E1. Fundamental concepts

Electrochemistry

An electrolyte is a substance which leads electric current in solution or in form of melt.

Electrolytic dissociation: In solutions neutral molecules decompose to charged particles – ions:

+: cation - : anion

E1. Fundamental concepts

Elementary charge: e = 1.602 · 10^-19 C

(The charge of an electron is –1.602 · 10^-19 C.) The charge of one mol ion:

z · e · N_A = z · 1.602·10^-19 C · 6.022·10²³ mol^-1=

= z · 96485 C mol^-1

N_A : Avogadro constant; z: charge number.

Faraday constant: F = 96485 C mol^-1 E.g. Ca²⁺ : z= +2

Fundamental units

The Faraday constant is equal to the charge of one mole singly charged positive ions

(e.g. Na⁺ or H⁺)

Composition units used in electrochemistry Concentration: c [mol dm^-3] (mol solute per dm³ solution, molarity)

Molality: m [mol kg^-1] (mol solute per kg solvent, Raoult concentration) Advantage of molality - more accurate

Fundamental concepts

E2. Equilibrium in electrolytes

Even very dilute solutions cannot be regarded ideal (because of the strong electrostatic interaction between ions).

Still K_c can be frequently used as equilibrium constant (It is assumed here that the activity coefficients are independent of concentration, so K_ is taken constant).

Dissociation equilibrium

CA = C⁺ + A^- C⁺: cation A^-: anion c₀(1-) c₀· c₀·

c₀ : initial concentration

 : degree of dissociation





  1

0 2

c

K

The degree of dissociation () is the number of dissociated molecules per the number of all molecules (before dissociation).

depends on concentration (it is higher in more dilute solutions).

0    1

Autoprotolitic equilibrium of water

H₂O+H₂O = H₃O⁺ +OH^- K_w = a(H₃O⁺)·a(OH^-)

The activity of water is missing because it is in great excess, its concentration is

practically constant, and can be merged into the equilibrium constant K_w.

(E2)

Ionization equilibrium of acids HA+H₂O = H₃O⁺ +A^-

) (

) (

) (

HA a

A a

O H

K

a









Ionization constant:

Its negative decimal logarithm is used:

pK_a = - lgK_a

(E4)

(E5)

Ionization equilibrium of bases B+H₂O = BH⁺ +OH^-

) (

) (

) (

B a

OH a

BH K_b a



 



Dissociation constant:

K_a is also frequently used for bases:

BH⁺ +H₂O = B + H₃O⁺ )

(

) ( )

( ₃



 

 a BH

B a O

H K_a a

pK_b = - lgK_b

(E6) (E7)

(E8)

E3.Chemical potentials and activities in electrolytes

The definition of chemical potential:

nj

p i T

i

n

G

, ,

 

 







 



Its dependence on composition:

i i

i

 

 RT ln a



(E10)

(E11)

i j 

In dilute liquid solutions molality (as already mentioned m = mol solute per kg solvent) or concentration (molarity) (c = mol solute per dm³ solution) are used instead of mole fraction.

The activity a is expressed as the product of a composition expressed quantity (m or c) and the corresponding activity coefficient (_mor_crespectively

Standard state: unit molality (or molarity) in case of infinitely diluted solution. So the standard state is a virtual state, result of extrapolation to infinitely diluted state.

Since the cations and anions are always together in the solution, the individual activities and therefore individual chemical potentials of the ions cannot be determined.

E.g. the chemical potentials in NaCl solution:

O Na H

O

Cl H Cl T p n n

Cl n

n p Na T

Na n

G n

G

2

2 , , ,

, ,

,

,











 













 















  



O Na H

O Cl H

O

H Na T p n n Cl T p n n

n p NaCl T

NaCl n

G n

G n

G

2

2 , , , 2 , , ,

,

, ^ _ ^  _













 















 



 







 



That is



 

 

Only the sum of the chemical potentials of the ions can be determined.

We can express the chemical pontentials as sum of their standards values and the activity dependence (E11):











Na



RT ln a

 

 

 RT ln a









  



NaCl



RT ln a 

RT ln a



)

0

ln(











NaCl

 RT a a



The geometric mean of the activities of the two ions is called mean activity:









 a

a

a

2

0



 



Another example is CaCl₂ solution:









CaCl

 



2

2

)

ln(

0

2 2

2







CaCl

 RT a a



Introducing the mean activity

3 2

2





 a

a

a

3

0

 ln

  RT a



In general, if the electrolyte dissociates to _A anions and _C cations:

,

C A

C A

z C

z

A

A C

C

A

     

where z_A and z_C are the charges of anions and cations, respectively. The mean activity

  _A _C

,

C

A

a

a

a

 

where  = _A + _C.

The chemical potential:







   



(E12b)

(E13) (E12a)

The (dimensionless) activities can be expressed in the following way (examples):

0

0

c

a c m ,

a

 

 m

 



_A, _C: activity coefficients m^o: unit molality (1 mol kg^-1), c⁰ unit molarity

The mean activity using molalities:

(E14)

(E15)

 





 

 











 







 

 





 

 





 m m m m

a

C A

C A

C C

A

A A C A C

Introducing the mean activity coefficient, _± and the mean molality m_±:

The mean activity:

m

a

 

 m

  



   



  



 m

 m

m

(E16) (E17)

(E18)

Example:

Calculation of the mean molality of 0.2 mol/kg Al₂(SO₄)₃ solution.

1 5

0 . 4

0 . 6

0 . 51 mol kg

m

   

E4. Debye-Hückel theory

The electrolytes do not behave ideally even if the solution is dilute. The reason is the electrostatic interaction between ions.

Debye and Hückel developed a theory in 1923, which explains the behaviour of dilute electrolytes. We discuss only the main

points of this theory.

Cations and anions are not distributed evenly in the solution. A selected anion is surrounded by more cations than anions and vica versa.

Around a cation there is an excess of anions. Around an anion there is an excess of cations.

The cloud of such ions has a spheric symmetry. It is called ionic atmosphere.

How does the cloud of ions of opposite charge influence the chemical potential?

The interionic attractive forces reduces the energy of ions, the chemical potential is reduced, too. So the activity coefficient is less than 1.

The result of a long derivation is a simple expression for the activity coefficient.

12

lg 

  z

 z

 A  I

where I is the ionic strength. It depends on the molalities (or concentrations) of all the ions in the solution.

The constant "A" depends on the permittivity (, D=E), density and temperature of the solvent. Its value is 0.509 in aqueous

(E19)

 



 0 5 m_i z_i² I .

  

 0 5 c_i z_i² I .

(E20a) (E20b)

12 A

C

z 0 509 I

z   







.

lg

According to the equation E19 the logarithm of the mean activity coefficient is a linear function of the square root of the ionic strength.

This law is valid in dilute solutions only.

Therefore E19 is called Debye-Hückel limiting law.

25^oC, aqueous solution

E5. The electrochemical potential

If dn_i mol neutral component is added to the

solution, the change of Gibbs free energy is  dn . The total differential of Gibbs free energy in an open system (material and energy

exchange with the surroundings are allowed) at constant T and p:

i i

i T

,

p

dn

dG ^  ^

If ions are moved to a place where the electrical potential is then electrical (non pV) work is done:

i

i

F dn

z

dW     

Then the total differential of the Gibbs free energy is:





T ,

p

z F dn

dG ^  ^{  }

The partial derivative of the Gibbs function with respect to the amount of substance in the presence of ions:









 



 







 z F

n G

i i

n p

i T _j



, ,

The flollowing quantity is called electrochemical potential:











z

F

i



 ^~

If ions take part in the processes, the condition of equilibrium is expressed in terms of electrochemical potentials.

Of course, for neutral atoms and molecules (z_i=0) the electrochemical potential is equal to the chemical potential:

i

i



 ~ 

The condition of phase equilibrium is the equality of the electrochemical potential of the component which is present in two phases.

Chemical equilibrium:



 ^

^{ ~}

^{ }

^{ ~}

that is



 ~  0

where -s are the stochiometric coefficients, A-s are for the reactants, B-

(E24a) (E24b)

Some practical applications of the electrochemical potential

1. Contact potential

Behaviour of electrons in metals. Their energies are different in different metals.

Therefore their electrochemical potentials are also different.

If two metals are brought into contact (welding, soldering), electrons flow from the metal where their electrochemical potential is higher to the metal where their electrochemical

As result the metal that gains net electrons, acquires negative charge, while the metal that loses net electrons, aquires positive charge.

.

Thus an electric potential difference 

occurs at the contact between the two metals.

The condition of equilibrium is the equality of electrochemical potential for electrons in the two metals.

b e a

e



 ^~  ^~

Substituting the expression E23 for E25, and considering that the charge of electron z_e= -1,

where a and b denotes the two metals.

b b

e a

a

e

 F      F  

 ^{F }  ^

^{ }

 ^{ }

^{ }

F

a e b

e a

b



 













(E25)

(E26)

 (E26) is called contact (electric)

potential, it is proportional to the difference of

A useful application of the contact potential is the thermocouple. Here the contact potential can measured directly. Consider for example a copper-constantan (60 % Cu, 40 % Ni) thermo- couple (copper leadings join the voltmeter)



 1 2

copper

V

copper

copper

At the contact point of the two metals (1) the potential difference is ₁. If we want to measure this, we have to attach a voltmeter. The voltmeter have to have high internal resistance (high /V value) for minimizing the electric current in the circuit.

Since the source of the current in the circuit is only the internal contact potential

₂=-₁

The contact potential depends on temperature (thermocouple!)

If points 1 and 2 are of different temperature, the resultant voltage is not zero any more.

It is the function of temperature difference:

U = f(t).

Themocouples are used for temperature measurement. They always measure temperature difference between two points. Practically one of the two points have fixed temperature (cold point).

2. Electrode reaction

Behaviour of a metal dips into a solution

containing ions of this metal. What happens in this case?

Let copper dips into CuSO₄ solution. The following equilibrium sets in:

Cu²⁺(solution) + 2e^-(metal) = Cu

The condition for equilibrium is



 ~  0

That is

   ~  2  ~  0

For the neutral copper atoms the chemical potential, for the charged particles (copper ions and electrons) the electrochemical potentials are used.

Substituting for E27 the expression of the electrochemical potential (E23) :

0 )

metal (

F 2 )

metal (

2

) solution (

F 2

e

Cu Cu²































Rearrange the equation E28 so that the electrical potential difference is on the left hand side





F  ( )  ( .)  2







We cannot measure this electrical potential difference, because for that we have to dip

another metal in the solution and the voltage of all the circuit can only be measured.

(E29)

E6. Electrochemical cells

CuSO₄ + Zn = ZnSO₄ + Cu Cu²⁺ + Zn = Zn²⁺ + Cu

Consider the following redox reaction:

In this reaction Cu²⁺ ions are reduced and Zn atoms are oxidized.

Two steps:

Cu²⁺ + 2 e^- = Cu reduction

Zn = Zn²⁺ + 2 e^- oxidation

In an electrochemical cell the oxidation and reduction are separated in space:

reduction means electron gain, oxidation means electron loss.

Galvanic cell: production of electrical energy from chemical energy.

Electrolytic cell: electrical energy is used to bring about chemical changes. (E.g.

The following slide shows a Daniell cell.

On the left hand side it works as a Galvanic cell, and produces electric current (spontanous process).

On the right hand side we use an external

power source to reverse the process. In that case the cell works as an electolytic cell.

Cathodes and anodes are electrodes.

Observe the change of anodic and cathodic functions comparing the galvanic and electrolytic

flow of electrons

 ^^^^ 



ZnSO4 CuSO4

Cu Zn



anode cathode

flow of electrons

cathode anode

 



ZnSO4 CuSO4

Cu Zn



resistor

Galvanic cell Electrolytic cell

Daniell cell

Galvanic cell Electrolytic cell

Zn Zn²⁺ + 2 e^- Zn²⁺ + 2 e^-  Zn Cu²⁺ + 2 e^-  Cu Cu  Cu²⁺ + 2 e^-   1.1 V

The potential difference of a cell in equilibrium conditions is called electromotive force (emf).

The electromotive force E of the cell can be measured as the limiting value of the electric potential difference , i.e. as the current through the cell goes to zero.

Oxidation on anodes, reduction on cathodes.

Simple measurement of emf:

- the emf is compensated by a

measurable and changable voltage source.

If the G meter measures zero, the two tensions are equal.

Cell diagram (description of the cell):

ZnZn²⁺ (aq.)  Cu²⁺ (aq)  Cu

Phase borders are denoted by vertical lines.

Two vertical lines are used for the liquid junction.

Usually a salt

bridge is used to prevent the liquids from mixing

(KNO₃/H₂O colloid in agar-agar, a sea

The electromotive force is the sum of the potential differences on the phase boundaries.

If the external leads are made of copper:

CuZn - contact potential

Zn Zn²⁺ - between solution and metal Zn²⁺ Cu²⁺ - between two solutions

Cu²⁺Cu - between metal and solution

Cu I Zn 2F [(Zn) - _left(Cu)] = 2_e (Zn) - 2_e (Cu) Zn I Zn²⁺ 2F [(sol) - (Zn)] = _Zn - _Zn²⁺ - 2_e(Zn)

Zn²⁺ I Cu²⁺ diffusion potential (_diff , neglected,equation E40, section E8)

Cu²⁺ I Cu 2F [_right(Cu) - (sol)] = 2_e (Cu) + _Cu²⁺ -_Cu

Summing up:

2F [_right(Cu) - _left(Cu)] = _Zn + _Cu²⁺ - _Cu – _Zn²⁺

In general:

Left hand side of E31: Gibbs free energy change for the reaction. (It depends on the concentration of participants)

Right hand side of E31: Electrical work done by the system.

(E31)

E F

z

rG    



Positive E - The reaction goes spontaneosly from left to right according to reaction equation.

(_rG < 0)

Negative E - The reaction goes spontaneosly in the opposite direction. (_rG > 0)

E = 0 - Chemical equilibrium. (_rG = 0) Sign dependence of the emf (E)

7. Thermodynamics of Galvanic cells, the Nernst equation

Find the relationship between the thermodynamic functions of reactions and the electromotive force.

1. _rG:

2. _rS: dG = V dp – S dT

G S



 

 

 

E F

z

r

G    



(E32)

Deriving E32

(

T S G

r p

r

   



 











p

r

T

zF E

S 



 







 



According to E33 we have to know for calculating  S the temperature

(E33)

_rH G = H -TS H = G + TS Therefore _r H = _r G + T _r S

_r H = -z F E + z F T

T p

E 



 









The changes of thermodynamic functions in chemical reactions can be accurately

determined by measuring the electromotive force and its temperature dependence.

Substituting E31 and E33 for E34

(E34)

(E35)

Nernst equation: Dependence of E on the composition

_r G =  _B _B -  _A _A According to E11

_i = _i⁰ + R T lna_i

Substituting this equation for E36

B

a

RT G

  ^







ln

(E36)

(E37)

 







_rG _B ⁰_{B A} ⁰_A RT _B ln a_B RT _A ln a_A

Dividing both sides by -z·F :

A B

A B r

a a F

z RT F

E z _

 



 

 

 ln

0

E38b is the original form of the Nernst equation. E⁰ is the standard electromotive force, it is the value of E when the activities

A B

A B

a a F

z E RT

E





 



ln

reactants (E38a)

(E38b)

From E⁰ the equilibrium constant of the reaction can be calculated (consider E11):

_r ⁰ = - RT lnK

F K z RT F

E z

ln

0

   



If K  1 , E⁰  0

(E39)

Example: Zn + Cu²⁺ = Cu + Zn²⁺







2 2

2

0

Cu Zn

a ln a F

E RT

E

V F K

z

RT ln  1 . 1 1

2 1 059

0 . lg K  . ^{37 . 29} 059

. 0

2 ,

lg K  2 

K  2·10³⁷, i.e. the reaction is very strong shifted in the direction of the products, Cu²⁺

E8. Electrode potentials

In a galvanic cell oxidation and reduction are separated in space (figure in section E).

Anode - oxidation Cathode - reduction

Many different electrodes can be

constructed. Electrodes can be combined

E.g. from 1000 different electrodes nearly half a million different galvanic cells can be made.

Instead of listing the electromotive forces of all the galvanic cells electrode potentials are defined. From the electrode potentials the

electromotive force can be calculated.

E = ε_right – ε_left (+ε_diff) (E40)



We have seen that the absolute values of electrode potentials cannot be determined.

Therefore a relative electrode potential scale was defined.

One electrode was selected as a standard, and the electrode potentials of all the electrodes are related to this standard.

The selected electrode is the standard hydrogen electrode

In a standard hydrogen electrode a platinum wire dips in a solution where the activity of hydrogen ions (H₃O⁺ ions) is 1, and hydrogen gas of 10⁵ Pa standard pressure bubbles around the platinum.

The adsorbed hydrogen molecules and the hydrogen (H₃O⁺) ions in the solution take part in the electrode reaction.

So the electrode potential of the electrode X is defined as the electromotive force of the following cell (if the electrode X is the cathode) :

  ^{p H  a  X X}

H

Pt

 1







 H e H

2

1

X e

X





X H

X

H 



 1

(E41)

(E42) The cell reaction is

The electrode potential of an electrode (



X H

X

H₂  ^  ^  2

1

 

 

0

p H ln a

F E RT

E



 



X: metal

X⁺: ion of the metal (E43)

In the standard hydrogen electrode

   

p and p

H

a

 



 ln a X

F E RT

E

E⁰= ⁰ is the electrode potential when the activity of X⁺ ions is 1, the standard electrode potential. This values are listed in electrode

and therefore, considering E43

(E44)

If the electrode potential is positive, the electrode oxidizes H₂.

If the electrode potential is negative, the electrode reduces H⁺.

In general, the value of the electrode potential shows the oxidizing-reducing ability of the system.

A system having more positive electrode potential oxidizes the system having more

The expression for electrode potential for a general electrode process:

1. Describe the electrode process, see (in the direction of reduction)

 _ox M_ox = _red M_red

2. Set up the Nernst equation (E38):

ox red

ox red

a a F

z RT



 

 

 

 ⁰ ln That is

ox

a

RT



 

 ln ^

(See E24a)

We apply the rules valid for the equilibrium of heterogeneos reactions.

The activities of pure metals and precipitates are left out.

For gases the activity is p_i/p^o (ideal gas).

E.g. AgCl electrode (see metal-insoluble solt electrode, section E9).

AgCl = Ag⁺ + Cl^- Ag⁺ + e^- = Ag

AgCl + e^- = Ag + Cl^-







Cl

AgCl F a

RT 1

0 ln



  _AgCl  a_Cl

F

RT ln





The electrode potentials at 25 ^oC:

10 ln lg ln a

a  ln a = lg a · ln 10, T = 298.15 K

] [

05916 .

0 10

ln Volt

F

RT  

] V a [

lg a z

.

red

oxox











 

 05916

0

0

E9. Types of electrodes

1. Metal electrodes

As already mentioned (section E6,

electrode reactions) a metal dips into solution containing ions of the same metal (Me).

Me^z+ + z·e^- = Me z, charge number (positive integer)















 z

z

Me o

Me

zF a RT a

zF

RT 1 ln

0 ln 





Amalgam electrode – metal dissolved in mercury.

(E46)

2.Gas electrodes

Indifferent metal (e.g. Pt) is surrounded by a gas (in form of bubbles).

The electrode reaction is that of gas molecules and ions.

E.g. PtCl₂Cl^-(aq) Cl₂ + 2 e^- = 2 Cl^-

 

p p 05916

36 0

1 ⁰

Cl₂





 . lg

. at 25 ⁰C

⁰=1.36 V

The most important gas electrode is the hydrogen electrode (see also equations

E41, E42 and E43).

The electrode: H₂ (Pt) H⁺ (aq) The reaction: H⁺ + e^- = ½ H₂

At 25 ^oC

0 H H

p p 05916 a

0 0

2







 . lg

If p = p^o  = - 0.059· pH

[V]

3.Redox electrodes

An indifferent metal dips in a solution

containing a species in two oxidation states.

E.g. Sn²⁺, Sn⁴⁺ Sn⁴⁺ + 2 e^- = Sn²⁺













2 4 4

2

ln

2

0

/

Sn Sn Sn

Sn

a

a F

 RT



4. Metal - insoluble salt electrodes

A metal electrode is covered by the layer of an insoluble salt of the metal. The electrolyte contains the anions of the insoluble salt.

E.g. Ag/AgCl electrode AgAgCl(s)  KCl(aq) Ag = Ag⁺ + e^-

Ag⁺ + Cl^- = AgCl

Ag + Cl^- = AgCl + e^-

red ox







Cl

AgCl

F a

RT 1

0

ln









a

F

RT ln





depends on the anion activity, „Cl^- electrode”.

Calomel electrode, HgHg₂Cl₂ (s)KCl (aq.) Hg₂Cl₂ = Hg₂²⁺ + 2Cl^-

Hg₂²⁺ + 2 e^- = 2 Hg

Hg₂Cl₂ + 2 e^- = 2 Hg + 2Cl^-

0

1

 ln



 

RT



(E49)







a

F

RT ln

0

2





The metal - insoluble salt electrodes are used as reference electrodes because their electrode potential is stable.

They are non-polarizable electrodes,

their electrode potential does not change if the current changes.

(E50)

E10.Membrane potentials, glass electrodes

Membranes are significant in biology. The walls of living cells behave as membranes.

Consider e.g. two solutions of different concentrations of KCl, and let the membrane dividing them be permeable only to K⁺.

The K⁺ ions tend to diffuse into the more dilute solution but the Cl^- ions cannot follow.





 









membrane

c^a c^b<c^a

KCl solutions





 









membrane

Diffusion through the membrane

K⁺

The diffusion does not continue infinitely since the charge separation causes retard migration of the cations (repulsion of the anions)

The system reaches equilibrium when the electrochemical potentials of the K⁺ ions are the same on each side of the membrane.

b K a

K ^

 

 ^{~ ~}

where the two solutions are denoted by (E51)

b b

K a

a

K 

 F    

 F  







a b

F  

 





   1











 

K



RT ln a



b K

a b K

K a

K

a

RT a









   ln



a

a

RT



 ln

Considering E23 and E26:

(E52)

Glass electrodes are used for measuring pH.

The main part of the glass electrode is a glass membrane of special composition. It consists of Si-O frame with negative charge compensated by positive ions (Li⁺, Na⁺, Ca²⁺).

The positive ions are mobile so the membrane slightly conducts electricity.

Thickness ~0.05 mm.

Resistance 10 - 10 ohm.

Before use the glass electrode is soaked in water. Some of the sodium ions are

replaced by hydrogen ions.

When we dip the glass electrode in a solution containing hydrogen ions, equilibrium of H⁺ ions is reached between the solution and the glass.

The condition for equilibrium:

)

~ ( )

~

( glass  

solution



) (

) (

) (

)

(glass F glass _H solution F solution

H       



The chemical potential of hydrogen ions in the solution depends on their activity:









 

H



RT ln a



) 1 (

) 1 (

) (

)

( solution

glass F glass F

solution    _H  _H

  





  



 _{H H} a_H

F solution RT

glass F

F 1 ( ) ln

)

1 ( ₀





Applying E23 and E26

since

At 25 ^oC a_H  a_H

F

RT ln 0.059lg and  lg a_H  pH

The electrode potential of the glass electrode depends on the pH of the solution.

1 unit pH change causes 59 mV change in the electrode potential.

The system for measuring pH consists of a glass electrode and a reference electrode.

The emf is measured. Solutions of known pH ]

V [ pH .

)

(0  0 059

 

 (E54)

E11. Conductivity of electrolytes

Ohm´s law:

I   R

ΔΦ: potential difference . [V]

R: resistance [Ώ]

In case of metallic conductors the resistance depends on the kind of material, the geometry and the temperature. For uniform cross section

A R  l

 

The resistivity is the resistance of a conductor of unit length and unit cross section.

 

 

 _or

m

m

m  



In case of electrolytes the reciprocals of R and

are used:

G R1

 Conductance [Ω^-1=S, Siemens]

  ¹

(E57)

(E58)