Physical and Interfacial Electrochemistry Electrochemical Thermodynamics. Module JS CH3304 MolecularThermodynamics and Kinetics

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1 Physical and Interfacial Electrochemistry 213 Lecture 4. Electrochemical Thermodynamics Module JS CH334 MolecularThermodynamics and Kinetics

Thermodynamics of electrochemical systems Thermodynamics, the science of possibilities is of general utility.

We can readily compute thermodynamic state functions such as G, H and S for a chemical reaction by determining how the open circuit cell

We can compute the thermodynamic efficiency of a fuel cell provided that G and H for the cell reaction can be evaluated.

2 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 thermodynamic state functions such as G, H and S for a chemical reaction by determining how the open circuit cell potential E cell varies with solution temperature. We can compute the thermodynamic efficiency of a fuel cell provided that G and H for the cell reaction can be evaluated. We can also use measurements of equilibrium cell potentials to determine the concentration of a redox active substance present at the electrode/solution interface. This is the basis for potentiometric chemical sensing. M Ox, Red Red',Ox' M ' - Anode Oxidation e - loss LHS + Cathode Reduction e - gain RHS

3 Standard Electrode Potentials Standard reduction potential (E ) 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) 2H 2 (1 atm) ) E = V Standard d hydrogen electrode (SHE)

4 Measurement of standard redox potential E for the redox couple A(aq)/B(aq). electron flow E e reference electrode SHE P t H 2 in H 2 (g) A(aq) Pt indicator electrode H + (aq) B(aq) salt bridge test redox couple E provides a quantitative measure e for the thermodynamic tendency of a redox couple to undergo reduction or oxidation.

5 Standard electrode potential E is for the reaction as written The more positive E the greater the tendency for the substance to be reduced The half-cell reactions are reversible The sign of E changes when the reaction is reversed Changing the stoichiometric coefficients of a half-cell reaction does not change the value of E 19.3

6 We should recall from our CH111 electrochemistry lectures that any combination of two redox couples may be used to fabricate a galvanic cell. This facility canthenbeusedtoobtainuseful thermodynamic information about a cell reaction which would be otherwise difficult to obtain. Herein lies the usefulness of electrochemical thermodynamics. Given any two redox couples A/B and P/Q we can readily use tables of standard reduction potentials to determine which of the two couples is preferentially reduced. Once this is known the galvanic cell can be constructed, the net cell potential can be evaluated, and knowing this useful thermodynamic information can be obtained for the cell reaction. The procedure is simple to apply. One determines the couple with the most positive standard d reduction potential (or the most positive equilibrium potential E e determined via the Nernst equation if the concentrations of the reactants differ from 1 mol dm -3 ). This couple will undergo reduction at the cathode. The other redox couple will consequently undergo oxidation at the anode. This information can also be used to determine the direction of electron flow, for upon placing a load on the cell electrons will flow out of the anode because of the occurrence of a spontaneous de-electronation (otherwise known as oxidation or electron loss) reaction, through h the external circuit and into the cathode causing a spontaneous electronation (aka reduction or electron gain) reaction to occur. Hence in a driven cell the anode will be the negative pole of the cell and the cathode the positive pole. Now according to the IUPAC convention if the cell reaction is spontaneous the resultant cell potential will be positive. We ensure that t 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 E e,rhs is more positive than E e, lhs a positive cell potential ti V e will be guaranteed

7 A P B Q E E E E E cell RHS LHS electron flow cathode anode Anode Oxidation M - V e + M Cathode Reduction P(aq) A(aq) E e,anode Q(aq) B(aq) E e,cathode A ne B Cathode salt bridge PQme Anode redox couple E E ( T ) G, H, S, ma np mb nq Q aa m n B Q R m n aa A P

8 E cell Relation between thermodynamics mics of cell reaction and observed cell potential 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 work W e. From thermodynamics we note that maximum electrical work done at constant temperature and pressure W e is equal to the change in Gibbs free energy G for the net cell reaction. We use basic physics to evaluate the electrical work W e done in moving n mole Electrons through a potential difference of E cell. W q E e E 1 electron : e cell cell W N e E 1 mole electrons : e A cell n mole electrons : G G W e ne N n F E A cell E cell W n N e A e E cell G W e

9 A ne B E A, B P me Q E P, Q ma mne mb np nme nq We assume that E A,B is more positive than E P,Q and so is assigned as the cathode reaction. 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 G,,,,,, G mg ng m nfe mfe nmf E E nmfe cell A B P Q A B P Q A B P Q cell Note that t nm denotes the number of electrons transferred per mole of reaction as written. ma np mb nq Net Cell Reaction A ne B Cathode m n ab aq QR m n PQme Anode a a A P

10 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 rr ng species s in the two phases s and in the establishment of a compensating two way flow of charge across the interface resulting in a definite equilibrium potential difference e or E e. e A single equilibrium potential difference may not be measured. Instead a potential is measured between two electrodes (a test or indicator electrode and a reference electrode). This is a potentiometric measurement. The potential of the indicator electrode is related to the activities of one or more of the components of the test solution and it therefore determines the overall equilibrium cell potential E e. 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.

11 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 l ti l 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.

12 Equilibrium condition between phases: chemical potential. It is well known from basic chemical thermodynamics that if two phases and with a common uncharged species, are brought together, then the tendency of species 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 ~ The standard thermodynamic definition of the chemical potential is: G n Alternatively we can view the chemical potential of a species s in a phase as a measure of the work that must be done for the reversible transfer of one mole of uncharged species from the gaseous state of unit fugacity (the reference state) into the bulk of phase. In electrochemistry we deal with charged species s and charged phases. n k, P, T Phase

13 Electrochemical Activity We consider the work done W i in transferring a species i from the interior of a standard phase to the interior of the phase of interest. We also assume that t the species i has a charge q i = z i e. i Standard Phase W i i Destination Phase The electrochemical activity can be defined in the following manner. exp q i i i exp z F a a a exp W i i i kt B RT kt B i a a i i If two phases and contain a species i with different electrochemical activities iii such that the electrochemical activity of species i in phase is greater than that of phase then there is a tendency for species i to leave phase and enter phase. The driving force for the transport t of species i is the difference in electrochemical activity between the two phases. i In the latter expression a i represents the activity of species i.

14 Now from the definition of electrochemical activity zf i a i ai exp RT zf i aiaiexp RT Hence ai ai zf i exp ai ai RT ai exp zf i a RT i We can follow the lead of Lewis and introduce the difference in electrochemical potential as follows. i i i We can immediately deduce a relationship between the electrochemical potential difference and the ratio of electrochemical activities between two phases and via the following relationships. a i ln a a i i RT ln RT zif ai i zf i i Hence the difference in electrochemical potential is split up into two distinct components. First, the difference in chemical potential i and second the difference in electrical potential. Hence we note that the electrochemical potential is defined as the work required to transfer 1 mole of charged species from infinity in vacuum into a material phase. This work consists of three separate terms. The first constitutes a chemical term which includes all short range interactions between species (such as an ion) and its environment (ion/dipole interaction, ion/induced dipole interactions, dispersion forces etc). This constitutes the chemical potential term. The second constitutes an electrostatic term linked to the crossing of the layer of oriented interfacial dipoles (z i F). The third constitutes an electrostatic term linked to the charge of the phase (z i F). The outer potential y is the work done in bringing a test charge from infinity up to a point outside a phase where the influence of short range image forces can be neglected. The surface potential c defines the work done to bring a test charge across the surface layer of oriented dipoles at the interface. Hence the inner Galvani potential f is then defined as the work done to bring the test charge from infinity to the inside of the phase in question and so we define:

15 Electrode q M Electrolyte Solution q S zf zf i i i i i RT ln a z F i i i Vacuum Charged Interface Dismantle interface. Remove all excess charge & oriented dipole layers Vacuum = = Dipole layer across metal No dipole layer Vacuum Uncharged metal Charge electrode Charge solution Uncharged solution No dipole layer Oriented dipole Layer on solution Charged electrode with Dipole layer qm qs Charged solution with Dipole layer

16 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 from a given reference state at infinity into the bulk of an electrically charged phase. G n n,, k P T ~ G Electrochemical potential Electrochemical Gibbs energy It is sometimes useful to separate the electrochemical potential into chemical and electrical l components as follows. zf q Species valence Chemical potential Galvani electrical potential Species charge Equilibrium involving charged species transfer between two adacent phases is attained when no difference exists between the electrochemical potentials of that species in the two phases.

17 Rigorous Analysis of Electrochemical Equilibrium E F Ener rgy Reference Vacuum Level e Filled electronic energy levels Before Contact e e e E F Ox e Red When two phases come into contact the electrochemical potentials of the species in each phase equate. For equilibrium at metal/solution interface the electrochemical potential of the electron in both phases equate. Metal Reference Vacuum Level After Contact e Via process of Charge transfer e Ener rgy E F e Metal Filled electronic energy levels Ox e Red

18 The Nernst Equation We consider the following ET reaction. Ox z ox ne Red n zox zred zo zr z red Simplifying we get ar nerortln zr zof ao ar nerortln nf a O At equilibrium We note the following Also n O e R e e F z F O O O zf R R R RT ln a RT ln a O O O R R R Hence RT a ln n n a R O R e F O Also e e F At equilibrium RT a ln n n a e R O R e O Simplifying we get e F Hence O R ne RT ao ln nf nf ar O RTln ao ne zof RT ao ln R RT ln ar zrf nf a R Equation. n This is the Nernst O R e nf

19 Review of Thermodynamics We recall that the Gibbs energy G is used To determine whether a chemical reaction Proceeds spontaneously or not. We consider the gas phase reaction A(g) B (g). We let denote the extent of reaction. Clearly l < < 1. When = we have pure A and when = 1 we have 1 mol A destroyed and 1 mol B formed. Also dn A = -d and dn B = + 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: dg dn dn A A B B d d d Furthermore A B B A (1) In the latter the symbol r G = reaction Gibbs free energy Since varies with composition Then so also does r G. G rg (1) p, T forward reaction spontaneous reverse reaction spontaneous G dg PT, d (2) Hence we get from eqn. 1 and 2 G PT, G r B A (3) 1 Extent of reaction

20 Reaction Gibbs energy G G G r p, T extent of reaction = K 1 K 1 K 1 G r G r G r 1 G is the slope of the G versus graph at any degree of r G is the slope of the G versus graph at any degree of advancement of the chemical reaction.

21 If A > B then A B is spontaneous and r G is negative. If A > B then B A is spontaneous and r G is positive. If A = m B then r G = and chemical equilibrium has been achieved. Since A and B are ideal gases then we write p A A RT ln p p B B RT ln p A B pb pa B A B A RT ln RT ln p p p B B A RT ln pa rg RTln QR Q R = reaction quotient K = Equilibrium constant QR rg RTln K RTln QR RTln K ln r G r G RTln QR At equilibrium Q R = K and r G= G RT r ln K

22 Gibbs energy and chemical equilibrium. G Reaction not spontaneous In forward direction G Equilibrium Q=K Q large, Q>K [P]>>[R] G positive Q small, Q<K [P]<<[R] G negative Reaction spontaneous In forward direction G G Standard state Q=1 lnq= G ln Q G RT ln Q

23 The expression ust derived for the special case A B can also be derived more generally. If we set n as the stoichiometric coefficient of species (negative for reactants and positive for products, we can derive the following. G dg d d PT, G G r PT, G RT ln a r rg RT ln a rgrtln a G G RTln Q r r R We can relate chemical potential to activity a as follows. RT ln a In the latter we have introduced the following definitions. QR a ln a Again at equilibrium v ln a G Q K r R r G RT ln K a G K f Standard free energy r G f G oof formation of species.

Potentiometric Measurements We now mention the

A two electrode electrochemical cell is used.

equilibrium cell potential without drawing any significant

defined das E = (i --> ) where Df denotes the Galvani

24 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 obect of the exercise is to make a measurement of the equilibrium cell potential without drawing any significant current since we note that the equilibrium cell potential is defined das E = (i --> ) where Df denotes the Galvani potential difference measured between the cell terminals.this obective is achieved either using a null detecting potentiometer or a high impedance voltmeter.

25 The second measurement protocol involves use of an electrometer. The latter is based on a voltage follower circuit. A voltage follower employs an operational amplifier. The amplifier has two input terminals called the summing point S (or inverting input) and the follower input F (or non-inverting input). Note that the positive or negative signs at the input terminals do not reflect the input voltage polarity but rather the non inverting and inverting nature respectively of the inputs. Now the fundamental property of the operational amplifier is that the output voltage V o is the inverted, amplified voltage difference V where V = V - -V + denotes the voltage of the inverting input with respect to the non inverting input. Hence we can write that V o AV AV AV In the latter A denotes the open loop gain of the amplifier. Although ideally A should be infinite it will typically be 1 5. V o AV A V o V i V o V i 1 V i 1 A Also ideally the amplifier should exhibit an infinite input impedance so that they can accept input voltages without t drawing any current from the voltage source. This is why we can measure a voltage without any perturbation. In practice the input impedance will be large but finite (typically 1 6 ). An ideal amplifier should also be able to supply any desired current to a load. The output impedance should be zero. In practice amplifiers can supply currents in the ma range although higher current output can also be achieved. Keeping these comments in mind we can now discuss the operation of the voltage follower circuit presented In this configuration the entire output voltage is returned to the inverting input.. If V i represents the input voltage then we see from the analysis outlined across that the circuit is called a voltage follower because the output voltage is the same as the input voltage. The circuit offers a very high input impedance and a very low output impedance and can therefore be used to measure a voltage without perturbing the voltage significantly.

26 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. G nfe G nfe rg rg RTln QR r nfe nfe RT ln Q R r T = 298K RT/F = 25.7 mv At T = 298 K RT E E ln Q R.592 nf E E log Q R n QR Reaction quotient products reactants

27 Zn( s) Cu E cell 2 2 Equilibrium G ( aq) E cell Zn.59 2 ( aq) Cu( s) 2 Zn 2 Cu E Q K cell W log 2 Cu Dead cell R Q 1 W large Q 1 E cell Ecell As cell operates 2 2 [ Zn ] [ Cu ] Q R E cell Q Zn Cu 2 2 Q 1 W small

28 Determination of thermodynamic parameters from E cell vs temperature data. Measurement of the zero current cell potential E as a function of temperature T enables thermodynamic quantities such as the reaction enthalpy H and reaction entropy S to be evaluated for a cell reaction. Gibbs-Helmholtz eqn. E a b 2 T T ct T a, b and c etc are constants, which can be positive or negative. T is a reference temperature (298K) E T G rh nfet r nfe T rh rg T T E nfe r r G P nfe nft T E H nfet T P P P Temperature coefficient of zero current cell potential obtained from experimental E=E(T) data. Typical values lie in range VK -1

29 Once H and G are known then S may be evaluated. G H T S r r r r S r H T r G 1 E rs nfe nft nfe T T P E rs nf TT P Electrochemical measurements of cell potential conducted d under conditions of zero current flow as a function of temperature provide a sophisticated method of determining useful thermodynamic quantities.

30 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 E e. Under ideal reference H 2 in indicator circumstances, electrode Pt electrode the response of SHE the indicator electrode to changes in analyte H 2 (g) A(aq) Pt species activity at the indicator electrode/ solution interface should be rapid, reversible and governed by the Nernst equation. H + (aq) B(aq) test analyte 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.

31 Hydrogen/oxygen fuel cell Anode reaction 2H 2 (g) 4H + + 4e - H 2 /O 2 fuel cell H + ion Conducting membrane Cathode reaction O 2 (g) + 4e - +4H + 2H 2 O Remember CH111 Electrochemistry: GnFE eq, cell E E E eq, cell eq, C eq, A

32 Ballard PEM Fuel Cell.

33 Efficiency of fuel cell(i). Efficiency =work output/heat input GnFE eq, cell E E E eq, cell eq, C eq, A For electrochemical or cold combustion: Heat input entalphy change H for cell reaction Work output Gibbs energy change G for cell reaction rg rh TrS TrS max 1 rg rh rh rh Usually nfe 1 eq, cell max H r r H H 2 (g) + ½ O 2 (g) H 2 O (l) G = kcal mol -1 ; H = kcal mol -1 n = 2, E eq = V, =.83.

34 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 value E eq,, but will be less 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 electrolyte. cell. nfe nf ( E eq p ) cell Losses. real H H reactions occurring at one or both electrodes. Ohmic losses through the Sum of all overpotential

35 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. E E RT a B ln m nf a Aa H This expression can be readily simplified to the following form E 233RT 2.33 a B m RT E E ln 2.33 ph nf aa n F S N RT m 2.33 F n ph Idea: Potentiometric measurements can give rise to accurate ph measurements (especially metal oxide electrodes Lyons TEECE Group Research 212/213). Hence we predict that a Nernst plot of open circuit or equilibrium Potential versus solution ph should be linear With a Nernst slope S N whose value is directly related to the redox stoichiometry of the redox reaction through h the m/n ratio. When m = n then we predict that S N = - 2/33RT/F Which is close to 6 mv per unit change in ph at 298 K.

36 Membrane Potential Since we are considering charged species we define equilibrium In terms of equality of elctrochemical potentials. M + ( ) zf zf a a M + X - M + X - a M + a zf zf zf Now RT ln a RT ln a Hence ln ln zf RT a RT a a RT ln a RT a ln zf zf a is the electric potential necessary To prevent equalization of ionic activities by Difusion across the membrane. Net - ve Of course a similar result for the membrane potential can be obtained by equalizing the ratio of electrochemical l activities and noting that the following pertains. Membrane permeable only to M + ion. Net + ve Now if we assume that We get the final expression for the membrane potential RT a M ln z F a a zf a a exp 1 a RT RT a M ln zf a

Lecture 14. Thermodynamics of Galvanic (Voltaic) Cells.

Lecture 14. Thermodynamics of Galvanic (Voltaic) Cells. Lecture 14 Thermodynamics of Galvanic (Voltaic) Cells. 51 52 Ballard PEM Fuel Cell. 53 Electrochemistry Alessandro Volta, 1745-1827, Italian scientist and inventor. Luigi Galvani, 1737-1798, Italian scientist