged species carrying the current usually increases as the concentration increases), it is
convenient to define a conductivity, called equivalent conductivity, which measures the conductivity
relative to the same equivalent concentration, thus allowing to compare different salts:
 eq 
  1000
c z
(18)
where c is the molar concentration and z is the total (absolute) charge of positive and negative ions.
The factor 1000 is the transformation factor for the concentration (which in chemistry is usually
measured in mole per liter, while the equivalent conductivity is measured in Scm2mol-1). The molar
conductivity has been more often used in the past years (in an effort to stop using the normal, or
equivalent, concentration, which is often a source of confusion), defined as:
c 
  1000

or  c 
c
c
(19)
(in the last relationship, one should remember that the concentration must be given in molecm-3 !).
We should also mention that all the quantities defined above for solutions can be used for
molten salts too, which are also ionic conductors. Selected values for  are shown in Table 2.
The large differences in conductivity between electronic and ionic conductors should be noted
and is due to the different conduction mechanism: in electronic conductors charge is carried by
electrons, which are small and consequently very fast charge carriers, while in ionic conductors, charge
is carried by mobile ions, which are massive and have therefore much smaller mobilities.
The conductivity  depends on the concentration of ions and their mobility: more ions means
more charge, i.e., larger conductivity, while faster ones means more charge can move in a given time;
10
–11–
we can relate  to the ion mobility by the following relationship:

z
2
i
(20)
Fui ci
i
Table 2. Electric conductivities for various conductors and electrolyte solutions.
Electronic conductors
Cu
Al
Pt
Pb
Ti
Hg
Graphite
Aqueous solutions
0.1 mole/L
0.011
0.025
0.048
0.0004
NaCl
KOH
H2SO4
CH3COOH
LiClO4 solutions
Water
Propylene carbonate
Dimethylformamide
, -1cm-1
5.6105
3.5105
1.0105
4.5104
1.8104
1.0104
2.5102
, -1cm-1
1 mole/L
0.086
0.223
0.246
0.0013
, -1cm-1
0.073 (1 M)
0.005 (0.66 M)
0.022 (1.16 M)
10 mole/L
0.247
0.447
0.604
0.0005
As the conductivity  is expected to depend linearly with concentration, it would appear that
the molar conductivity does not depend on concentration. This is not true however; for weak
electrolytes, which are not totally dissociated when dissolved, this is obvious, as the concentration of
free ions depends on the total concentration in a non-linear manner. For strong electrolytes, like NaCl,
it is less obvious, but similar effects occur due to interaction between ions at relatively large
concentrations. Only for totally non-interacting ions would the molar conductivity be constant with
concentration, but this is only an ideal situation; real electrolyte solutions approach this behavior only
in the limit of extremely dilute solutions.
For weak electrolytes it is easy to obtain a dependence of the molar conductivity on the
concentration. Let us consider for example a weak acid, HA, dissolved in water and write down the
equilibrium:
HA
+
initial:
c
equilibrium:
(1 – )c
H2O

H3O+ +
A
0
0
c
c
where  is the dissociation degree (0 <   1). The equilibrium constant (assuming that the water
11
–12–
concentration is very large and almost constant) is:
 2c
K
1 
(20)
Figure 6. Dependence of the molar conductivity on the square root of concentration for a strong
(HCl) and a weak (CH3COOH) electrolyte.
Figure 7. Plot showing the validity of Ostwald's law for CH3COOH.
Thus, for weak electrolytes the conductivity depends on the concentration because the ion
12
–13–
concentration is only c, with  depending on concentration according to eq. (20). At the limit of very
low concentrations (c  0) the dissociation degree is one (  1); we can define a limiting molar
conductivity, 0, corresponding to c  0, and we can write:
c = 0 or  
c
0
(21)
(note that from eq. 21, the molar conductivity for weak electrolytes decreases as the concentration
increases, but the total conductivity, , usually increases. In many cases  has a maximum at some
concentration, after which it starts to decrease, as an increase in the total concentration, c, will actually
lead to a much larger decrease in c – see Figure 6)
Using eq. 21 we can write eq. 20 as follows:
K = K + 2c or

 c
1
c
 1
or 0  1  c

K
c
0 K
(22)
from where, dividing by 0, we obtain:
cc
1
1


 c  0 ( 0 ) 2 K
(23)
which is known as the law of dilution (or Ostwald's law). A plot of 1/c vs. cc will give a straight
line (Figure 7) with an intercept of 1/0 and a slope of 1/(02K), allowing thus to determine both the
limiting molar conductivity, 0, and the acidity constant, K.
For strong electrolytes the theoretical treatment giving the conductivity dependence on
concentration is quite complicated and involves elaborate computations. Ionic interactions and the
"electrophoretic effect" are considered, in order to give a complex dependence on the concentration.
The electrophoretic effect (which occurs also during the electrophoretic motion of charged colloidal
particles in an electric field – whence its name) is due to the simultaneous movement of ions and their
ionic atmosphere: while the central ion moves in one direction, the counterions surrounding it move in
the opposite direction. All ions, including the central one, carry some solvent along (their solvation
shell), the net result being a slow down of the central ion. Thus, the molar conductivity decreases as
the concentration increases. In the limit of zero concentration, where ions are far apart and do not
interact with each other, the movement of cations and anions are totally independent: the presence of
cations does not influence in any way the movement of anions (and vice-versa). As a result, in this
region the molar conductivity of any strong electrolyte can be described as the sum of contributions
from its individual ions (the law of the independent migration of ions):
c = ++ + --
(24)
where i are the numbers of cations and anions per formula unit (+ = - = 1 for NaCl and CuSO4 while
13
–14–
+ = 1 and - = 2 for MgCl2). This simple result allows one to calculate to calculate the limiting molar
conductivities of any strong electrolyte. In this concentration range, it was found empirically (by
Kohlrausch) that the conductivity of strong electrolytes varies with the square root of the
concentration:
c = 0 – Kc1/2
(25)
which is known as the Kohlrausch's law; the theoretical description leading to the same equation was
made later by Onsager.
As the measurement of conductivity for a salt yields the total conductivity, c, the individual
contributions from anions and cations, or ionic conductivities (eq. 24) are obtained from transference
numbers measurements:
t 


and t  
  
  
(26)
The measurement of electrolyte conductivity was initiated (and extensively performed
afterwards) by Kohlrausch and his coworkers, between 1860 and 1880. They used a Wheatstone bridge
(which is still used as principle for measuring conductivities even in modern electronic devices). As
d.c. voltages may often cause electrode reactions (thus introducing large errors), a.c. voltage is usually
employed when measuring conductivities, as it allows better accuracy. Thus, an a.c. voltage, having a
frequency of about 1 – 2 kHz, is applied in an a.c. bridge arrangement and the adjustable capacitance is
changed until the bridge is balanced and the impedance of the cell (from which the resistance can be
easily extracted) is determined.
Water is by and large a unique solvent for electrolytes, as it has several, quite important
features:
(a) water molecules are able to bond with its neighbors through hydrogen bonds, leading to a
highly structured solvent;
(b) it self-ionizes to a small extent, containing thus a small concentration of H+ and OH ions; it
can act as both a proton donor and proton acceptor;
(c) water is a small molecule, having a substantial dipole (this is why water is a very polar
solvent, with a high dielectric constant), interacting strongly with charged species and thus being able
to solvate most ions; this is actually why most of the salts are dissociated in ions when dissolved in
water. Non-aqueous solvents are not able to solvate ions to the same extent as water (even when their
dielectric constant is higher, such as for dimethylformamide, they are much larger molecules and
therefore interact much less with ions), and incomplete ionization (or ion pairing) commonly occurs
in such solvents.
(d) it is found virtually everywhere on earth, and it is the most common and cheapest solvent
14
–15–
available.
When comparing solvents for ionic substances, two factors should be considered first:
(a) the ability of the solvent to interact with ions, which is related to its dielectric constant and
the size of the solvent molecule. Solvents with high dielectric constant and small molecules will
solvate ions better and will provide larger conductivities. The ion-solvent interaction is however very
important also, and in some cases, even solvents with very low dielectric constant (such as ethers) may
give reasonable conductivities when very specific ions are dissolved. For example, ions (I) and (II)
give reasonably high conductivities in solvents like tetrahydrofuran (THF) and tert-butyl methyl ether
(both having low dielectric constants), electrochemistry being thus accessible in such solvents.
F
CF3
CF3
F
F
O
4
THF
F
F
4
B
B
I
II
(b) the solvent's viscosity, as it determines the ionic mobility. For example, propylene
carbonate has a high dielectric constant, and thus would be expected to give high conductivity
solutions, but as it is a rather viscous solvent, its solutions have quite low conductivities.
From a practical point of view, aqueous solutions are always preferred, whenever possible, as
they have better conductivities (and thus will lead to lower ohmic losses), while pure water is readily
available at only a fraction of the price needed for other solvents. For many applications though, water
electrochemistry is not possible and one must use other solvents, including molten salts (e.g., for
aluminum and silicon electrodeposition).
2.4. Practical applications of conductivity measurements
Determination of solubility by conductance measurements
If s is the solubility (in mole/L) of a sparingly soluble salt and  is the specific conductivity of
this saturated solution, then:
 c  1000

s
(27)
The salt being only sparingly soluble, the saturated solution will be so dilute, that c will not
differ appreciably from the limiting value at infinite dilution, 0, hence:
 0  1000


 s  1000
s
0
(28)
The specific conductivity, , can be determined experimentally, while 0 may be derived from
ion conductivities; thus, it is possible to calculate the solubility of the salt from eq. (28). This method
15
–16–
can be used only if the solute undergoes simple dissociation into ions of known conductivity.
Conductivity titrations
When a solution of a strong acid, e.g. HCl, is gradually neutralized by a strong base, e.g.,
NaOH, the protons of the former are replaced by metal ions (Na+), which have a much lower
conductivity. The conductivity will therefore decrease steadily as the base is added. When
neutralization is complete, further addition of the base does not remove any more ions, but instead will
bring more ions, and thus the conductivity will start to increase. The conductivity change with
equivalents of base added has thus a minimum at the equivalent point (or end point), when the acid is
neutralized. In practice, the neutralization point is determined from the intersection of the two straight
lines that give the conductivity in the regions with excess of acid and excess of base (Figure 8).
If the acid is moderately weak or very weak, the conductivity curve shows a different shape,
depending on the relative strength of the acid. If the acid is moderately weak (such as CH3COOH), the
salt formed during the neutralization usually dissociates better than the free acid, and after a small
decrease, an increase is observed again. After the neutralization, the conductivity increases again, but
with a different slope. If the acid is very weak (such as boric acid or phenol), the conductivity
increases steadily, but again with a different slope after neutralization. In this case it is better to titrate
the very weak acid with a weak base, for which, due to its low conductivity, a (almost) constant
conductivity is reached after neutralization.
Figure 8. Conductivity titration curve for the neutralization of a strong acid with a strong base.
Conductivity titrations are rarely used nowadays, but the principle is used for ion
chromatography detectors, widely used as they allow an easy conversion of the concentration into an
16
–17–
electric signal; conductivity measurements are also quite sensitive to low amount of ionic substances.
Precipitation titrations
When a NaCl solution is added slowly to an AgNO3 one (or viceversa), AgCl, a sparingly
soluble salt, is formed. AgCl, being sparingly soluble will have a very small (almost negligible)
contribution to the total conductivity. As a result, the conductivity will remain almost constant until the
neutralization point is reached, after which increases sharply as the total ionic concentration increases.
2. Electrodics
Electrodics is a fundamental part of electrochemistry, and it deals with electrodes and
electrochemical reactions. Before the advent of various materials for electrodes, the electrode was
viewed as a metal in contact with an electrolyte, with current flowing at the interface
electrode/electrolyte. As now there are many non-metallic electrodes, we shall define an electrode as a
system comprised of an electronic conductor (metal, semiconductor, graphite or conducting organic
materials – such as conducting polymers) and an electrolyte (not necessary liquid!) in contact with it.
A more sophisticated, and somewhat more rigorous, definition identifies an electrode as a system
consisting of two or more electronically and ionically conducting phases, switched in series, between
which charge carriers (electrons and ions) can be exchanged, one of the terminal phases being an
electronic conductor (e.g. metal) and the other an ionic conductor (e.g. electrolyte). The electrode can
be schematically denoted by these two terminal phases, e.g., Cu/CuSO4 solution.
2.1. Electrode potentials
When two phases, either of them containing charged species, a (electric) potential difference is
established between the bulk of these phases. According to electrostatics, the electric potential at a
point in space is defined by the work required to move a unit electric charge from infinity to that point.
 (Galvani potential
 (Volta potential
 (surface
potential
17
–18–
Figure 9. Fundamental electrode potentials used in electrochemistry.
In electrochemistry, there are several types of electrode potentials in use, in order to better
understand and define its behavior.
The Galvani potential (or inner potential), , is the work required to move a unit charge
from infinity into the given phase.
The Volta potential (or outer potential), , is the electric potential of an electrical charged
body which is defined as the work required to move a unit (electric) charge just outside the phase. The
term "just outside" is somewhat vague, but it can be viewed as a distance of about a thousand
nanometers outside the surface. The distance is chosen as to make the Volta potential not to have any
influence from the surface.
The surface potential, , is the work required to pass the charge across the surface layer. Th
main contribution to this potential arises from the electric double layer, which is always formed at the
interface between two phases containing charged species.
It is obvious that the sum between the Volta and surface potential must give the Galvani
potential:
=+
(29)
The Volta potential, and the difference of such potentials between two electrodes, is directly
measurable and thus accessible to experimental data. By contrast, the Galvani potential cannot be
measured and thus it is inaccessible through experiments. However, Galvani potentials are vey
important in electrochemistry, since the "true" electrode potential is the difference between the Galvani
potentials of the electrode phase and the electrolyte phase. As the Volta potential can be measured, one
can say that the surface potential is also important, as one can obtain the Galvani potential from it.
Even though the Galvani potential cannot be measured, it can be estimated theoretically with a
margin of about 0.2 V: the error is quite large for most practical applications, but the estimates are
still useful in comparing various systems.
2.2. The electrochemical potential
The work associated with the transfer of charged species (electrons, ions) is composed of two
parts:
(a) First, the chemical environment of the particle is changed, regardless of the electric
potential difference at the phase boundary. The corresponding work (referred to 1 mole of component)
represents the chemical potential, i of the species in the given medium;
18
–19–
(b) On the other hand, regardless of the change in the chemical environment of the particle, the
transfer across the potential difference is accompanied by electrical work.
~,
The total quantity, combining the two above quantities, is the electrochemical potential, μ
i
which is the total work associated with the transfer of 1 mole of the i-th component (having the charge
z), from infinity into the given phase (Butler in 1926 and Guggenheim in 1930):
~  μ  z F
μ
i
i
i
(30)
~ , can be defined also as:
The electrochemical potential, μ
i
~
 G 
~


μi  


n
 i  T , p ,n j
~
where G contains an
(31)
electric component (namely, zF; actually it includes the sum for all
components).
The electrochemical potential is thus a work (i.e., an energy), not an electric potential, and it
should be stressed out that the electric potential and the electrochemical potential, although related to
each other, are fundamentally different quantities.
In order to understand better the physical significance of the electrochemical potential, let us
consider a simple example: a Zn electrode immersed in a ZnCl2 aqueous solution and let us focus on
the Zn2+ ions in both metallic zinc and in solution. In the metal phase, the Zn2+ ions are fixed in the
metal lattice, with electrons freely moving throughout the lattice. In solution, the Zn2+ ions are
hydrated, thus interacting with the water (more generally, with the solvent), while also interacting with
the Cl- ions. The energy state of the Zn2+ ions at any location clearly depends on the chemical
environment (solvent and counterions), which is manifested through short-range interaction forces. In
addition to this energy, there is also an energy required simply to move the +2 charge (disregarding
any chemical effects) to different locations, which may have different electric potentials. This energy
is clearly dependent on the electric potential  at that specific location, hence it depends on the
electrical properties of both the environment and the ion (its charge).
2.3. More about electrode potentials
As we have already seen, the term "electrode potential" is a complex quantity, and it's meaning
is not so obvious only from its name. We can think of the electrode potential as the potential difference
between the electrode's surface and the region in the solution adjacent to the electrode.
All the practical methods of measuring the electrode potential involve the completion of an
electric circuit and, therefore, require a second electrode-solution interface. Thus, these measurements
always give the difference between potential differences at the two interfaces.
As the electrode potential is such a complex quantity, and their absolute values being
19
–20–
experimentally inaccessible, electrode potentials are therefore expressed as the measured potential
difference between the electrode of interest and an arbitrarily selected standard. The electrode that
serves as the standard for potential is the Pt electrode at which an equilibrium between protons and
hydrogen is established (the activity of the protons in solution is chosen to be 1 mole/L):
H+ + e
1/2H2
This electrode is called the normal hydrogen electrode (or NHE) and serves as the reference
point for potential measurements in electrochemistry. The NHE consists of a platinized-Pt electrode (to
ensure a fast reaction and thus attaining the equilibrium fast) immersed in a solution with proton
activity equal to unity, saturated with hydrogen gas at unit fugacity (close to 1 atm pressure). By
definition, as this electrode serves as standard, its potential is 0 V at all temperatures. The sign of the
electrode potential is always the observed sign of the polarity when coupled with a NHE. Thus, the
term anodic of NHE denotes an electrode whose potential is positive. More recently, it was proposed
that the platinized-Pt type NHE should be replaced by a palladium electrode saturated with palladium
hydride (PdH0.3), which proves to be more stable, its potential being +50 mV vs. NHE.
To demonstrate the relation between the difference
2.4. The Nernst equation
The electromotive force of a cell reaction has also a thermodynamical interpretation. The link
between the electromotive force and the free enthalpy is:
G = –nFE
(32)
with G < 0 for E > 0. If all substances are at unity activities, then:
G0 = –nFE0
(33)
where E0 is the standard electrode potential.
Now, from a thermodynamic point of view, the free enthalpy change for a chemical reaction
can be expressed as (van't Hoff isotherm):
G = G0 + RTln(Q)
(34)
in which Q indicates the ratio of activities of products to those of reactants (Q is also called the
activity quotient). If we substitute for G and G0, we obtain:
–nFE = –nFE0 + RTln(Q)
(35)
which can be rearranged to give:
E  E0 
RT
ln Q
nF
(36)
20
–21–
the well-known Nernst equation.
Thus, for a simple reversible oxidation-reduction process:
Ox + ne
Red
where Ox and Red represent the oxidized and reduced forms, respectively, of a given species, one can
write:
E  E0 
RT aO
ln
nF aR
(37)
where aO and aR are the surface (i.e. near the electrode) activities of Ox and Red species. E0 is the
value of the electrode potential when the surface activities are equal to one. From a practical point a
view, the use of standard electrode potential is somewhat restricted, as the knowledge about activities
in solution is quite limited. For this reason, E. H. Swift advocated the use of formal potentials,
denoted by E0', to replace the standard potential in practice. If one writes the activities as ai = ici, then:
E  E0 
RT γ O RT cO
RT cO
ln

ln
 E 0 '
ln
nF γ R nF cR
nF cR
(38)
The formal potential, E0' is experimentally accessible, but it depends on the concentration of
Ox and Red, contrary to E0, as it contains the ratio of activity coefficients.
2.5. The thermodynamics of interfaces
Let us suppose that we have an interface of surface area A separating two phases  and 
(Figure 10). The region between the solid lines represents the interfacial zone. To the right we have
pure phase , while to the right we have pure phase . The intermolecular forces are short-range
forces, so the interfacial zone extends only over a few hundred angstroms. As this region is very
narrow, we can regard the perturbation of the properties of the pure phases  and  within this region
as properties of a surface, or interfacial properties.
A
Dividing surface
B
Pure 
Pure 
21
A'
B'
Interfacial zone
–22–
Figure 10. Schematic diagram of an interfacial region separating two phases,  and . The
phases  and  can be any phases.
Let us now compare the real interfacial zone with an imaginary reference interfacial zone. In
the reference zone, we shall define a dividing surface, shown with a dotted line in Figure 10. The
position of the dividing surface is arbitrary and does not influence in any way the final results; it is
convenient though to consider that it coincides with the actual interfacial surface. With respect to this
reference, we shall consider that phase  lies to the left from the dividing surface, while phase  lies to
the right. The reason for defining the reference system is that the properties of the interface are
governed by excesses and deficiencies in the concentrations of components, i.e., we are concerned with
differences between quantities of various species in the actual interfacial region, with respect to the
quantities we would expect if the existence of the interface did not perturb the pure phases. These
differences are called surface excess quantities. For example, the surface excess in the number of
moles of any species, such as ions or electrons, would be:
niσ  niS  niR
(39)
where niσ is the excess quantity and niS and niR are the numbers of moles of species i in the interfacial
region for the actual system and the reference system, respectively. Surface excess quantities can be
defined for any extensive variable.
One of these variables is the electrochemical free enthalpy. For the reference system the
electrochemical free energy depends on the usual variables: temperature, pressure and the molar
~
~
~
concentrations of all components, i.e., G R  G R (T , p, niR ) . The surface area has no impact on G R
because the interface does not perturb the phases  and . Therefore, there is no energy of interaction.
On the other hand, we know from experience that real systems have a tendency to minimize (or
~
maximize) the interfacial area; hence the free enthalpy of the actual system, G S , must depend on the
~
~
area. Thus, G S  G S (T , p, A, niS ) . If we write the total differentials:
~
~ R  G R

dG  
 T
~

 G R
dT  

 p



dp 



i
~
 G R
 R
 n
 i
 R
dni


22
(40)
–23–
~
~ S  G S

dG  
 T
~

 G S
dT  

 p


~

 G S
dp  

 A



dA 



i
~
 G S
 S
 n
 i
 S
dni


(41)
If we deal with experiments at constant temperature and pressure, we can drop the first two
~
~.
terms in each expression. The partial derivatives ( G R / niR ) are the electrochemical potentials, μ
i
Since the system is considered at equilibrium, the electrochemical potential is constant throughout the
system for any given species. Since the electrochemical potential is the same in all regions, i.e. in the
pure phases  and , it must be the same in the interfacial region:
~
~
 G R   G S 
~
(42)
μ i   R    S 
 ni   ni 
~
We can also define the partial derivative ( G S /A), namely as the surface tension, . The
surface tension is a measure of the energy required to produce a unit area of new surface, e.g. by
dividing the system more finely. Doing this requires that atoms or molecules previously in the bulk of
their phases be brought to the new interface. They have fewer binding interactions with neighbors in
their original phase, but may have new ones with neighbors in the opposite phase. Thus, the surface
tension depends on the identity of both phases,  and .
Now we can write the differential excess free enthalpy as:
~
~
~
~ d (n S  n R )
dG σ  dG S  dG R  γ dA 
μ
i
i
i

(43)
i
and from (39) we have:
~
~ dn σ
dG σ  γ dA 
μ
i
i

(44)
i
Eq. (44) tells us that the interfacial free enthalpy can be described (at constant pressure and
temperature) by the variables A and ni, all of which are extensive.
23
–24–
24
–25–
25
–26–
Basic Principles of the Kinetics of Electrode Processes
Electrode Processes as Heterogeneous Chemical Reactions
Electrode processes are heterogeneous chemical reactions, which occur at the interface of an
electrode (not necessarily metallic) and an electrolyte, accompanied by the transfer of electric charge
through this interface. The simplest electrode reaction involves an inert electrode (surface), two
electroactive species, O and R, completely stable and soluble in the chosen solvent and an excess of
inert, electroinactive, electrolyte:
O + ne  R
O is an oxidized species while R is its reduced form. In general, even this simple
electrochemical process consists in fact of several steps, such as:
(a) electron transfer at the electrode surface;
(b) mass transfer (e.g., of O from the bulk solution to the electrode surface);
(c) chemical reactions preceding or following the electron transfer. Such chemical reactions
may be either homogeneous reactions, such as protonation (e.g., the dissociation of a weak acid) and
dimerization (when the species formed by electron transfer undergoes chemical change to form a more
stable product, e.g., 2H  H2), or heterogeneous ones, as is the case with the catalytic decomposition
on the electrode surface;
(d) other surface processes, such as adsorption, desorption or phase formation. Adsorption
plays an important role in electrocatalytic reactions (e.g. the evolution of H2 on Pt electrode), as the
adsorption of reaction intermediates provides alternative lower energy pathways. Also, adsorption of
species which are not directly involved in the electron transfer process is sometimes used to modify the
net electrode reaction (e.g. additives used in electroplating and corrosion inhibition). The electrode
process may involve the formation of a new phase, e.g. the electrodeposition of metals in plating,
refining and winning (the electro crystallization step) or bubble formation when the product is a gas; a
transformation of one solid phase to another can also occur, for example the reaction:
PbO2 + 4 K + 30 ~ + 2 e  PbSO4 + 2H2O
The formation of a new phase is itself a multistep process, requiring both nucleation and
subsequent growth; crystal growth may involve both surface diffusion and 3-D lattice growth.
According to Figura, the overall electrode process will consist of the following consecutive
steps:
1. - Mass transfer from the bulk of the solution to the layer in contact with the electrode to replace the
26
–27–
ions or molecules. This takes place partly by ion migration, partly by diffusion (the replacement of
neutral molecules occurs by diffusion, only). Convection due to spontaneous or external mixing may
also contribute to the mass transfer.
2. The localization of ions or molecules in the region of the electrochemical double layer, dehydration
(in general, desolvation) and chemical reactions (possible in several steps) leading to intermediates
formation.
3. Adsorption of the intermediates.
4. Electron transfer, i.e., neutralization or formation of ions, or alteration of the ionic charge – by
electron gain or loss. This is actually the electrochemical step.
5. The removal of primary products by desorption or product incorporation into the crystalline lattice
of the electrode (electrocrystallization, diffusion into the bulk of amalgam electrode, etc.)
6. Secondary conversion of the primary products in a reaction.
7. The departure of the products from the surface of electrode by mass transport. Diffusion is always
involved in this final step (convection as well when the solution is stirred).
From the above discussion it follows that the simplest electrode process involves three steps
only: mass transfer of the reactant, the heterogeneous electron transfer and a final step of mass transfer
of the product or electrocrystallization, etc.). A representative reaction of this sort is the reduction of an
aromatic hydrocarbon in an aprotic solvent, e.g.:
9,10-DPA + e-  9,10-DPA (dimethyl formamide as solvent)
9,10-DPA = 9,10-diphenylanthracene =
The electrode process is a special kind of heterogeneous reaction. The typical feature of
electrode processes, as opposed to other chemical reactions, is, the dependence of the activation energy
for the electron transfer step on the potential difference between electrode and solution. It follows that
in an electrochemical process, changing the potential implies changing the activation energy.
Moreover, as the potential can be easily adjusted, it means that we have an easy way of changing the
activation energy in a controllable manner, which is a great advantage. The second important feature
is that the rate of electrode processes is influenced by the structure of double layer at the metal/solution
interface.
Obviously, since the steps of general mechanism presented in the above figure are consecutive,
the rate of the overall process will be controlled by the "slowest step".
27
–28–
Note that heterogeneous reactions at the electrode are described differently than homogeneous
reactions in chemistry: the reaction rate v has dimensions of moles-1cm-2, as this is a surface
reaction and not a bulk one (for which the reaction rate is expressed in moless-1cm-3). It is assumed
that electrode reactions are first-order processes, so v = kc. The heterogeneous rate constant, k must
be measured in cm2/s if the concentrations are expressed in mole/cm3. The rate constant k is dependent
on the electric field close to the surface, and hence on the applied electrode potential.
Note also that the concentrations entering rate expressions are always surface concentrations,
CO(0,t) and CR(0,t), where t is time. Their values may differ from those in the solution bulk, CO(,t)
and CR(,t), and in many cases this difference is significant.
electron
transfer
mass transfer
ELECTRODE
chemical reaction
and/or desolvation
adsorption
or desorption
desorption
or adsorption
chemical reaction
and/or solvation
electron
transfer
mass transfer
28
–29–
29
–30–
30
–31–
IONICS
Migration
Transference numbers
The drift speed
Electrical conductivity of solutions
Debye-Huckel
ELECTRODICS
Electrode potentials
The electrochemical potential
Potentials and Thermodynamics of cells
Electrode potential – the Nernst equation
The electrical double layer
Electrode kinetics – basic principles
Electrode kinetics – BV and microscopic treatment
Adsorption phenomena
Various special chapters and applications.
31