CHEM465/865, 2004-3, Lecture 5-7, 20th Sep., 2004
Electrochemical Equilibrium – Electromotive Force
Relation between chemical and electric driving forces
Electrochemical system at constant T and p: consider Gibbs free energy G%%%%
Consider electrochemical reaction:
(1) A B e C D eR LA B e C D eν ν ν ν ν νν ν ν ν ν νν ν ν ν ν νν ν ν ν ν ν− −− −− −− −+ + + ++ + + ++ + + ++ + + +����
reactants products
Note:
Ø Stoichiometric coefficients (ννννi) of reactants (A, B) have negative sign,
those of products (C,D) are positive
Ø Electrons are written explicitly in this equation since they will appear
in the condition of electrochemical equilibrium!
Objective: use thermodynamic arguments to derive electrical potential (the
so-called electromotive force, EMF) of a cell and relate this EMF to the
composition of electrochemical cells
Reaction (1) advances by small amount dξξξξ from left to right
� amount of A changes by Ad 0ν ξν ξν ξν ξ <<<<
amount of B changes by Bd 0ν ξν ξν ξν ξ <<<<
amount of C changes by Cd 0ν ξν ξν ξν ξ >>>>
Thermodynamics Composition
Electrical potential, EMF
Amount of D changes by Dd 0ν ξν ξν ξν ξ >>>>
Example:
2 3 2
A B C D
2 ( ) 3 ( ) 12 3 ( ) 4 ( ) 12
2, 3, 3, 4
R LAl O s Si s e SiO s Al s e
ν ν ν νν ν ν νν ν ν νν ν ν ν
− −− −− −− −+ + + ++ + + ++ + + ++ + + += − = − = == − = − = == − = − = == − = − = =
����
Change in electrochemical Gibbs free energy:
,
i i
i
i i
i
G Nα αα αα αα α
ααααµµµµ
ν µ ξν µ ξν µ ξν µ ξ
====
====
∑∑∑∑
∑∑∑∑
%%%% %%%%
%%%%
d d (distinguish species and phases)
d (i: all species and all phases)
and, thus,
,
r i i
iT p
GG ν µν µν µν µ
ξξξξ ∂∂∂∂∆ = =∆ = =∆ = =∆ = = ∂∂∂∂
∑∑∑∑%%%%
%%%% %%%%
(general relation, involves all species, all phases)
Condition of electrochemical equilibrium: 0rG∆ =∆ =∆ =∆ =%%%%
Not in chemical equilibrium! 0rG∆ <∆ <∆ <∆ < (spontaneous reaction)
this is the chemical driving force of the reaction
Let’s consider some examples first:
1. Metal|solution interface, solution contains ions of the metal
Ion-transfer: Mz+ + ze- � M (cathodic reaction, ϕϕϕϕM > ϕ > ϕ > ϕ > ϕS)
Write equilibrium condition: 0µ µ µµ µ µµ µ µµ µ µ− − =− − =− − =− − =% % %% % %% % %% % %z+
M M S
M e Mz where
( )µ µµ µµ µµ µ====%%%%M 0,M
M M standard value, unit activity ,
Fµ µ ϕµ µ ϕµ µ ϕµ µ ϕ= −= −= −= −%%%%M 0,M M
e e ,
(((( ))))lnRT a Fµ µ ϕµ µ ϕµ µ ϕµ µ ϕ= + += + += + += + +%%%% z+ z+ z+
S 0,S S
M M Mz
M S
ϕϕϕϕ Sϕϕϕϕ M
Here, i
ααααµµµµ0, is the standard chemical potential of species i in phase αααα,
referring to standard conditions:
Ø unit activities of solvated species
Ø partial pressures of 1 bar
Ø concentrations: 1 mol/l
Metals, solids, liquids: for practical purposes always at standard
conditions!
Insert all the relations into condition of electrochemical equilibrium gives
an expression for the electrode potential E or EMF of the Mz+|M redox
couple:
(((( ))))lnRT
E aF F
µ µ µµ µ µµ µ µµ µ µϕ ϕ ϕϕ ϕ ϕϕ ϕ ϕϕ ϕ ϕ
+ −+ −+ −+ −= ∆ = − = += ∆ = − = += ∆ = − = += ∆ = − = +z+
z+
0,S 0,M 0,M
e MM M S MS M
z
z z
and, thus,
(((( ))))0ln
RTE E a
F= += += += + z+Mz
E depends on activities of potential determining ions.
E0 is the standard EMF: 0EF
µ µ µµ µ µµ µ µµ µ µ+ −+ −+ −+ −====
z+
0,S 0,M 0,M
e MMz
z
2. Non-metal in contact with its ions on surface of an inert, conducting
substance, e.g. H2|H+ on Pt
H2 (g) � 2 H+ + 2 e- (anodic reaction)
(((( )))) (((( ))))0
2 2 ln 2 2 2 ln 0
rG
RT a F F RT pµ ϕ µ ϕ µµ ϕ µ ϕ µµ ϕ µ ϕ µµ ϕ µ ϕ µ
∆ =∆ =∆ =∆ =
⇒⇒⇒⇒ + + + − − − =+ + + − − − =+ + + − − − =+ + + − − − =
%%%%
+ +2 2
0,S S 0,Pt Pt 0,g
e H HH H
chemical potential of hydrogen gas: (((( ))))2
lnH
RT pµ µµ µµ µµ µ= += += += +2 2
g 0,g
H H
Electrode potential:
+2
+
0,g 0,S 0,Pt 1/2
H e HS S Pt HPt
Hz
22 2
lnpRT
EF F a
µ µ µµ µ µµ µ µµ µ µϕ ϕ ϕϕ ϕ ϕϕ ϕ ϕϕ ϕ ϕ
− −− −− −− −= ∆ = − = += ∆ = − = += ∆ = − = += ∆ = − = +
+
1/2
H
H
20ln
pRTE E
F a
= += += += +
3. Consider the following galvanic cell: PtL|H2(g)|HCl(m)|AgCl(s)|Ag(s)|PtR
m is the molality of the electrolyte solution
Left electrode: ANODE, oxidation
H2 (g) � 2 H+ + 2 e- (at PtL)
Right electrode: CATHODE, reduction
2 AgCl(s) + 2e- (at PtR) � 2 Ag(s) + 2 Cl-
overall: H2 (g) + 2 AgCl(s) + 2e- (at PtR) � 2
HCl (m) + 2 Ag(s) + 2 e- (at PtL)
Electrochemical equilibrium:
2 2 2 2 2 0µ µ µ µ µ µµ µ µ µ µ µµ µ µ µ µ µµ µ µ µ µ µ+ + − − − =+ + − − − =+ + − − − =+ + − − − =% % % % % %% % % % % %% % % % % %% % % % % %L R
2
Pt Ptaq s g S
HCl Ag e H AgCl e
Again: solids, metals, liquids unit activity
Only electrochemical potentials of electrons in Pt-wires are potential
dependent.
Why can µµµµ%%%% aq
HCl be considered as an electroneutral species?
Insert expressions for electrochemical potentials:
(((( ))))0
1/ 2, 2
2 2 2
ln2
2 2 2 0
2 0
HClrr r
r
aGE G G RT
F p
F F
G F
µ µ µ µ µ µ ϕµ µ µ µ µ µ ϕµ µ µ µ µ µ ϕµ µ µ µ µ µ ϕ
ϕϕϕϕ
ϕ ϕϕ ϕϕ ϕϕ ϕ
ϕϕϕϕ
ϕϕϕϕ
+ −+ −+ −+ −
∆∆∆∆= − = − ∆ = ∆ += − = − ∆ = ∆ += − = − ∆ = ∆ += − = − ∆ = ∆ +
− + − − + =− + − − + =− + − − + =− + − − + =
⇒⇒⇒⇒ ∆ + =∆ + =∆ + =∆ + =
⇒⇒⇒⇒
−−−−
L R L R
2
R L
R L
2
Pt P
Pt P
t Pt Ptaq s g S
t
HCl Ag H AgCl e e
Pt Pt
H
with
General relationship between EMF E and reaction Gibbs free energy ∆∆∆∆rG:
How is EMF determined by the chemical composition of the system?
this relationship is determined by Nernst’s equation
Again: consider
A B e C D eR LA B e C D eν ν ν ν ν νν ν ν ν ν νν ν ν ν ν νν ν ν ν ν ν− −− −− −− −+ + + ++ + + ++ + + ++ + + +����
The relation between the reaction Gibbs free energy and the composition of
the mixture is:
0ln
r rG G RT Q∆ = ∆ +∆ = ∆ +∆ = ∆ +∆ = ∆ +
The relation between EMF E and rG∆∆∆∆ is
e e e
e
or
0
0
ln
ln
R L r rG G RT
E QF F F
RTE E Q
F
ϕ ϕϕ ϕϕ ϕϕ ϕν ν νν ν νν ν νν ν ν
νννν
∆ ∆∆ ∆∆ ∆∆ ∆= − = − = − −= − = − = − −= − = − = − −= − = − = − −
= −= −= −= −
This is Nernst’s equation!
Here, Q is the so-called reaction quotient:
[[[[ ]]]] [[[[ ]]]][[[[ ]]]] [[[[ ]]]]
C D
A B
C DQ
A C
νννν νννν
ν νν νν νν ν====
Note: instead of concentrations, there could be activities or partial
pressures in the reaction quotient as well. It depends on, how the amount
of a certain species that participates in the equilibrium has to be specified.
The Nernst’s equation is just another from to formulate the condition of
electrochemical equilibrium.
Now: what would be the equivalent to the condition of chemical
equlibrium??? 0E ====
If you make a connection between the two electrodes, then the system will
go to chemical equilibrium. This is what happens during the discharge of a
battery. A discharged battery is in chemical equilibrium. In a fuel cell, one
makes sure that the system cannot reach chemical equilibrium by
continuously supplying the reactants. A fuel cell is an open system.
Electrode configurations and reference electrodes
Nernst equation: in principle possible to calculate and measure EMFs for
half reactions and electrochemical cells
(examples in problem sets)
In general, the EMF
e
0ln
RTE E Q
Fνννν= −= −= −= −
has to parts:
Ø Standard EMF 0
E
standard conditions are:
§ 25°C
§ partial pressures of gaseous species: 1 bar
§ 1M solutions
§ unit activities of ions
Ø a part that depends on the composition of the system, i.e. deviations
from standard conditions, involving dependence on temperature
If standard potentials and composition of the system are known, then in
principle EMFs of all systems could be determined!
However: there is no absolute scale, only differences in electrode
potentials can be determined. Is that bad? No, only those differences are of
practical relevance.
Electrode potential of single electrode configuration:
Metal|solution interface
Electrode potential: M S
el (cathodic)E ϕ ϕϕ ϕϕ ϕϕ ϕ= −= −= −= −
Remember: this is proportional to the amount of work
required to move a test charge across the metal|solution
interface
Measurement: at least two electrodes are required
Electrochemical Cell
EMF: C A
cellE ϕ ϕϕ ϕϕ ϕϕ ϕ= −= −= −= −
Electrical work that the system can perform in
bringing single electron from anode to cathode.
This value can be calculated if the electrode
potentials are known at the two distinct M|S
interfaces
C A
cell el elE E E= −= −= −= −
This is measurable. In order to determine it, a metal connection has to be
established between the two electrodes. A voltmeter with high electronic
resistance will allow only a very small current to flow, so that the
measurement can be performed practically under conditions of
electrochemical equilibrium.
Remember: the EMF of a cell only depends on initial and final states of the
system.
M S
ϕϕϕϕ Sϕϕϕϕ M
Anode Cathode
ϕϕϕϕA ϕϕϕϕC
ϕϕϕϕS
Classification of electrodes
Classification based on nature of species involved in electrochemical
equilibria, i.e. redox couples
Conventionally, four types of electrodes are distinguished
1) Electrodes of the first type
Ø Metal electrode in contact with solution of its ions,
Mz+ + ze- M, (((( ))))z+el Mz
0ln
RTE E a
F= += += += +
e.g. Cu|Cu2+ or Au|[Au(CN)2]-
Ø Nonmetal in contact with its ions on the surface of an inert
metal electrode, e.g. the
HYDROGEN ELECTRODE (see picture)
Pt,H2|H+,
+
1/2
H
H
20ln
pRTE E
F a
= += += += +
or Pt,Cl2|Cl- (these are gaseous electrodes)
2) Metal electrodes in contact with solution containing anions that form
a poorly soluble salt with the metal ions, e.g. the
CALOMEL ELECTRODE (see picture)
Hg|Hg2Cl2|Cl-, (((( ))))-Cl
0ln
RTE E a
F= −= −= −= −
The salt is almost entirely in solid form. It can thus be considered at
unit activity. The electrode potential is a function of the anion activity
only. Due to the low solubility of the salt, these electrodes are very
stable. They are good reference electrodes. Another example:
Ag|AgCl|Cl-
3) Redox or inert electrodes: simultaneous equilibria with respect to
anions and cations, e.g. Quinhydrone electrode
4) Other, e.g. pH sensitive glass electrode.
The most important reference electrode is the standard hydrogen electrode
(SHE). Its potential is fixed at 0
SHE 0E ==== at all temperatures. Tabulated
standard electrode potentials correspond to the standard EMFs of an
electrochemical cell in which the standard hydrogen electrode is on the left
(ANODE), the considered electrode is on the right (CATHODE) and all
components of the system are at unit activity.
If you want to read more about electrochemical equilibria, electrode
potentials, EMFs, measurements of potentials, I would recommend the
following literature:
Ø Encyclopedia of Electrochemistry, Edited by A.J. Bard and M.
Stratmann, vol. 1, ch. 1, Weinheim, Wiley-VCH, 2002-...
From R.A. Silbey and R.J. Alberty, Physical Chemistry, Wiley, NY, 2001.
From: Encyclopedia of Electrochemistry, Edited by A.J. Bard and M.
Stratmann, vol. 1, ch. 1, Weinheim, Wiley-VCH, 2002-...
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