Batteries Slides

Energy StorageBatteries I

Anis Allagui

University of SharjahSustainable and Renewable Energy Engineering

[email protected]

Last Updated: February 15, 2016

mailto:[email protected]

Introduction Sort History Applications How Does it Work? Thermodynamics Electrochemistry

Outline

1 Introduction

2 A Short History

3 Applications

4 How Does it Work?

5 ThermodynamicsChemical ThermodynamicsElectric Properties

6 ElectrochemistryCell PolarizationElectrode Processes and Reaction RatesElectrical Double-Layer

Anis Allagui Energy Storage | Batteries I 2/61


Introduction I

Electrochemical devices for energy storage and conversion:1 batteries,2 fuel cells, and3 electrochemical supercapacitors (ESs)



Galvani’s Frog Experiment

28 February 2013 9

Short history electrochemical cells

Shown here is an illustration of Luigi Galvani's famous frog experiments taken from his work, De Viribus - Electricitatis in Motu Musculari. 1792. A chance observation led Luigi Galvani (1737-98) to discover animal electricity in 1871. When the nerve of a frog that Galvani's wife was preparing for soup was accidentally touched with a knife a muscle contraction occurred despite the frog not being connected to an electrical machine.

Figure 2.1: Luigi Galvani’s famous frog experiments (De Viribus - Electricitatis inMotu Musculari. 1792). A chance observation led Luigi Galvani to discover animalelectricity in 1871, when the nerve of a frog that Galvani’s wife was preparing forsoup was accidentally touched with a knife; a muscle contraction occurred despitethe frog not being connected to an electrical machine.



Voltaic Pile

Figure 2.2: Voltaic pile (the first elec-trical battery): alternately stacked cop-per and zinc plates with a piece of pa-per soaked in salt solution between thetwo plates. This "voltaic pile" (named"artificial electrical organ" by Volta) pro-vided electricity when the plates wereconnected with a wire, GNU Free Doc-umentation License.

RemarkThe connection between chemical reactions and electrical energy wasquickly recognized. A battery is a device that converts the chemical energycontained in its active materials directly into electric energy by means ofan electrochemical oxidation-reduction (redox) reaction



Planté Cell

28 February 2013 11

The lead/sulfuric acid/lead dioxide system was discovered by Gaston Planté (1834 to 1889) in 1859: the lead-acid battery was born. This system was used for storing electricity fortelegraphy, e.g. he used lead plates as electrodes. These batteries were not yet suitable for industrial production.


Figure 2.3: The lead/sulfuric acid/leaddioxide system was discovered by GastonPlanté in 1859: the lead-acid recharge-able battery was born. This system wasused for storing electricity for telegraphy,e.g. he used lead plates as electrodes.These batteries were not yet suitable forindustrial production.



Electrical Car is not New

28 February 2013 12

Electrical Car is not newFirst car with combustion engine 1885 (Karl Benz)

First electrical car 1881

Around 1900, competing options electric versus combustion.

Speed record 1899-1901 Camille JenatzyFigure 2.4: "La Jamais Contente", first automobile to reach 100 km/h in 1899.First car with combustion engine 1885 (Karl Benz), but the first electrical car wasbuilt in 1881. Around 1900, competing options electric versus combustion.



Fuel Cells

28 February 2013 13

Anode (-, ox): H2(g) → 2 H+ + 2e– H° = 0 v

Cathode (+, red): ½ O2 + 2 H+ + 2e– → H2O(l) H° = +1.23 v

net: H2(g) + ½ O2(g) → H2O(l) H° = +1.23 v

The Fuel Cell was first demonstrated in 1839 by Sir William Grove, a lawyer and amateur chemist. At the time, it was already known that water could be decomposed into hydrogen and oxygen by electrolysis; Grove tried recombining the two gases in a simple apparatus, and discovered what he called "reverse electrolysis"— that is, the recombination of H2 and O2 into water— causing a potential difference to be generated between the two electrodes:


Figure 2.5: The Fuel Cell was first demonstrated in 1839 by Sir William Grove.At the time, it was already known that water could be decomposed into H2 and O2by electrolysis; Grove tried recombining the two gases in a simple apparatus, anddiscovered what he called "reverse electrolysis": recombination of H2 and O2 intowater causing a potential difference to be generated between the two electrodes



Nickel-metal Hydride Cells

Figure 2.6: Nickel-metal hydride cells(1960’s): The discovery that certaincompounds such as LaNi5 can act as"hydrogen sponges" (LaNi5H6) made itpractical to employ metal hydrides as acathode material. These reversible (sec-ondary) batteries are widely used in cellphones, computers, and portable powertools

Anode: MH + OH− → M + H2O + e−

Cathode: NiOOH + H2O + e− → Ni(OH)2 + OH−

Net (1.2 V): MH + NiOOH→ M + Ni(OH)2



Lithium-ion Batteries I

28 February 2013 15

Lithium cells (1990’s Sony Corp.) Lithium is an ideal anode material owing to its low density and high reduction potential, making Li-based cells the most compact ways of storing electrical energy. Lithium cells are used in wristwatches, cardiac pacemakers, digital cameras, laptops etc.


Example

Anode (ox): LixC -> C+ xLi+ + xe-

Cathode (red): CoO2 + xLi+ + xe- -> LixCoO2

Net: LixC + CoO2 -> C + LixCoO2

VCell = ~3.6 V

Figure 2.7: Lithium cells (1990’s SonyCorp.) Lithium is an ideal anode mate-rial owing to its low density and high re-duction potential, making Li-based cellsthe most compact for storing electri-cal energy. Lithium cells are used inwristwatches, cardiac pacemakers, digitalcameras, laptops etc.

Anode: LixC→ C + cLi+ + xe−

Cathode: CoO2 + xLi+ + xe− → LixCoO2

Net (2.6 V): LixC + CoO2→ C + LixCoO2



Lithium-ion Batteries II

28 February 2013 16

109

108

107

106

105

104

103

1975 1980 1985 2000199519901970

Num

ber

of t

rans

isto

rs p

er in

tegr

ated

cir

uit

1900 196519601850

Lead-Acid

NiCdNiMH 102

101

100

Energy density [Wh/kg]Li-ion Improved

Li-ion

Nano-Li-ion

?

Moore’s law for Li-ion batteries?

Figure 2.8: Moore’s law for Li-ion batteries?



mWh to kWh Storage Applications ImWh-kWh Storage applications

50 kWh100 Wh1 Wh1 mWh

Efficient use of Energy

HEV

PHEV

EV

Mobile Energy Storage

21

EV

Figure 3.1: mWh to kWh storage applications; HEV: hybrid electric vehicle –PHEV: plug-in hybrid electric vehicle



mWh to kWh Storage Applications II

Figure 3.2: Nissan Leaf

Range: 175 kmCapacity: 24 kWhPower: 90 kWWeight: 300 kgFull charge time: 8 hoursLifetime: 160000 kmBattery: Li-ion, (+) LiMn2O4 based,(-) Carbon?



MWh to GWh Storage ApplicationsMWh – GWh StorageApplications

10000 MWh10 MWh100 kWh

Strategic reserves

Energy: Enable Solar and Wind, store energy on a scale of hours/days

Power: Backup, Grid stabilisation, Efficiency, timescale minutes/seconds

22

Efficient use of EnergyNot for batteries

Figure 3.3: MWh to GWh storage applications



Common Battery Types: Energy = f (size) I

Table 1: Classification of batteries vs. size [1]

Type Energy ApplicationsMiniature batteries 100 mWh – 2 Wh Watches, calculators, implanted

medical devicesBatteries for portableequipment

2 Wh – 100 Wh Flashlights, toys, portable radio andTV, phones, laptops

Starting, lighting and igni-tion (SLI) batteries

100 – 600 Wh Cars, trucks, buses, lawn mowertraction

Vehicles traction batteries 20 – 630 kWh Fork-lift trucks, locomotives3 MWh Submarines

Stationary batteries 250 Wh – 5 MWh Emergency power, local energy stor-age, remote relay stations

Load leveling batteries 5 – 100 MWh Spinning reserve, peak shaving, loadleveling



Ragone PlotRagone Plot Energy vs Power density

27

Figure 3.4: Energy vs. Powerdensity.0.28 kWh = 1 MJ: energy to ac-celerate a car of 1 ton to 161km/h (no air resistance);10 kWh = 36 MJ: energy contentof 1 L of gasoline1 W: power to burn 1 Watt light(bike);3 MW: large wind turbine;1 GW: large power plant



Simple Chemical and Electrochemical Reactions I

Chemical reaction: A + B = AB;The driving force for this reaction isthe difference in the values of the stan-dard Gibbs free energy of the products(AB), and the standard Gibbs free en-ergies of the reactants, A and B [2]:

∆G0r =

∑∆G0

f (Products)

−∑

∆G0f (Reactants) (1)

2 Advanced Batteries

If A and B are simple elements, this is called a formation reaction, and sincethe standard Gibbs free energy of formation of elements is zero, the value of theGibbs free energy change that results per mol of the reaction is simply the Gibbsfree energy of formation per mol of AB, that is:

∆Gr◦ = ∆G f

◦(AB) (1.3)

Values of this parameter for many materials can be found in a number ofsources [1–3].

While the morphology of such a reaction can take a number of forms, considera simple 1-dimensional case in which the reactants are placed in direct contact andthe product phase AB forms between them. The time sequence of the evolution ofthe microstructure during such a reaction is shown schematically in Fig. 1.1.

It is obvious that for the reaction product phase AB to grow, either A or B mustmove (diffuse) through it, to come into contact with the other reactant on the otherside. If, for example, A moves through the AB phase to the B side, additional AB willform at the AB/B interface. Since some B is consumed, the AB/B interface will moveto the right. As the amount of A on the A side has decreased, the A/AB interface willlikewise move to the left. AB will grow in width in the middle. The action will bethe same when species B, rather than species A, moves through the AB phase in thisprocess. There are experimental methods to determine the identity of the movingspecies, but that is not relevant here.

A B

A

A

A AB

B

B

B

AB

AB

Fig. 1.1 Simple schematic model of chemical reaction of A and B to form AB, indicating how themicrostructure of the system varies with time

Figure 4.1: Schematic model of chem-ical reaction of A and B to form AB



Simple Chemical and Electrochemical Reactions II

Electrochemical mechanism: like thechemical reaction case, AB must formas the result of a reaction between Aand B, but with an additional element:an electrolyte (passage of ionic, butnot electronic species).

A and B are connected to an externalelectrical circuit.

RemarkDriving force is the same as for thechemical mechanism (Eq. 1)

1 Introductory Material 3

In case this process occurs by an electrochemical mechanism, the time depen-dence of the microstructure is illustrated schematically in Fig. 1.2. Like the chemicalreaction case, product AB must form as the result of a reaction between the reactantsA and B; but an additional phase is present in the system- an electrolyte.

The function of the electrolyte is to act as a filter for the passage of ionic, but notelectronic species. The electrolyte must contain ions of either A or B, or both, andbe an electronic insulator.

The reaction between A and B involves not just ions but electrically neutralatoms. Hence for the reaction to proceed there must be another path whereby elec-trons can also move through the system. This is typically an external electricalcircuit connecting A and B. If A is transported in the system, and the electrolytecontains A+ ions, negatively charged electrons, e−, must pass through the externalcircuit in equal numbers, or at an equal rate, to match the charge flux due to thepassage of A+ ions through the electrolyte to the other side.

For an electrochemical discharge reaction of the type illustrated in Fig. 1.2 thereaction at the interface between the phase A and the electrolyte can be written as

ElectrolyteA B

A BABElectrolyte

BElectrolyte

BABElectrolyte

ABElectrolyte

A

A

AB

Electronic Path

Fig. 1.2 Simple schematic model of time evolution of the microstructure during the electrochemi-cal reaction of A and B to form AB, a mixed conductor. In this case it is assumed that A+ ions arethe predominant ionic species in the electrolyte. To simplify the figure, the external electronic pathis shown only at the start of the reactionFigure 4.2: Schematic model of electro-chemical reaction of A and B to form AB;A = A+ + e− (left) and A+ + e− = A(right)



Simple Chemical and Electrochemical Reactions III

Review of Thermodynamic Concepts Reversible Fuel Cell Potential Fuel Cell E�ciency

Introduction III

Figure 1.1: Thermodynamic relations

Anis Allagui Fuel Cells | Thermodynamics 5/35

Figure 4.3: Thermodynamic relations



Simple Chemical and Electrochemical Reactions IV

The balance between the chemical and electrical forces upon the ions underopen circuit conditions can be simply expressed as an energy balance:

∆G0r = −zFE (2)

where z is the charge number of the mobile ionic species, E the voltagebetween the electrodes, and F is the Faraday constant (96500 C/mol).



Direction of Redox Reaction IGalvanic cell (Discharge)

28 February 2013 38

Electrochemical cells, Definitions

anod

e

cath

ode

- +

Zn2+2e- 2e-

Cu2+

NO3-

porousbarrier

Cu(NO3)2Zn(NO3)2

cath

ode

anod

e

- +

Zn2+2e-

2e-

Cu2+

NO3-

porousbarrier

Cu(NO3)2Zn(NO3)2

DC supplye-

Ie- I

Galvanic cell (Discharge)

Chemical -> Electrical energySpontaneous redox reaction 'G<0+ Cathode (red) / - Anode (ox)

Electrolytic cell (Charge)

Electrical -> Chemical EnergyForced redox reaction 'G>0+ Anode (ox) / - Cathode (red)

Note: oxidation always occurs at the anode, reduction at the cathode (not related to the sign of the electrode!)Figure 4.4: Chemical → electrical

Spontaneous redox reaction ∆G < 0+ Cathode (red) / - Anode (ox)


28 February 2013 38

Electrochemical cells, Definitions

anod

e

cath

ode

- +

Zn2+2e- 2e-

Cu2+

NO3-

porousbarrier

Cu(NO3)2Zn(NO3)2

cath

ode

anod

e

- +

Zn2+2e-

2e-

Cu2+

NO3-

porousbarrier

Cu(NO3)2Zn(NO3)2

DC supplye-

Ie- I

Galvanic cell (Discharge)

Chemical -> Electrical energySpontaneous redox reaction 'G<0+ Cathode (red) / - Anode (ox)


Electrical -> Chemical EnergyForced redox reaction 'G>0+ Anode (ox) / - Cathode (red)

Note: oxidation always occurs at the anode, reduction at the cathode (not related to the sign of the electrode!)Figure 4.5: Electrical → chemical

Forced redox reaction ∆G > 0+ Anode (ox) / - Cathode (red)

RemarkOxidation always occurs at the anode, reduction at the cathode.



Direction of Redox Reaction II

Anodic (Oxidation) Reaction:

Reductant→ Oxidant + ze−

Zn(s)→ Zn2+ + 2e−

Zn is oxidized under the loss of elec-trons; by definition I > 0

28 February 2013 39

Electrochemical cells, Introduction


Reductant -> Oxidant + ze-

Zn(s) -> Zn2+ + 2e-

Zn is oxidized under the loss of electrons

Definition I>0

anod

e

e-

I

Zn2+

Cathodic (Reduction) Reaction:

Oxidant + ze- -> Reductant

Cu2+ + 2e- -> Cu(s)

Cu2+ is reduced by the gain of electrons

Definition I<0

cath

ode

e-

I

Cu2+

Figure 4.6:


Oxidant + ze− → ReductantCu2+ + 2e− → Cu(s)

Cu2+ is reduced by the gain of elec-trons; by definition I < 0

28 February 2013 39

Electrochemical cells, Introduction


Reductant -> Oxidant + ze-

Zn(s) -> Zn2+ + 2e-

Zn is oxidized under the loss of electrons

Definition I>0

anod

e

e-

I

Zn2+


Oxidant + ze- -> Reductant

Cu2+ + 2e- -> Cu(s)

Cu2+ is reduced by the gain of electrons

Definition I<0

cath

ode

e-

I

Cu2+

Figure 4.7:Anis Allagui Energy Storage | Batteries I 22/61


Chemical Thermodynamics

Voltage Generated by a Redox Reaction I

Laws of Thermodynamics introduce three state variables:0th law: introduces the absolute temperature T1st law: introduces the internal energy U by conservation of energy:

dU = δQ + δW

2nd law: introduces the entropy S :

dS >δQT




Voltage Generated by a Redox Reaction II

State variable introduced by different forms of work δW in the form:

δW =∑

iYidXi

where Yi intensive "forces" and Xi are extensive "displacements", such as:1 Mechanical pressure work: δW = −pdV2 Electrical work: δW = EdQ (E electric field, and Q electric charge)3 Adding mass work: δW = µdn (µ chemical potential, and n number

of moles added)




Voltage Generated by a Redox Reaction IIIFundamental equation of reversible thermodynamics :

dU = δQ + δW = TdS +∑

iYidXi (3)

With the new variable G = U − TS + pV ∗ (Gibbs free energy), we write:

dG = dU − TdS − SdT + pdV + Vdp

= TdS − pdV +∑

iµidni − TdS − SdT + pdV + Vdp

which simplifies to:

Definition

dG = Vdp − SdT +∑

iYidXi (4)




Voltage Generated by a Redox Reaction IV

RemarkAs the battery electrochemically converts chemical energy into electricenergy, it is not subject, as are combustion or heat engines, to thelimitations of the Carnot cycle dictated by the second law ofthermodynamics.Batteries are capable of having higher energy conversion efficiencies!

∗G = H − TSAnis Allagui Energy Storage | Batteries I 26/61



In a Battery . . .

We have mass flow in the battery (ions) that cause the charge flow. Themass flow is represents chemical work (moving ions) which is converted toelectrical work outside the battery.

28 February 2013 50

Equilibrium electrochemistry

During discharge

Think of Battery: we have mass flow in the battery (ions) that cause the charge flow. The mass flow is represents chemical work (moving ions) which is converted to electrical work outside the battery.

chem j jj

W dnP ¦In the battery:

In the Resistance outside the battery:

Elec cellW E dQ �

The battery does chemical work which is converted to useable electrical work outside:

Chem ElecW WG G

anod

e

cath

ode

e-

Cathions (+)

Anions (-)

- +Resistance

Figure 5.1: The battery does chemical work which is converted to useableelectrical work outside: δWchem = δWelec



Electric Properties

On the Cell Voltage I

∑jµjdnj = −EcelldQ (5)

NoteIn general chemical reactions can be written as:

ν1A1 + ν2A2 + . . .+ νnAn = 0

where Aj is the chemical species and νj the coefficient (negative ifreactant and positive if product).

Example: in H2 + Cl2 → 2HCl, we have A1 = H2, A2−−Cl2, A3 = HCl,ν1 = −1, ν2 = −1 and ν3 = 2



Electric Properties

On the Cell Voltage II

So we can write:

∑jµjdnj =

(∑jνjµj

)dn = −EcelldQ (6)

With dQ = zFdn (Faraday’s law):

Ecell = −∑

j νjµj

zF (7)

where F is Faraday’s constant (96500 C or 26.8 Ah) and z the number ofelectron involved in stoichiometric reaction.



Electric Properties

On the Cell Voltage III

ExampleCalculate the theoretical cellpotential of the Daniel battery shownin figure 5.2.

28 February 2013 52

Application on Daniel element:

Anode (-): oxidation Zn(s) -> Zn2+ + 2e-

Cathode (+): reduction Cu2+ + 2e- -> Cu(s)

Total: Cu2+ + Zn(s) -> Cu(s) + Zn2+

The potential of an electrochemical cell is equal to the difference of the potential of the half cells, which is determined by the chemical potential difference of the redox couple.

Equilibrium electrochemistry

anod

e

cath

ode

- +

Zn2+2e- 2e-Cu2+

NO3-

porousbarrier

Cu(NO3)2Zn(NO3)2

V

� � � �

� � � �

2 2

2 2

2 2/ /

2

2 2

j jCu Znj Zn Cu

Cell C A

Cu ZnCu ZnCell

Cell Cu Cu Zn Zn

EzF F

EF F

E

Q P P P P PI I

P P P P

H H

� �

� �

� �

� � � � � �

� � �

�

¦

I$ IC

Figure 5.2:

At the anode (-) oxidation of Zn:Zn(s)→ Zn2+ + 2e−

At the cathode (+) reduction of Cu2+:Cu2+ + 2e− → Cu(s)

Overall reaction:Cu2+ + Zn(s)→ Cu(s) + Zn2+

Ecell = φc − φa = −∑

j νjµj

zF

= −µCu + µZn2+ − µCu2+ − µZn

2F

=µCu2+ − µCu

2F −µZn2+ − µZn

2F= E0

Cu/Cu2+ − E0Zn/Zn2+



Electric Properties

On the Cell Voltage IVRemarkThe potential of an electrochemical cell is equal to the difference of thepotential of the half cells, which is determined by the chemical potentialdifference of the redox couple.

Discharge Discharge

28 February 2013 53

Electrochemical cells, driving force

Standard reduction potentials are commonly denoted by the symbol H°. (means under standard conditions 293 K and 1 bar, 1 molair concentrations)H° values for hundreds of electrodes have been determined and are usually tabulated in order of increasing tendency to accept electrons (increasing oxidizing power.)

The more positive the half-cell EMF, the smaller the tendency of the reductant to donate electrons, and the larger the tendency of the oxidant to accept electrons

oxidant(electron acceptor)

reductant(electron donor) E°, volts

Na+ + e- -> Na(s) –2.71Zn2+ + 2e- -> Zn(s) –.76Fe2+ + 2e- -> Fe(s) –.44Cd2+ + 2e- -> Cd(s) –.40Pb2+ + 2e- -> Pb(s) –.126

2 H+ + 2e- -> H2(g) 0.000AgCl(s) + e- -> Ag(s) + Cl–(aq) +.222

Hg2Cl2(s) + 2e- -> 2Cl–(aq) + 2Hg(l) +.268Cu2+ + 2e- -> Cu(s) +.337I2(s) + 2e- -> 2 I–(s) +.535Fe3+ + e- -> Fe2+(aq) +.771Ag+ + e- -> Ag(s) +.799

O2(g) + 4H+ + 4e- -> 2 H2O(l) +1.23Cl2(g) + 2e- -> 2 Cl–(g) +1.36

Zn(s) + Cu2+ -> Zn2+ + Cu(s), (Rcell = HRCu/Cu2+– HRZn/Zn2+ = 0.337 – (-0.76) = 1.097 V

Mor

e no

ble

Mor

e lik

ely

to r

un t

o th

e rig

ht

What is the potential difference H (Volt, V=J/C) build up?

For the Daniel cell, we would have acell potential:

Ecell = E0Cu/Cu2+ − E0

Zn/Zn2+

= 0.337− (−0.76)= 1.097 V



Electric Properties

On the Cell Voltage V

28 February 2013 55

Anod

e

cath

ode

- +

Zn2+2e- 2e-

Cu2+NO3

-

porousbarrier

Cu(NO3)2Zn(NO3)2

P�Zn/Zn2+) P�Cu/Cu2+)

P(Zn2+)-P�Zn)-2F(IA-IE)=-2FHZn/Zn2+

2F(IC-IE)=2FHCu/Cu2+P(Cu2+)-P(Cu)

Balancing field and chemical potential

I�x)

IA

IE

IC

HZn/Zn2+

HCu/Cu2+

x

x

(ocv = IC – IA =

= (PCu2+ - PCu)/2F - (PZn2+ - PZn)/2F

= HCu/Cu2+ – HZn/Zn2+

Figure 5.3: Balancing field and chemical potential; EOCV = ECu/Cu2+−EZn/Zn2+



Electric Properties

Cell Voltage Variation with Concentration: theNernst Equation I

We have already found that:

Ecell = −∑

j νjµj

zFBy introducing equation µ = µ0 + RT ln C (derived for ideal gases, dilutedsolutions), we obtain the Nernst equation:

Ecell = E0 − RTzF ln

[∏i

C νii

](8)

where Ci is the concentration of species i, νi is the stoichiometric coefficientof species i, and E0 is the reversible voltage at standard conditions.K =

∏i C νi

i is the equilibrium constant of the redox reaction.



Electric Properties

Cell Voltage Variation with Concentration: theNernst Equation II

Example

1 How do the electrode po-tentials of the reactionsCu2+ + 2e− → Cu(s) andZn2+ + 2e− → Zn(s) depend onthe concentration.

2 Does the open cell voltage goup, down or remains unchangedif we add Cu(NO3)2 and lessZn(NO3)2?

3 When does the Daniel reactionstop?

Question Electrochemistry 2

Anod

e

cath

ode

-

e-

++++

porousbarrier

Zn2+

Zn2+

e-

e-

e-NO3

-

NO3-

NO3-

NO3-

IA IE IE ICEA EC

+

Add less Zn(NO3)2 and more Cu(NO3)2

H = HOCu/Cu2+ - RT/2F [ln (cCu(s)/cCu2+)]= 0.337 - 0.013 [ln (cCu(s)/cCu2+)] > 0.337

H = HOZn/Zn2+ - RT/2F [ln (cZn(s)/cZn2+)]= -0.76 - 0.013 [ln (cZn(s)/cZn2+)]< -0.76

10. Does the open cell voltage goa) up or b) down?c) no clue

(cell = HCu/Cu2+– HZn/Zn2+ = (0.337+x) – (-0.76-y) = 1.097+x+y V => a) up

Figure 5.4:



Electric Properties

Cell Voltage Variation with Concentration: theNernst Equation III

1 For Cu2+ + 2e− → Cu(s), K = C 1Cu(s) × C−1

Cu2+ , hence:

E = E0Cu/Cu2+ −

RT2F ln

(CCu(s)

CCu2+

)= 0.337− 0.013 ln

(CCu(s)

CCu2+

)Similarly for Zn2+ + 2e− → Zn(s)

E = −0.760− 0.013 ln(CZn(s)

CZn2+

)



Electric Properties

Cell Voltage Variation with Concentration: theNernst Equation IV

2 When we add more Cu(NO3)2, we have

0.337− 0.013 ln(CCu(s)

CCu2+

)= 0.337 + x > 0.337 V

and with less Zn(NO3)2 we have:

−0.760− 0.013 ln(CZn(s)

CZn2+

)= −0.760− y < 0.760 V

The Ecell is:

Ecell = ECu/Cu2+ − EZn/Zn2+

= (0.337 + x)− (−0.760− y) = 1.097 + x + y > 1.097 V⇒ Voltage goes up.



Electric Properties

Cell Voltage Variation with Concentration: theNernst Equation V

3 The condition for the Daniel reaction to stop is if:

ECu/Cu2+ = EZn/Zn2+

0.337− 0.013 ln(CCu(s)

CCu2+

)= −0.760− 0.013 ln

(CZn(s)

CZn2+

)which would give CZn2+/CCu2+ = 4.4× 1036



Electric Properties

Cell Voltage Variation with Concentration: theNernst Equation VI

ExampleDoes the cell voltage of the Daniel battery remain constant duringdischarge?

The battery reaction is Cu2+ + Zn(s)→ Cu(s) + Zn2+.When it proceeds, Cu2+ is consumed and Zn2+ in produced. Therefore,from Nernst equation, the voltage cannot remain unchanged:

Ecell = E0Cu/Cu2+ −

RT2F ln

(CCu(s)

CCu2+

)− E0

Zn/Zn2+ + RT2F ln

(CZn(s)

CZn2+

)



Electric Properties

Cell Voltage and Gibbs Free Energy I

From equation 4, we have

dG = Vdp − SdT +∑

iµidni (9)

At constant pressure and temperature, it reduces to:

dG =∑

iµidni (10)

but∑

j µjdnj = −EcelldQ (from δWchem = δWelec), thus:

∆G = −E∆Q and ∆G0 = −zFE0 (@ standard conditions) (11)



Electric Properties

Cell Voltage and Gibbs Free Energy II

ExampleCalculate the Gibbs free energy of the Daniel battery shown in figure 5.2.

Overall reaction: Cu2+ + Zn(s)→ Cu(s) + Zn2+

∆G0 = −nFE0 = −2× 96500× 1.097 = −211731 J/mol

∆G0 is negative⇒ reaction occurs spontaneously (battery discharge, exother-mic reaction)

RemarkIn general we have:

1 ∆G < 0 : E > 0 → discharge exothermic2 ∆G > 0 : E > 0 → charge endothermic



Electric Properties

Energy Density I

The energy density of the battery Cu2+ + Zn(s)→ Cu(s) + Zn2+ is calcu-lated from Gibbs free energy:

∆G0 = −nFE0 = −211731 J/mol

With just considering the metals with the molar mass of Cu (63.5 g/mol)and the molar mass of Zn (65.4 g/mol), the energy density is:

Energy density = 211731(63.5 + 65.4)× 10−3 = 1643 kJ/kg

RemarkFor proper calculation of the energy density (and other electricalproperties), we need to account for the masses of the electrolyte, thecurrent collectors, and the packing of the cell, not just the active materials!



Electric Properties

Energy Density II

1.14 CHAPTER ONE

FIGURE 1.3 Components of a cell.

1.4.4 Theoretical Energy*

The capacity of a cell can also considered on an energy (watthour) basis by taking both thevoltage and the quantity of electricity into consideration. This theoretical energy value is themaximum value that can be delivered by a specific electrochemical system:

Watthour (Wh) ! voltage (V) " ampere-hour (Ah)

In the Zn/Cl2 cell example, if the standard potential is taken as 2.12 V, the theoreticalwatthour capacity per gram of active material (theoretical gravimetric specific energy ortheoretical gravimetric energy density) is:

Specific Energy (Watthours /gram) ! 2.12 V " 0.394 Ah/g ! 0.835 Wh/g or 835 Wh/kg

Table 1.2 also lists the theoretical specific energy of the various electrochemical systems.

1.5 SPECIFIC ENERGY AND ENERGY DENSITY OF PRACTICALBATTERIES

The theoretical electrical properties of cells and batteries are discussed in Sec. 1.4. In sum-mary, the maximum energy that can be delivered by an electrochemical system is based onthe types of active materials that are used (this determines the voltage) and on the amountof the active materials that are used (this determines ampere-hour capacity). In practice, onlya fraction of the theoretical energy of the battery is realized. This is due to the need forelectrolyte and nonreactive components (containers, separators, electrodes) that add to theweight and volume of the battery, as illustrated in Fig. 1.3. Another contributing factor isthat the battery does not discharge at the theoretical voltage (thus lowering the average

*The energy output of a cell or battery is often expressed as a ratio of its weight or size.The preferred terminology for this ratio on a weight basis, e.g. Watthours /kilogram(Wh/kg), is ‘‘specific energy’’; on a volume basis, e.g. Watthours / liter (Wh/L), it is ‘‘energydensity.’’ Quite commonly, however, the term ‘‘energy density’’ is used to refer to eitherratio.

Figure 5.5: Component of a cell



Electric Properties

Energy Density III

BASIC CONCEPTS 1.15

FIGURE 1.4 Theoretical and actual specific energy of battery systems.

voltage), nor is it discharged completely to zero volts (thus reducing the delivered ampere-hours) (also see Sec. 3.2.1). Further, the active materials in a practical battery are usuallynot stoichiometrically balanced. This reduces the specific energy because an excess amountof one of the active materials is used.

In Fig. 1.4, the following values for some major batteries are plotted:

1. The theoretical specific energy (based on the active anode and cathode materials only)2. The theoretical specific energy of a practical battery (accounting for the electrolyte and

non-reactive components)3. The actual specific energy of these batteries when discharged at 20!C under optimal

discharge conditions

These data show:

• That the weight of the materials of construction reduces the theoretical energy density orof the battery by almost 50 percent, and

• That the actual energy delivered by a practical battery, even when discharged under con-ditions close to optimum, may only be 50 to 75 percent of that lowered value

Thus, the actual energy that is available from a battery under practical, but close to optimum,discharge conditions is only about 25 to 35 percent of the theoretical energy of the activematerials. Chapter 3 covers the performance of batteries when used under more stringentconditions.

Figure 5.6: Theoretical (based on the active anode and cathode materials only)and actual specific energy of battery systems when discharged at 20℃ at optimalconditions. The theoretical specific energy of a practical battery accounts for theelectrolyte and non-reactive components



Electric Properties

Energy Density IVExample

1 Write and plot the voltage of the Daniel element as a function ofCZn2+ concentration, assuming starting concentration CZn2+(t = 0) =0.1 mol/l and CCu2+(t = 0) = 1.0 mol/l, CZn(s) = CCu(s) = 1, andthe cell volume = 10 ml.

2 Calculate the energy delivered by the battery.

We have CCu2+ = 1.1− CZn2+ (0.1 6 CZn2+ 6 1.1), therefore:

Ecell = 1.097− 0.013 ln(

CZn2+

1.1− CZn2+

)Note that at t = 0, Ecell is:

Ecell(t = 0) = 1.097− 0.013 ln(

CZn2+(t = 0)CCu2+(t = 0)

)= 1.127 V



Electric Properties

Energy Density V

0.9

0.95

1

1.05

1.1

1.15

0 0.2 0.4 0.6 0.8 1 1.2

Ece

ll /

V

CZn2+ / mol/l

Figure 5.7:



Electric Properties

Energy Density VI

On the other hand, we have from equation 11 (with∫

ln(ax + b)dx =(x + b

a)

ln(ax + b)− x, a 6= 0):

dG = −EdQ = −E × zF V dCZn2+

∆G = −2× 96500× 0.01×1.1∫

0.1

[1.097− 0.013 ln

(CZn2+

1.1− CZn2+

)]dCZn2+

= −2117×1.1∫

0.1

dCZn2+ + 27.5×1.1∫

0.1

ln(

CZn2+

1.1− CZn2+

)dCZn2+

= −2117×[CZn2+

]1.10.1 + 27.5×

[CZn2+

{ln(CZn2+

)− CZn2+

}]1.10.1

− 27.5×[(

CZn2+ − 1.1/1)

ln(1.1− CZn2+

)− CZn2+

]1.10.1

= −2113.3 J



Cell Polarization

Cell Polarization I

We know that the maximum electric energy that can be delivered by thechemicals that are stored within the electrodes in the battery dependson ∆G of the electrochemical couple.Within any cell undergoing charge or discharge, at least three forms ofcharge transmission can be considered:Electron flow in the electronic conductors such as electrode materials

Ion flow in the electrolyte (aqueous, solid or molten salt)Charge transfer at the electrode/electrolyte interfaces (e.g. in the

reaction: Zn(s)→ Zn2+(aq))



Cell Polarization

Cell Polarization IICharge transport, kinetics

(1) (3)

(2)

(1) ionic conduction A+

(2) Charge transport over the electrolyte-electrode interface(3) ionic conduction A or B (4) electronic conduction

20 March 2013 16

(4)

Figure 6.1: (1) Ionic conduction A+ (2) Charge transport over the electrolyte-electrode interface (3) Ionic conduction A or B (4) electronic conduction

RemarkAt the steady state, the condition of electroneutrality should be verified:rate of electron flow in the external circuit should be equal to the rate ofcharge transfer at each electrode/electrolyte process.



Cell Polarization

Cell Polarization III

It would be desirable if during the discharge all of this energy could beconverted to useful electric energy, but there are losses:

1 activation losses, needed to drive the electrochemical reaction at theelectrode surface,

2 concentration losses, which arises from the concentration differences ofthe reactants and products at the electrode surface and in the bulk(mass transfer), and

3 ohmic losses (IR drop)



Cell Polarization

Cell Polarization IV

2.2 CHAPTER TWO

trodes, and the contact resistance between the active mass and the current collector. Theseresistances are ohmic in nature, and follow Ohm’s law, with a linear relationship betweencurrent and voltage drop.

When connected to an external load R, the cell voltage E can be expressed as

E ! E " [(! ) # (! ) ] " [(! ) # (! ) ] " iR ! iR (2.1)0 ct a c a ct c c c i

where E0 ! electromotive force or open-circuit voltage of cell(!ct)a, (!ct)c ! activation polarization or charge-transfer overvoltage at anode and cathode(!c)a, (!c)c ! concentration polarization at anode and cathode

i ! operating current of cell on loadRi ! internal resistance of cell

As shown in Eq. (2.1), the useful voltage delivered by the cell is reduced by polarizationand the internal IR drop. It is only at very low operating currents, where polarization andthe IR drop are small, that the cell may operate close to the open-circuit voltage and delivermost of the theoretically available energy. Figure 2.1 shows the relation between cell polar-ization and discharge current.

FIGURE 2.1 Cell polarization as a function of operating current.

Although the available energy of a battery or fuel cell depends on the basic electrochem-ical reactions at both electrodes, there are many factors which affect the magnitude of thecharge-transfer reaction, diffusion rates, and magnitude of the energy loss. These factorsinclude electrode formulation and design, electrolyte conductivity, and nature of the sepa-rators, among others. There exist some essential rules, based on the electrochemical princi-ples, which are important in the design of batteries and fuel cells to achieve a high operatingefficiency with minimal loss of energy.

1. The conductivity of the electrolyte should be high enough that the IR polarization is notexcessively large for practical operation. Table 2.1 shows the typical ranges of specificconductivities for various electrolyte systems used in batteries. Batteries are usually de-signed for specific drain rate applications, ranging from microamperes to several hundredamperes. For a given electrolyte, a cell may be designed to have improved rate capability,with a higher electrode interfacial area and thin separator, to reduce the IR drop due toelectrolyte resistance. Cells with a spirally wound electrode design are typical examples.

2. Electrolyte salt and solvents should have chemical stability to avoid direct chemical re-action with the anode or cathode materials.

Figure 6.2: Cell polarization as a function of operating current.



Cell Polarization

Cell Polarization V

To achieve a high operating efficiency with minimal loss of energy:1 high conductivity of the electrolyte to minimize the IR polarization,2 chemically stable electrolyte to avoid direct chemical reaction with

the electrodes materials,3 fast rate of electrode reaction at both electrodes to minimize the

charge-transfer polarization (porous electrodes),4 adequate electrolyte transport to facilitate the mass transfer and

avoiding building up excessive concentration polarization (porosity, thick-ness, structure of separator, concentration of reactants),

5 compatible current collector material with the electrode material andthe electrolyte without causing corrosion problems.



Electrode Processes and Reaction Rates

Relationship between Current and Reaction Rate I

For the general electrode process:

Ox + ne− ↔ Red

the forward and backward reactions can be described by heterogeneous rateconstants kf and kb, respectively:

The rates of the forward (kf CO) and backward (kbCR) reactions cab begiven by:

if = n FAkf CO (12)ib = n FAkbCR (13)

where A is the area of the electrode.




Relationship between Current and Reaction Rate II

The role of electrons in the process is established by assuming that the rateconstants depend on the electrode potential, E : a fraction αE† is involvedin driving the reduction process, while the fraction (1 − αE) is effective inmaking the reoxidation process more difficult.

The potential-dependent forward and backward reaction rate coefficients canbe written:

kf = k0,f exp(−αn FE

RT

)(14)

kb = k0,b exp(

(1− α)n FERT

)(15)




Relationship between Current and Reaction Rate III

With the equilibrium condition if = ib = i0‡, where no net current flows,we obtain from equations 12 to 15:

COk0,f exp(−αn FEeq

RT

)= CRk0,b exp

((1− α)n FEeq

RT

)(16)

or

Eeq = RTnF ln

(k0,f

k0,b

)+ RT

nF ln(

CO

CR

)(17)

= E0 + RTnF ln

(CO

CR

)(18)




Relationship between Current and Reaction Rate IVRemark

Equation 18 is consistent with the Nernst equation, expect that it is writtenin terms of concentrations rather than activities.From equations 12 and 14, at equilibrium conditions, we have:

i0 = if = nFAC0k0,f exp −αnFEeq

RT (19)

The exchange current i0 is a parameter of interest in the battery field thatmeasures the rate of exchange of charge between oxidized and reduced speciesat any equilibrium potential without net overall change, and can be expressedusing equations 12, 14, 17, and 19 as:

i0 = nFAkC 1−α0 Cα

R (20)

The rate constant k = k0,f exp −αnFE0

RT is defined for a particular potential,the standard potential of the system E0.




Relationship between Current and Reaction Rate VWhen the net current is not zero, i.e when the potential is sufficiently dif-ferent from the equilibrium potential, the net current approaches the netforward current (or, for anodic overvoltages, the backward current):

i = nFAkC0 exp −αnF(E − Eeq)RT (21)

If E − Eeq = 0, then i = i0, and:

i = i0 exp −αnF(E − Eeq)RT (22)

A transformation into a net current would give (using equation 12, 15, 17–18):

i = if − ib = nFAk[C0 exp −αnFE0

RT − CR exp (1− α)nFE0

RT

](23)




Relationship between Current and Reaction Rate VI

RemarkRemember that in equation 23, CO and CR are concentrations at thesurface of the electrode, or are the effective concentrations. These are notnecessarily the same as the bulk concentrations.Concentrations at the interface are often (almost always) modified bydifferences in electric potential between the surface and the bulk solution.

†α is called also the transfer coefficient‡Exchange current



Electrical Double-Layer

Electrical Double-Layer and Ionic Adsorption I

ELECTROCHEMICAL PRINCIPLES AND REACTIONS 2.13

FIGURE 2.10 Orientation of water molecules inelectrical double layer at a negatively charged elec-trode.

FIGURE 2.11 Typical cation situated in electricaldouble layer.

Consider a negatively charged electrode in an aqueous solution of electrolyte. Assumethat at this potential no electrochemical charge transfer takes place. For simplicity and clarity,the different features of the electrical double layer will be described individually.

Orientation of solvent molecules, water for the sake of this discussion, is shown in Fig.2.10. The water dipoles are oriented, as shown in the figure, so that the majority of thedipoles are oriented with their positive ends (arrow heads) toward the surface of the electrode.This represents a ‘‘snapshot’’ of the structure of the layer of water molecules since theelectrical double layer is a dynamic system which is in equilibrium with water in the bulksolution. Since the representation is statistical, not all dipoles are oriented the same way.Some dipoles are more influenced by dipole-dipole interactions than by dipole-electrodeinteractions.

Next, consider the approach of a cation to the vicinity of the electrical double layer. Themajority of cations are strongly solvated by water dipoles and maintain a sheath of waterdipoles around them despite the orienting effect of the double layer. With a few exceptions,cations do not approach right up to the electrode surface but remain outside the primarylayer of solvent molecules and usually retain their solvation sheaths. Figure 2.11 shows atypical example of a cation in the electrical double layer. The establishment that this is themost likely approach of a typical cation comes partly from experimental AC impedancemeasurements of mixed electrolytes and mainly from calculations of the free energy ofapproach of an ion to the electrode surface. In considering water-electrode, ion-electrode,and ion-water interactions, the free energy of approach of a cation to an electrode surfaceis strongly influenced by the hydration of the cation. The general result is that cations ofvery large radius (and thus of low hydration) such as Cs can contact /adsorb on the electrode!

surface, but for the majority of cations the change in free energy on contact absorption ispositive and thus is against the mechanism of contact adsorption.6 Figure 2.12 gives anexample of the ion Cs! contact-adsorbed on the surface of an electrode.

It would be expected that because anions have a negative charge, contact adsorption ofanions would not occur. In analyzing the free-energy balance of the anion system, it is foundthat anion-electrode contact is favored because the net free-energy balance is negative. Bothfrom these calculations and from experimental measurements, anion contact adsorption isfound to be relatively common. Figure 2.13 shows the generalized case of anion adsorption

Figure 6.3: Orientation of water molecules in ECDL at a negatively charged elec-trode (left) and cation situation close to the ECDL (right). This would influencethe real (actual) concentration of electroactive species at an electrode surface




Electrical Double-Layer and Ionic Adsorption IIELECTROCHEMICAL PRINCIPLES AND REACTIONS 2.15

FIGURE 2.14 Potential distribution of posi-tively charged electrode.

As mentioned previously, the bulk concentration of an electroactive species is often notthe value to be used in kinetic equations. Species which are in the electrical double layerare in a different energy state from those in bulk solution. At equilibrium, the concentrationC e of an ion or species that is about to take part in the charge-transfer process at the electrodeis related to the bulk concentration by

e!zF!e BC " C exp (2.28)RT

where z is the charge on the ion and !e the potential of closest approach of the species tothe electrode. It will be remembered that the plane of closest approach of many species isthe outer Hemholtz plane, and so the value of !e can often be equated to !o. However, asnoted in a few special cases, the plane of closest approach can be the inner Helmholtz plane,and so the value of !e in these cases would be the same as !i. A judgment has to be madeas to what value of !e should be used.

The potential which is effective in driving the electrode reaction is that between thespecies at its closest approach and the potential of the electrode. If E is the potential of theelectrode, then the driving force is E ! ! e. Using this relationship together with Eqs. (2.26)and (2.28), we have

e ei !z E! !"nF (E ! ! )O" C exp expOnFAk RT RTe e!z F! (1 ! ")nF (E ! ! )R! C exp exp (2.29)R RT RT

where zO and zR are the charges (with sign) of the oxidized and reduced species, respectively.Rearranging Eq. (2.28) and using

z ! n " z (2.30)O R

Figure 6.4: Schematic of potential distribution of positively charge electrode.The potential at the Outer Helmholtz plane φ0 is referred to the potential of thebulk electrode.




Electrical Double-Layer and Ionic Adsorption III

As mentioned in Remark 11, it is the actual concentration at the interfaceand not the bulk solution that we have to use in equation 23.At equilibrium, the concentration C e of an ion or species that is about totake part in the charge-transfer process at the electrode is related to thebulk concentration by:

C e = C B exp −zFφe

RT (24)

where φe is the potential of the closest species to the electrode (at the outerHelmholtz plane) and z the charge on the ion.Equations 23 and 24 would give:

inFAk

= C0 exp−zOFφe

RTexp

−αnF(E − φe)RT

−CR exp−zRFφe

RTexp

(1 − α)nF(E − φe)RT

(25)




References

[1] C. A. Vincent and B. Scrosati.Modern Batteries.Butterworth-Heinemann, Oxford, UK, 1997.

[2] Robert Huggins.Advanced Batteries.Springer US, Boston, MA, USA, 2009.


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