<<

 jj  0

jj 

Physical and Interfacial 2013

Lecture 4. Electrochemical

Module JS CH3304 MolecularThermodynamics and Kinetics

Thermodynamics of electrochemical systems

Thermodynamics, the science of possibilities is of general utility. The well established methods of thermodynamics may be readily applied to electrochemical cells. We can readily compute functions such as G, H and S for a by determining how the open circuit potential

Ecell varies with . We can compute the thermodynamic efficiency of a cell provided that G and H for the cell reaction can be evaluated. M Ox,Red Red',Ox' M ' We can also use measurements of equilibrium cell potentials to - + determine the concentration of Anode a active substance present Cathode Oxidation Reduction at the /solution interface. e- loss This is the basis for potentiometric e- gain chemical sensing. LHS RHS

1 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 atm.

Reduction Reaction

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

E0 = 0 V

Standard hydrogen electrode (SHE)

Measurement of standard redox potential E0 for the redox couple A(aq)/B(aq).

electron flow

Ee

reference indicator H2 in electrode Pt electrode SHE P t A(aq) H2(g) H+(aq) B(aq)

test redox couple salt bridge E0 provides a quantitative measure for the thermodynamic tendency of a redox couple to undergo reduction or oxidation.

2 Standard

• E0 is for the reaction as written •The more positive E0 the greater the tendency for the substance to be reduced • The half-cell reactions are reversible •The sign of E0 changes when the reaction is reversed • 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 electrochemistry lectures that the couple with the most positive standard any combination of two redox couples may reduction potential (or the most positive be used to fabricate a . This equilibrium potential Ee determined via the Nernst facility can then be used to obtain useful equation if the concentrations of the reactants -3 thermodynamic information about a cell differ from 1 mol dm ) . This couple will undergo reaction which would be otherwise difficult reduction at the cathode . The other redox couple to obtain. Herein lies the usefulness of will consequently undergo oxidation at the anode . electrochemical thermodynamics. This information can also be used to determine the direction of electron flow, for upon placing a Given any two redox couples A/B and P/Q load on the cell electrons will flow out of the we can readily use tables of standard anode because of the occurrence of a spontaneous reduction potentials to determine which of de-electronation (otherwise known as oxidation or the two couples is preferentially reduced. electron loss) reaction , through the external Once this is known the galvanic cell can be circuit and into the cathode causing a spontaneous constructed, the net cell potential can be electronation (aka reduction or electron gain) evaluated, and knowing this useful reaction to occur. Hence in a driven cell the anode thermodynamic information can be obtained will be the negative pole of the cell and the for the cell reaction. cathode the positive pole. Now according to the IUPAC convention if the cell reaction is spontaneous the resultant cell potential will be 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 positive cell potential Ve will be guaranteed

3 0 0 0 0 0 A  P  B  Q Ecell  ERHS  ELHS  Ecathode  Eanode

electron flow

Ve Anode + - Oxidation Cathode M Reduction M P(aq) A(aq)

Q(aq) B(aq) Ee,anode Ee,cathode

 A ne B Cathode salt bridge mA nP mB nQ  mn PQmeAnode aaBQ QR  mn redox couple aaAP E  E(T )  G,H,S,

• When a spontaneous reaction takes place in a Galvanic cell, electrons are deposited in one electrode (the site of oxidation or anode) and collected from another (the site of reduction or cathode), and so there is a net flow of current which can be used

to do electrical We. • From thermodynamics we note that Ecell maximum electrical work done at constant temperature and We is equal to the change in Gibbs free G for the net cell reaction. • We use basic physics to evaluate the

electrical work We done in moving n Electrons through a potential difference of Ecell . W  q E  e E 1 electron : e cell cell

1 mole electrons : We  N Ae Ecell

We  n N A e Ecell n mole electrons :

Relation between thermodynamics G  We  ne N A Ecell of cell reaction and observed G  We cell potential  n F Ecell

4 A ne B E0  A, B  0 0 P me Q E P, Q We assume that E A,B is more positive than E0 and so is assigned as the cathode reaction. mA mne mB P,Q nP nme nQ

We subtract the two reactions to obtain the following result. mA nP mB  nQ mA nQ mB nP

To proceed we subtract the corresponding thermodynamic state functions In this case G0

00 0 0 0 00 0 GmGnGmnFEmFEnmFEEcell A,, B P Q A , B  P , Q  A ,, B  P Q  nmFE cell

Note that nm denotes the number of electrons transferred per mole of reaction as written. mA nP mB nQ  A ne B Cathode aamn Net Cell Reaction BQ  QR  mn P Q me Anode aaAP

The establishment of equilibrium does not imply the cessation of redox activity at the interface . The condition of equilibrium implies an equality in the electrochemical potentials of the transferring species in the two phases and in the establishment of a compensating two way flow of charge across

the interface resulting in a definite equilibrium potential difference e or Ee . A single equilibrium potential difference may not be measured. Instead a potential is measured between two (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

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 should be rapid, reversible and governed by the Nernst equation. The ET reaction involving the analyte species should be kinetically facile and the ratio of the analyte/product concentration should depend on the interfacial potential difference via the Nernst equation.

5 The potentiometric measurement. Potential reading Device (DVM) In a potentiometric measurement two electrodes are used. These consist of the indicator or sensing electrode, and a reference electrode. A Electroanalytical measurements relating potential to analyte B concentration rely on the response of one electrode only (the indicator electrode). Reference Indicator The other electrode, the reference electrode electrode electrode is independent of the Solution containing solution composition and provides analyte species A stable constant potential. The open circuit cell potential is measured using a potential measuring device such as a potentiometer, a high impedance voltameter or an electrometer.

Equilibrium condition between phases: . It is well known from basic chemical thermodynamics that if two phases  and  with a common uncharged species 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 Potential between the two phases. The condition for equilibrium is Phase  Phase    0 ~ jj  j    j  

jj  The standard thermodynamic definition of the chemical j potential is:  G    j     n j  nPTk ,,

Alternatively we can view the chemical potential of a species j in a phase  as a measure of the work that must be done for the reversible transfer of one mole of uncharged species j from the gaseous state of unit (the reference state) into the bulk of phase  . In electrochemistry we deal with charged species and charged phases .

6 Electrochemical Activity   We consider the work done W in transferring i i a  a  a species i from the interior of a standard i   i   i phase to the interior of the phase of interest. We also assume that the species i has a charge

qi = zie. If two phases  and  contain a species i with different electrochemical i activities such that the Wi electrochemical activity of  species i in phase  is i 0  greater than that of phase  then there is a tendency for species i to leave phase Standard Phase Destination Phase  and enter phase . The driving for the The electrochemical activity can be defined in the transport of species i is the following manner. difference in electrochemical activity  qzFW iii00 between the two phases. aaiiexp  a i exp  exp kTBB RT kT

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

Hence the difference in electrochemical Now from the definition of electrochemical potential is split up into two distinct activity zF components. First, the difference in chemical aaexp i  0 ii   potential i and second the difference in RT electrical potential . zF i 0 aaii exp   Hence we note that the electrochemical RT potential is defined as the work required to Hence transfer 1 mole of charged species from infinity in vacuum into a material phase. aaii zFi This work consists of three separate terms. exp  The first constitutes a chemical term which aaii  RT includes all short range interactions between species (such as an ) and its environment a  zF i exp i  (ion/dipole interaction, ion/induced dipole  interactions, dispersion etc). This aRTi   constitutes the chemical potential term j. The second constitutes an electrostatic term We can follow the of Lewis and introduce the linked to the crossing of the layer of oriented difference in electrochemical potential as follows. interfacial dipoles (ziF). The third constitutes an electrostatic term

linked to the charge of the phase (ziF). ii    i  The outer potential y is the work done in bringing a test charge from infinity up to a We can immediately deduce a relationship point outside a phase where the influence of between the electrochemical potential difference short range image forces can be neglected. and the ratio of electrochemical activities The surface potential c defines the work done to bring a test charge across the surface between two phases  and  via the following layer of oriented dipoles at the interface. relationships. Hence the inner Galvani potential f is then defined as the work done to bring the test aa   RTlnii  RT ln  z F  charge from infinity to the inside of the phase ii   in question and so we define: aaii 

ii zF    

7 Electrode Solution qM

qS

 zF  zF  Charged iii iiInterface 0 iiiRTln a  z F Dismantle interface. Remove all excess charge & oriented dipole layers =

Vacuum Vacuum =

Dipole layer No dipole Uncharged Uncharged No dipole across metal layer metal solution layer Oriented dipole Layer on solution Charge electrode Charge Vacuum solution

Charged electrode Charged solution with Dipole layer with Dipole layer qM qS

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 . Electrochemical potential  G   j    j    ~  n j  G  Electrochemical Gibbs energy nPTk ,,

It is sometimes useful to separate the electrochemical potential into chemical and electrical components as follows.

 jjj zF  jj q

Species valence Galvani electrical potential Chemical potential Species charge  Equilibrium involving charged species transfer jj    0 between two adjacent phases is attained when  no difference exists between the electrochemical jj  potentials of that species in the two phases .

8 Rigorous Analysis of Electrochemical Equilibrium When two phases come into contact the electrochemical Reference Vacuum Level potentials of the species in Before Contact each phase equate. For equilibrium at metal/solution 0  0  e  ee    interface the electrochemical 0 potential of the electron in E eF E  F  both phases equate. Ox Filled electronic e   Energy energy levels Red

Metal Reference Vacuum Level

After Contact  0  0 e   ee  Ox Via process of Energy E Charge transfer F e   Red Filled electronic  Metal energy levels

The Nernst Equation Simplifying we get

00 aR   nRTzzFeRO   ln   RO  We consider the following ET reaction. aO 

zox z red Oxne Red 00 aR   nRTnFeRO   ln    aO  nzox z red  z O z R

At equilibrium Hence 00 OeRn         RT a    ROln RF We note the following e  nnaO  0 ee   F  Also 0 ee   F  OOO zF 

RRR zF  At equilibrium ee    Also 00 0 RO     RT a R  0 OO RTln a O  ln e F  nnaO  0  RR RTln a R   Simplifying we get 00 0 Hence OR   na  e  RT  O    ln  nF nF a  0 R  OOeO RTln a  n  z F  0 RT aO    ln This is the Nernst 0 nF a  RRRRTln a    z F  R  Equation. 00  n  0   0 OR e  nF

9  Review of Thermodynamics In the latter the symbol  We recall that the Gibbs energy G is used  To determine whether a chemical reaction  G = reaction Proceeds spontaneously or not. r We consider the gas phase reaction A(g)  B (g). We let  denote the . Since  varies with composition Clearly 0 <  < 1. When  = 0 we have pure A and j when  = 1 we have 1 mol A destroyed and Then so also does rG. 1 mol B formed.

Also dnA = -d and dnB = + d where n 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 as follows: (1)

dGAA dn BB dn (1) ABdd    BA   d  Furthermore G dG  d (2)  PT,

Hence we get from eqn. 1 and 2 G rBAG   (3)  PT,

10 If A > B then A  B is spontaneous and rGis 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 write p 0 RT ln A QR = reaction quotient AA p0 K = p 0 RT ln B BB p0 Q pp  R 00 BArRGRTKRTQRTln  ln  ln BA  BARTln00  RT ln   pp  K

00 pB BART ln  pA 0 rRGRTQ  ln

0 rrGGRTQln R

At equilibrium QR = K and rG= 0

rGRTK ln

Gibbs energy and chemical equilibrium. Reaction not spontaneous In forward direction G P  R

Equilibrium Q large, Q>K Q=K [P]>>[R] G  0 G positive

Q small, Q

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

G We can relate chemical potential  to activity a as follows. dG jj d d  j j j  PT, 0 G jjRTaln j rjjG    PT, In the latter we have introduced the following

definitions.  j QaRj  j 0 rjjjjGRTa   ln    v  jj jj lnaajj ln   0  j j j rjGRTa  ln j Again at equilibrium rRGQK0  0  j GRTKln rjGRT  ln  a r j  j 0 Ka  j 0  f G  j rrGGRTQln R 00 Standard free energy rjfGG  j oOf formation of species j.

Potentiometric Measurements

We now mention the practicalities of conducting a potentiometric measurement . A two electrode electrochemical cell is used . This consists of a reference electrode and an indicator electrode . The object of theexerciseistomakeameasurement of the equilibrium cell potential without drawing any significant current since we note that the equilibrium cell potential is defined as

E= (i --> 0) where Df denotes the Galvani potential difference measured between the cell terminals .This objective is achieved either using a null detecting potentiometer or a high impedance voltmeter .

12 The second measurement protocol involves use of an Also ideally the amplifier should exhibit electrometer . The latter is based on a voltage an infinite input impedance so that they follower circuit . A voltage follower employs an can accept input voltages without drawing operational amplifier . The amplifier has two input any current from the voltage source . terminals called the summing point S (or inverting This is why we can measure a voltage input) and the follower input F (or non-inverting without any perturbation . In practice input) . Note that the positive or negative signs at the input impedance will be large but the input terminals do not reflect the input voltage finite (typically 106 ). An ideal amplifier polarity but rather the non inverting and inverting should also be able to supply any desired nature respectively of the inputs . Now the current to a load. The output impedance fundamental property of the operational amplifier is should be zero. In practice amplifiers can that the output voltage Vo is the inverted, amplified supply currents in the mA range although voltage difference V where V = V- -V+ denotes the higher current output can also be voltage of the inverting input with respect to the achieved. non inverting input. Hence we can write that

V o  AV   AV  AV  Keeping these comments in mind we can now discuss the operation of the voltage In the latter A denotes the open loop gain of the follower circuit presented In this amplifier. Although ideally A should be infinite it will configuration the entire output voltage is typically be 105. returned to the inverting input. .IfVi represents the input voltage then we see from the analysis outlined across that the circuit

V o  AV  AVo  V i is called a voltage follower because the output voltage is the same as the input V V  i  V voltage. The circuit offers a very high o 1 i input impedance and a very low output 1  impedance and can therefore be used to A measureavoltagewithoutperturbingthe voltage significantly.

The Nernst equation.

The potential developed by a Galvanic cell depends on the composition of the cell. From thermodynamics the Gibbs energy change for a Nernst eqn. holds chemical reaction G varies with composition of the for single redox reaction mixture in a well defined couples and net cell manner. We use the relationship reactions. between G and E to obtain the Nernst equation. 0 00GGRTQln rrG nFE  G  nFE rr R 0 nFE nFE RTln QR Reaction RT/F = 25.7 mV At T = 298 K quotient T = 298K products QR  0 RT reactants EEln Q 0 0.0592  R EE log Q nF n R

13 Zn(s)  Cu 2 (aq)  Zn2 (aq)  Cu(s) EquilibriumG 0 EQKcell0 R 2 W  0 0 0.059 Zn  Ecell  Ecell  log 2  2 Cu  Dead cell

Q 1 W large 0 Q 1 Ecell  Ecell

Zn2 Q    Cu 2 Q 1 W small As cell operates [][]Zn22 Cu

QER cell

Determination of thermodynamic parameters

from Ecell vs temperature data.

Measurement of the zero current cell 2 E  a  bT T  cT T   potential E as a 0 0 function of temperature T enables a, b and c etc are constants, which thermodynamic quantities can be positive or negative. such as the reaction H T0 is a reference temperature (298K) and reaction S to be evaluated for a cell reaction.  E    Temperature  T P coefficient of Gibbs-Helmholtz eqn. zero current    cell potential H nFE T  nFE  G r   obtained from r T experimental rrHGT T P E E=E(T) data. nFE  nFT  T Typical values P lie in range E 10-4 –10-5 VK-1 r HnFET r GnFE T P

14 • Once H and G are known then S may be evaluated.

rrGHTS r HG S rr r T 1  E  r SnFEnFTnFE     TT P E r SnF T P

• Electrochemical measurements of cell potential conducted under conditions of zero current flow as a function of temperature provide a sophisticated method of determining useful thermodynamic quantities.

Fundamentals of potentiometric measurement : the Nernst Equation.

The potential of the indicator electrode is related to the activities of one or more of the components of electron flow the test solution and it therefore determines the overall E equilibrium cell e potential Ee . reference indicator Under ideal H2 in circumstances, electrode Pt electrode the response of SHE the indicator electrode to changes in analyte A(aq) Pt H2(g) species activity at + the indicator electrode/ H (aq) B(aq) solution interface should be rapid, reversible and test analyte governed by the Nernst equation. salt bridge redox couple The ET reaction involving the analyte species should be kinetically facile and the ratio of the analyte/product concentration should depend on the interfacial potential difference via the Nernst equation.

15 Hydrogen/oxygen fuel cell

GnFE eq, cell Remember CH1101 Electrochemistry: Eeq,,, cellEE eq C eq A

Ballard PEM Fuel Cell.

16 Efficiency of fuel cell(I) .

GnFE eq, cell Efficiency  =work output/ input EEEeq,,, cell eq C eq A For electrochemical or ‘cold’ : Heat input  entalphy change H for cell reaction Work output  Gibbs energy change G for cell reaction  G HT S T S   r rr1 r  GH max  H HH rr r r r Usually   1 nFEeq,clel  max  r H

0 -1 0 -1 H2(g) + ½ O2(g)  H2O (l) G = -56.69 kcal mol ; H = - 68.32 kcal mol 0 n = 2, E eq = 1.229 V ,  = 0.830.

Efficiency of fuel cell (II).

• For all real fuel cell • This occurs because of: systems terminal cell – Slowness of one or more potential E does not intermediate steps of equal the equilibrium reactions occurring at one or both electrodes. value E , but will be less eq – Slowness of mass than it. transport processes • Furthermore E will either reactants to, or decrease in value with products from, the increasing current electrodes. drawn from the fuel – Ohmic losses through the electrolyte. cell. Sum of all overpotential nFE nF() E    cell  eq  Losses. real HH

17 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 mH ne B

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

RT  a  Idea: 0 B Potentiometric measurements EE ln  m  nF a a  can give rise A H  to accurate pH measurements This expression can be readily simplified to the following form (especially metal oxide electrodes Lyons TEECE Group Research 2012/2013). 0 2.303RTaB  m RT E EpHln  2.303 nF a n F E A

RTm Hence we predict that a Nernst plot of open circuit or SN 2.303 equilibrium F n Potential versus solution pH should be linear

With a Nernst slope SN whose value is directly related to the redox of the redox reaction through the m/n ratio.

When m = n then we predict that SN = - 2/303RT/F pH Which is close to 60 mV per unit change in pH at 298 K.

Membrane Potential +  Since we are considering charged species we define equilibrium  M  In terms of equality of elctrochemical potentials. M+X- M+X- jj()    zF   zF jj jj a j   a   M+ j aajj         0  

Net - ve Net + ve zFjj zF zF j   j  j      0 RTln a Membrane permeable Now jj    j  only to M+ion. 0 jj  RTln a j   Hence  00 Now if we assume that jj    zF 00 RTln a  RT ln a  jjjj j    We get the final expression for the membrane potential 00 aj   j  RT ln j a   aj  RT  j    M ln   00  zFjj a  j j  RT aj    ln  zF zF a Of course a similar result for jjjthe membrane potential can azF   aa jjexp  1 be obtained by equalizing the jj   aRTj    is the electric potential necessary ratio of electrochemical a  To prevent equalization of ionic activities activities and noting that the RT j  M ln  following pertains. by Difusion across the membrane. zFjj a

18