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-...