j j 0

Physical and Interfacial

j j

Physical and Interfacial Electrochemistry 2013

Lecture 4Lecture 4.Electrochemical

Th d iThermodynamics

Module JS CH3304 MolecularThermodynamics and Kinetics

Thermodynamics of electrochemical systems

Thermodynamics the science ofThermodynamics, the science ofpossibilities is of general utility.The well established methods of thermodynamicsymay be readily applied toelectrochemical cells.We can readily compute thermodynamic state functionsthermodynamic state functionssuch as G, H and S for achemical reaction by determininghow the open circuit cell potentialEcell varies with solution temperature.We can compute the thermodynamicefficiency of a fuel cell providedefficiency of a fuel cell providedthat G and H for the cell reactioncan be evaluated.We can also use measurements of

'Ox',Red'RedOx, MM+equilibrium cell potentials to

determine the concentration ofa redox active substance presentat the electrode/solution interface

+-AnodeOxidation

l

CathodeReductionat the electrode/solution interface.

This is the basis for potentiometricchemical sensing.

e- lossLHS

Reductione- gainRHS

Standard Electrode Potentials

Standard reduction potential (E0) is the voltage associated with a reduction reaction at an electrode when all solutes are 1 M and all gases are at 1 atmgases are at 1 atm.

2 + 2H+ (1 M) 2H (1 t )

Reduction Reaction

E0 = 0 V

2e- + 2H+ (1 M) 2H2 (1 atm)

0

St d d h d l t d (SHE)Standard hydrogen electrode (SHE)

Measurement of standard redox potential E0 forMeasurement of standard redox potential E forthe redox couple A(aq)/B(aq).

Ee

electron flow

PtH2 inreference

electrodeindicatorelectrode

H ( )

Pt

Pt

A(aq)

SHE

H2(g)

H+(aq)

A(aq)

B(aq)

salt bridge test redox couple

E0 p o ides a q antitati e meas e foE0 provides a quantitative measure forthe thermodynamic tendency of a redox couple to undergo reduction or oxidation.

Standard electrode potential

• E0 is for the reaction as written

• The more positive E0 the greater the tendency for the substance to be reducedbe reduced

• The half-cell reactions are reversible

• The sign of E0 changes when the reaction is reversed

• Changing the stoichiometric • Changing the stoichiometric coefficients of a half-cell reaction does not change the value of E0

19.3

We should recall from our CH1101 The procedure is simple to apply. One determines th l ith th t iti t d d electrochemistry lectures that

any combination of two redox couples maybe used to fabricate a galvanic cell . Thisfacility can then be used to obtain useful

the couple with the most positive standard reduction potential (or the most positive equilibrium potential Ee determined via the Nernst equation if the concentrations of the reactants diff f 1 l d 3) Thi l ill d

ythermodynamic information about a cellreaction which would be otherwise difficultto obtain. Herein lies the usefulness ofelectrochemical thermodynamics.

differ from 1 mol dm-3) . This couple will undergo reduction at the cathode . The other redox couple will consequently undergo oxidation at the anode . This information can also be used to determine th di ti f l t fl f l i

y

Given any two redox couples A/B and P/Q we can readily use tables of standard reduction potentials to determine which of

the direction of electron flow, for upon placing a load on the cell electrons will flow out of the anode because of the occurrence of a spontaneous de-electronation (otherwise known as oxidation or l t l ) ti th h th t l

pthe two couples is preferentially reduced. Once this is known the galvanic cell can be constructed, the net cell potential can be evaluated, and knowing this useful

electron loss) reaction , through the external circuit and into the cathode causing a spontaneous electronation (aka reduction or electron gain) reaction to occur. Hence in a driven cell the anode

ill b th ti l f th ll d th g

thermodynamic information can be obtained for the cell reaction.

will be the negative pole of the cell and the cathode the positive pole. Now according to the IUPAC convention if the cell reaction is spontaneous the resultant cell potential will be

iti W th t h i th b positive. We ensure that such is the case by writing the cathode reaction on the rhs , and the anode reaction on the lhs of the cell diagram . Then since Ee,rhs is more positive than Ee,lhs a

siti ll t ti l V ill b t dpositive cell potential Ve will be guaranteed

QBPA 00000anodecathodeLHSRHScell EEEEE

electron flow

VeAnodeOxidation Cathode- +

P( )M

M Reduction

P(aq)

Q(aq)

A(aq)

B(aq)EE d

( q)

salt bridge

Ee,cathodeEe,anode

mA nP mB nQ A ne B Cathode salt bridge

redox couple)( SHGTEE

m nB Q

R m nA P

mA nP mB nQa a

Qa a

P Q me Anode

,,,)( SHGTEE

• When a spontaneous reaction takes placein a Galvanic cell, electrons are depositedin one electrode (the site of oxidation or( fanode) and collected from another (the siteof reduction or cathode), and so there isa net flow of current which can be usedto do electrical work We.• From thermodynamics we note that maximum electrical work done at constantEcelltemperature and pressure We is equal tothe change in Gibbs free energy G forthe net cell reaction.

W b i h i l h • We use basic physics to evaluate the electrical work We done in moving n mole Electrons through a potential difference of Ecell .

1 electron : cellcelle EeEqW

1 mole electrons : cellAe EeNW 1 mole electrons : cellAe

n mole electrons : cellAe EeNnW

ENWGR l ti n b t n th m d n mics

cell

cellAe

EFnENenWG

eWG Relation between thermodynamicsof cell reaction and observedcell potential

0

0

,A ne B E A B

h E0 h 0 ,P me Q E P Q

mA mne mB

We assume that E0A,B is more positive than

E0P,Q and so is assigned as the cathode reaction.

nP nme nQ

We subtract the two reactions to obtain the following result.

mA nP mB nQmA nQ mB nP

To proceed we subtract the corresponding thermodynamic state functionsIn this case G0

0 0 0 0 0 0 0 0, , , , , ,cell A B P Q A B P Q A B P Q cellG m G n G m nFE mFE nmF E E nmFE

N t th t nm d n t s th n mb f l t ns t nsf d p m lNote that nm denotes the number of electrons transferred per moleof reaction as written.

A ne B Cathode m n

mA nP mB nQa a

Net Cell ReactionA ne B CathodeP Q me Anode

B Q

R m nA P

a aQ

a a

The establishment of equilibrium does not imply the cessation of redoxThe establishment of equilibrium does not imply the cessation of redoxactivity at the interface .The condition of equilibrium implies an equality in the electrochemicalpotentials of the transferring species in the two phases and inpot nt a s of th transf rr ng sp c s n th two phas s an nthe establishment of a compensating two way flow of charge acrossthe interface resulting in a definite equilibrium potential difference e orEe .eA single equilibrium potential difference may not be measured.Instead a potential is measured between two electrodes(a test or indicator electrode and a reference electrode).This is a potentiometric measurement.

The potential of the indicator electrode is related to the activities of one or more of the components of the test solution and it therefore one or more of the components of the test solution and it therefore determines the overall equilibrium cell potential Ee . Under ideal circumstances, the response of the indicator electrode to changes in analyte species activity at the indicator electrode/solution interface analyte species activity at the indicator electrode/solution interface should be rapid, reversible and governed by the Nernst equation.The ET reaction involving the analyte species should be kineticallyfacile and the ratio of the analyte/product concentration should depend facile and the ratio of the analyte/product concentration should depend on the interfacial potential difference via the Nernst equation.

The potentiometric measurement. Potential readingDevice (DVM)

In a potentiometric measurementtwo electrodes are used.These consist of the indicator

A

These consist of the indicatoror sensing electrode, and areference electrode.El t l ti l t A

BElectroanalytical measurementsrelating potential to analyteconcentration rely on the

Indicatorelectrode

Reference electrode

response of one electrodeonly (the indicator electrode).The other electrode, the reference

Solution containinganalyte species

, felectrode is independent of thesolution composition and providesA stable constant potentialA stable constant potential.The open circuit cell potentialis measured using a potential measuring device such as apotentiometer a high impedance voltameter or an electrometerpotentiometer, a high impedance voltameter or an electrometer.

Equilibrium condition between phases: chemical potential.It is well known from basic chemical thermodynamics that if two phases and with a common unchargedspecies j, are brought together, then the tendency of species j to pass from phase to phase will be determined by the difference in the chemical

Phase Phase

~

determined by the difference in the chemicalPotential between the two phases.The condition for equilibrium is

0 j~ j

j

j j

j j

0

The standard thermodynamic definition of the chemical jThe standard thermodynamic definition of the chemicalpotential is:

j

Gn

j

j n P Tn

k

, ,

Alternatively we can view the chemical potential of a species j in a phase as a measure of the work that sp c s j n a phas as a m asur of th wor that must be done for the reversible transfer of one mole of uncharged species j from the gaseous state of unit fugacity (the reference state) into the bulk of phase .I l t h ist d l ith h d s i s d In electrochemistry we deal with charged species and charged phases .

Electrochemical Activity

We consider the work done Wi in transferringa species i from the interior of a standard phase to the interior of the phase of interest.W l th t th i i h h

i i ia ia

We also assume that the species i has a charge qi = zie.

i

If two phases and contain a species i with different electrochemical

i i i h h h

0i

iiW

activities such that the electrochemical activity of species i in phase is greater than that of phase

h h i d

Standard Phase Destination Phase

The electrochemical activity can be defined in the

then there is a tendency for species i to leave phase and enter phase .The driving force for the t t f i i i th The electrochemical activity can be defined in the

following manner.

0 0exp exp expi i ii i i

q z F Wa a ak T RT k T

transport of species i is the difference in electrochemical activity between the two phases. p p pi i i

B Bk T RT k T

In the latter expression ai represents the activity of species i.

0exp iz Fa a

Now from the definition of electrochemical activity

Hence the difference in electrochemical potential is split up into two distinct components. First, the difference in chemical potential and second the difference in

0

exp

exp

i i

ii i

a aRTz Fa aRT

Hence

potential i and second the difference in electrical potential .

Hence we note that the electrochemical potential is defined as the work required to Hence

expi i i

i i

a a z Fa a RT

transfer 1 mole of charged species from infinity in vacuum into a material phase.This work consists of three separate terms. The first constitutes a chemical term which includes all short range interactions between

expi i

i

a z Fa RT

includes all short range interactions between species (such as an ion) and its environment (ion/dipole interaction, ion/induced dipole interactions, dispersion forces etc). This constitutes the chemical potential term j.The second constitutes an electrostatic term

We can follow the lead of Lewis and introduce the difference in electrochemical potential as follows.

The second constitutes an electrostatic term linked to the crossing of the layer of oriented interfacial dipoles (ziF).The third constitutes an electrostatic term linked to the charge of the phase (ziF).

h l h k d i i i

We can immediately deduce a relationship between the electrochemical potential difference and the ratio of electrochemical activities

The outer potential y is the work done in bringing a test charge from infinity up to a point outside a phase where the influence of short range image forces can be neglected.The surface potential c defines the work done and the ratio of electrochemical activities

between two phases and via the following relationships.

l li ia aRT RT F

f p fto bring a test charge across the surface layer of oriented dipoles at the interface.Hence the inner Galvani potential f is then defined as the work done to bring the test charge from infinity to the inside of the phase

ln lni ii i

i i

i i

RT RT z Fa a

z F

charge from infinity to the inside of the phase in question and so we define:

ElectrodeElectrodeElectrolyteSolution

MqSq

Charged f z F z F Interface

Dismantle interface.Remove all excesscharge & oriented

0 ln

i i i i i

i i i

z F z F

RT a z F

Vacuum Vacuum

=

=

charge & orienteddipole layers

Vacuum Vacuum

Unchargedsolution

Unchargedmetal

No dipolelayer

No dipolelayer

Dipole layeracross metal

Oriented dipoleL r n s luti n

Vacuum

Charge electrode Charge

solution

Layer on solution

Charged electrode with Dipole layer

Mq SqCharged solution with Dipole layer

In electrochemistry we deal with charged species and charged phases and one

Electrochemical Potential

In electrochemistry we deal with charged species and charged phases, and one introduces the idea of the electrochemical potential which is defined as the work expended in transferring one mole of charged species j from a given reference state at infinity into the bulk of an electrically charged phase . y y g p

Gj

~

j

j

Gn

Electrochemical potential

Electrochemical Gibbs energyGj n P Tk

, ,

It is sometimes useful to separate the electrochemical potential into chemicald l t i l t f ll

gy

and electrical components as follows.

j j j j jz F q

Chemical potential

Species valenceGalvani electrical potential

Species chargeSpecies charge

j j

j j

0Equilibrium involving charged species transferbetween two adjacent phases is attained whenno difference exists between the electrochemical j jno difference exists between the electrochemicalpotentials of that species in the two phases .

Rigorous Analysis of Electrochemical Equilibrium When two phases come into contact the electrochemical

Reference Vacuum Level

0e

potentials of the species in each phase equate. For equilibrium at metal/solution interface the electrochemical

Before Contact 0

e e

EF

Ox

e

rgy

Fill d

potential of the electron in both phases equate.

0e FE

Red e

Ener Filled

electronicenergy levels

MetalReference Vacuum Level

Ox 0

e rg

yAfter Contact

0e e

EF

Ox

Red e En

er

Filled

Via process ofCharge transfer

Metal

Filled electronicenergy levels

The Nernst Equation

W id h f ll i ET i

0 0 ln Re R O R O

an RT z z F

a

Simplifying we get

We consider the following ET reaction.

Ox Redox redz zne

Oa

0 0 ln Re R O

O

an RT nF

a

ox red O Rn z z z z

At equilibrium

O e Rn

Oa

0 0

lnR O RaRT F

Hence

We note the following

0e e F

z F

lneO

Fn n a

Also 0e e F

O O O

R R R

z F

z F

Also

e e

At equilibrium e e

0

0

ln

lnO O O

R R R

RT a

RT a

0 00lnR O Re

O

aRT Fn n a

Simplifying we getHence

0 lnO O e ORT a n z F

Simplifying we get

0 0 0

0

ln

ln

O R e O

R

O

n aRTnF nF a

aRT

This is the Nernst 0 lnR R RRT a z F

0 0 00

lnR

O R e

nF a

nnF

This is the NernstEquation.

We recall that the Gibbs energy G is usedT d t i h th h i l ti

In the latter the symbol

Review of Thermodynamics

To determine whether a chemical reaction Proceeds spontaneously or not.We consider the gas phase reaction A(g) B (g).We let denote the extent of reaction.Cl l 0 1 Wh 0 h A d

rG = reaction Gibbs free energy

Since j varies with compositionClearly 0 < < 1. When = 0 we have pure A and when = 1 we have 1 mol A destroyed and 1 mol B formed.Also dnA = -d and dnB = + d where n d t th tit ( l) f t i l

Since j varies with compositionThen so also does rG.

denotes the quantity (mol) of material used up or formed. By definition the change in Gibbs energy dG at constant T and P is related to the chemical potential f ll

Tpr

GG,

forward reaction

spontaneousas follows:

A A B BdG dn dnd d d

(1)reverse reactionspontaneous

spontaneous

(1) A B B Ad d d Furthermore

G

(1)

,P T

GdG d

Hence we get from eqn. 1 and 2

(2)

Extent of reaction

0 1

,r B A

P T

G G

(3)Extent of reaction

Reaction Gibbs energy GG

extent of Reaction Gibbs energyTp

rG,

reaction =

1K 1K1K

0 Gr

0 1

0 Gr0 Gr

0 1

G is the slope of the G versus graph at any degree of rG is the slope of the G versus graph at any degree of advancement of the chemical reaction.

If A > B then A B is spontaneous and rG is negative.If A B then A B is spontaneous and rG is negative.If A > B then B A is spontaneous and rG is positive.If A = mB then rG = 0 and chemical equilibrium has been achieved.

Since A and B are ideal gases then we writeSince A and B are ideal gases then we write

00ln A

A ApRTp

QR = reaction quotientK = Equilibrium constant

00

0 00 0

ln

ln ln

BB B

B AB A B A

pRTp

p pRT RT

ln ln ln R

r RQG RT K RT Q RTK

0 0

0 0

0

ln BB A

A

p p

pRTp

K

0 lnr RG RT Q

0 lnG G RT Q lnr r RG G RT Q

At equilibrium QR = K and rG = 0

lnrG RT K

Gibbs energy and chemical equilibrium. Reaction not spontaneousIn forward direction

GIn forward direction

EquilibriumQ=K

Q large, Q>K[P]>>[R]

0G[P]>>[R]G positive

Qln0GG

Q small, Q<K[P]<<[R]G negativeG negative

Standard stateQ=1lnQ=0lnQ 0

QRTGG ln0 Reaction spontaneousI f d di tiIn forward direction

The expression just derived for the special case A B can also be derived more generally If we set nj as thecan also be derived more generally. If we set nj as thestoichiometric coefficient of species j (negative for reactants and positive forproducts, we can derive the following.

,j j

j P T

GdG d d

G

We can relate chemical potential jto activity aj as follows.

0 lnj j jRT a

,r j j

P T

GG

j j j

jQ aIn the latter we have introduced the followingdefinitions.

0 lnr j j j jj j

G RT a ln lnj j

R jj

vj j

Q a

a a

0 ln j

r jj

G RT a

j jj j

Again at equilibrium 0

0

lr RG Q K

G RT K

0

ln

ln

jr j

j

r r R

G RT a

G G RT Q

0

0 0

lnj

r

jj

G RT K

K a

G G

0f G

0 0r j f

jG G Standard free energy

oOf formation of species j.

Potentiometric Measurements

We now mention the practicalities ofconducting a potentiometricmeasurement . A two electrodeelectrochemical cell is used . Thisconsists of a reference electrode andan indicator electrode . The object ofthe exercise is to make a measurementthe exercise is to make a measurementof the equilibrium cell potentialwithout drawing any significantcurrent since we note that the

l ll l d f dequilibrium cell potential is defined asE = (i --> 0) where Df denotes theGalvani potential difference measuredbetween the cell terminals Thisbetween the cell terminals .Thisobjective is achieved either using anull detecting potentiometer or a highimpedance voltmeter .

The second measurement protocol involves use of an electrometer . The latter is based on a voltage follower circuit A voltage follower employs an

Also ideally the amplifier should exhibit an infinite input impedance so that they

t i t lt ith t d i follower circuit . A voltage follower employs an operational amplifier . The amplifier has two input terminals called the summing point S (or inverting input) and the follower input F (or non-inverting input) Note that the positive or negative signs at

can accept input voltages without drawing any current from the voltage source . This is why we can measure a voltage without any perturbation . In practice th i t i d ill b l b t input) . Note that the positive or negative signs at

the input terminals do not reflect the input voltage polarity but rather the non inverting and inverting nature respectively of the inputs . Now the fundamental property of the operational amplifier is

the input impedance will be large but finite (typically 106 ). An ideal amplifier should also be able to supply any desired current to a load. The output impedance h ld b I ti lifi fundamental property of the operational amplifier is

that the output voltage Vo is the inverted, amplified voltage difference V where V = V- - V+ denotes the voltage of the inverting input with respect to the non inverting input Hence we can write that

should be zero. In practice amplifiers can supply currents in the mA range although higher current output can also be achieved.

Keeping these comments in mind we cannow discuss the operation of the voltagefollower circuit presented In this

non inverting input. Hence we can write that

V o AV AV AV

In the latter A denotes the open loop gain of the configuration the entire output voltage isreturned to the inverting input. . If Virepresents the input voltage then we see fromthe analysis outlined across that the circuit

ll d l f ll h

In the latter A denotes the open loop gain of the amplifier. Although ideally A should be infinite it will typically be 105.

is called a voltage follower because theoutput voltage is the same as the inputvoltage. The circuit offers a very highinput impedance and a very low output

d d h f b d

V o AV A Vo Vi V o

V i

1 1 Vi

impedance and can therefore be used tomeasure a voltage without perturbing thevoltage significantly.

1

A

The Nernst equation.

The potential developed by a Galvanic cell d p nds ndepends onthe composition of the cell.From thermodynamics the Gibbs energy change for achemical reaction G varies with composition of the Nernst eqn. holdschemical reaction G varies with composition of thereaction mixture in a well definedmanner. We use the relationship between G and E to obtain the

qfor single redoxcouples and net cellreactions.between G and E to obtain the

Nernst equation. 0 lnr r RG G RT Q 0 0r rG nFE G nFE

Reactionquotient

0 lnr r

RnFE nFE RT Q RT/F = 25.7 mVAt T 298 K quotient

productsreactantsRQ

0 RTT = 298K

At T = 298 K

reactants0 ln RRTE E QnF

0 0.0592 log RE E Qn

)()()()( 22 sCuaqZnaqCusZn Equilibrium 00ll R

GE Q K

2

20 log

2059.0

CuZnEE cellcell

00

cell RE Q KW

Dead cell 2 Cu Dead cell

large1 WQ 01 cellcell EEQ

2Zn 2CuZnQ

small1 WQ

As celloperates

2 2

Q

2 2[ ] [ ]

R cell

Zn CuQ E

Determination of thermodynamic parameters y pfrom Ecell vs temperature data.

M t f th t ll Measurement of the zero current cell potential E as afunction of temperature T enables

200 TTcTTbaE

a b and c etc are constants whichthermodynamic quantitiessuch as the reaction enthalpy H and reaction entropy S

a, b and c etc are constants, whichcan be positive or negative.T0 is a reference temperature (298K)

pyto be evaluated for a cell reaction.

PTE

Temperature

coefficient of tGibbs-Helmholtz eqn.

rGH G T r H nFE T nFE

T

zero currentcell potentialobtained fromexperimentalr r

P

H G TT

P

EnFE nFTT

E

experimentalE=E(T) data.Typical valueslie in range10 4 10 5 VK 1

r G n F E rP

EH nF E TT

10-4 – 10-5 VK-1

• Once H and G are known then S may beyevaluated.

G H T S r r r

r rr

G H T SH GS

T

1r

TES nFE nFT nFE

T T P

r

T T

ES nFT

PT

• Electrochemical measurements of cell potentiald t d d diti f t flconducted under conditions of zero current flow

as a function of temperature provide a sophisticatedmethod of determining useful thermodynamic quantities.

Fundamentals of potentiometric measurement : the Nernst Equationthe Nernst Equation.

The potential of the indicator electrode is related to the activitiesof one or more of the components ofof one or more of the components ofthe test solution and it thereforedetermines the overallequilibrium cell

Ee

electron flow

equilibrium cellpotential Ee .Under idealcircumstances, Pt

H2 inreferenceelectrode

indicatorelectrodec rcumstances,

the response ofthe indicatorelectrode to

PtSHE

changes in analytespecies activity atthe indicator electrode/

H2(g)

H+(aq)

A(aq)

B(aq)

Pt

solution interface shouldbe rapid, reversible andgoverned by the Nernst equation.Th ET l h l h ld b k ll

B(aq)

salt bridgetest analyteredox couple

The ET reaction involving the analyte species should be kineticallyfacile and the ratio of the analyte/product concentration should dependon the interfacial potential difference via the Nernst equation.

Hydrogen/oxygen fuel cell

2H2(g) 4H+ + 4e-

Anode reaction

H2/O2 fuel2H2(g) 4H+ + 4e-

Anode reaction

H2/O2 fuel2H2(g) 4H+ + 4e-

Anode reaction

H2/O2 fuelH2/O2 fuelcell

H+ ionConducting

H2/O2 fuelcell

H+ ionConducting

H2/O2 fuelcell

H+ ionConductingConductingmembraneConductingmembraneConductingmembrane

Cathode reactionCathode reaction

O2(g) + 4e- +4H+ 2H2OO2(g) + 4e- +4H+ 2H2OO2(g) + 4e- +4H+ 2H2O

,eq cellG nFE Remember CH1101

, , ,eq cell eq C eq AE E E emember H

Electrochemistry:

Ballard PEM Fuel Cell.

Efficiency of fuel cell(I) Efficiency of fuel cell(I) .eq cellG nFE

Efficiency =work output/heat input

For electrochemical or ‘cold’ combustion:

,

, , ,

eq cell

eq cell eq C eq A

G n

E E E

Heat input entalphy change H for cell reactionWork output Gibbs energy change G for cell reaction

max 1r

r

r r

r

r

r

H T S TGH

SH H

Usually 1

r rG H

,max

eq c lel

r

nFEH

1

H2(g) + ½ O2(g) H2O (l) G0 = -56.69 kcal mol-1; H0 = - 68.32 kcal mol-1n = 2 E0 = 1 229 V = 0 830n 2, E eq 1.229 V , 0.830.

Efficiency of fuel cell (II)Efficiency of fuel cell (II).Thi b fF ll l f l ll • This occurs because of:– Slowness of one or more

intermediate steps of

• For all real fuel cell systems terminal cell potential E does not intermediate steps of

reactions occurring at one or both electrodes.Sl f

potential E does not equal the equilibrium value Eeq, but will be less

– Slowness of mass transport processes either reactants to, or

eq,than it.

• Furthermore E will products from, the electrodes.

– Ohmic losses through the

decrease in value with increasing current d n f m th fu l Ohmic losses through the

electrolyte.drawn from the fuel cell.

( )nF EFE Sum of all overpotential( )eqcellreal

nF EnFEH H

p

Losses.

Nernst Equation involving both an electron and proton transfer.

We consider the following reaction which involves both the transfer of m protons and n electrons.This is a situation often found in biochemical and organic reactions.

A H B A mH ne B

The Nernst equation for this type of proton/electron transfer equilibrium is given bythe following expressionthe following expression.

0 ln Bm

A H

aRTE EnF a a

Idea:Potentiometric measurements can give riseA H

This expression can be readily simplified to the following form

2 303RT RT

gto accurate pH measurements(especially metal oxide electrodesLyons TEECE Group Research 2012/2013).

0 2.303 ln 2.303B

A

aRT m RTE E pHnF a n F

RT m

EHence we predict that a Nernst plot of open circuit or

)

2.303NRT mSF n

Hence we predict that a Nernst plot of open circuit or equilibriumPotential versus solution pH should be linearWith a Nernst slope SN whose value is directly related to the redox stoichiometryf h d h h h /

pH

of the redox reaction through the m/n ratio.When m = n then we predict that SN = - 2/303RT/FWhich is close to 60 mV per unit change in pH at298 K.

Membrane Potential M+Since we are considering charged species we define equilibrium

In terms of equality of elctrochemical potentials

ja

M+X- M+X-

In terms of equality of elctrochemical potentials.

( )j j

j j j jz F z F

ja M+

Net + veNet - ve

0

j ja a

Membrane permeable only to M+ion.

Net + veNet - ve j j j j jz F z F z F

Now

0

0

ln

lj j jRT a

RT

0 lnj j jRT a Hence

0 0 ln lnjjj j jz F RT a RT a Now if we assume that 0 0

j j

0 0 ln

jj

jj

j j j

j

j

aRT

a

We get the final expression for the membranepotential

ln jM

aRTF a

0 0

lnjj j

j j j

aRTz F z F a

j jz F a Of course a similar result for the membrane potential can be obtained by equalizing the

i f l h i l

exp 1j jj j

j

a z Fa a

a RT

ratio of electrochemical

activities and noting that the following pertains.

ln

j

jM

j j

aRTz F a

is the electric potential necessaryTo prevent equalization of ionic activities by Difusion across the membrane.