Chapter 10: Equilibrium Electrochemistry

1. Chapter 10: Equilibrium Electrochemistry Homework: Exercises(a only):5, 6,7,12, 18, 20, 25, 26 Self Test: 8 and 9

2. Thermodynamic Properties of Ions in SolutionEnthalpy and Gibbs Energy • Enthalpy and Gibbs energy of formationValues of DfHø and DfGø refer to formation of ions from reference state of parent ions • Individual enthalpies and Gibbs energies for ions not directly measurable Ag(s) ® Ag+(aq) + 1e- DfHø = ?, DfGø = ? • Only measure overall reactions Ag(s) + 1/2Cl2(g)® Ag+(aq) + 2Cl- (aq) DfHø = -61.58 kJ mol-1 • Since H and G state functions, the overall reaction is the sum of component reactions • DrHø =DfHø(Ag+, aq) + DfHø (Cl-,aq) • Could measure a number of reactions with similar components and by difference get DfHø • Need to know one DfHø however • Define reference ion and assign value of zero for DfHø and DfGø for it • Ion is H+: DfHø(H+(aq)) = 0 and DfGø(H+(aq)) = 0 definition • Measure other ions relative to H+ • Chlorine: DfHø(Cl-, aq) = -167.2 kJ mol-1 + DfGø(Ag+, aq) = -131.2 kJ mol-1 • Determine ions by difference from reaction enthalpy /Gibbs energy • Silver: DrHø =DfHø(Ag+, aq) + DfHø (Cl-,aq) or =DfHø(Ag+, aq) = DrHø - DfHø (Cl-,aq) DfHø(Ag+, aq) =-61.58 kJ mol-1 - (-167.2 kJ mol-1 ) = 105.6 kJ mol-1

3. 3) Dissociate Cl2(g) 4) Add e- from Cl(g) 2) Remove e- from H(g) 5) Solvate Cl- 1) Dissociate H2(g) 6) Solvate H+ 6) Form H2(g) and Cl2(g) from solvated ions (-DrG) Contribution to DfGø • Contributions can be seen by constructing a thermodynamic cycle • Sum around cycle is zero (G is state function) • DfGø of an ion includes contribution from dissociation, ionization and solvation of hydrogen • All Gibbs energies except DG of solvation estimated from standard tables • DfGø of two ions is related to DsolvGø DfGø(Cl-, aq)= 1272 kJ mol-1 + DsolvGø(Cl-) + DsolvGø(H+) • Can be estimated from Born equation

4. Born Equation - Solvation Gibbs Energies • Solvation Gibbs energy estimated from the electrical work required to transfer an ion to a solvent - Born Equation • Solvent treated as a dielectric with permittivity, eG • Good example of how work need not be PV work to calculate Gibbs energies {look at Justification 10.1, p.255} • For water, Born equation becomes DsolGø = 6.86 x 104(zi2/ri) kJ mol-1 ri is radius in pm • Values turn out to be in reasonable agreement with experimental

5. + Standard Entropies of Ions in Solution • Like G and H can’t measure entropies of isolated ions in solution • Assume DfSø for H+ in solution is 0 • Derive values for other ions relative to it • Some ions have positive DfSø and some negative relative to H+ • DfSø (Cl-,aq) = 57 J K-1 mol-1 DfSø (Mg2+,aq) = -128 J K-1 mol-1 • Means that relative to water Mg2+ induces more order and Cl- less • Can be rationalized in terms of charge on ion affecting local order around ion • Small highly charged ions (Mg2+) induce more order than bigger less highly charged ions • Depends a bit on your model for the liquid state how you think about it • Various models include distorted ice-like structure, flickering clusters (Frank-Wen clusters), glassy water (J. Gibbs) • For structure breaking ions Zone B can encroach on Zone A • Estimate of DfSø (H+) on absolute (3rd Law scale) is -21 J K-1 mol-1 • H+ induces order Zone B: Structure somewhat broken Zone C: Structure unaffected Zone A: Structure Highly Affected

6. Ion Activities Definition • The activity of a solution, a, is related to the chemical potential, µ µ = µø + RTln(a) • Tend to associate activity with concentration (molality) • Its related but not equivalent • Replacement valid in very dilute solution (<10-3 mol/kg total ion concentration) • Given a solution whose ions behave ideally with a molality, bø of 1 mol/kg a = g(b/bø) • g is the activity coefficient • Depends on composition, concentration (molality) and temperature • g ® 1 and a ® b/bø as b ® 0 • From [1] µ = µø + RTln(b) + RTln(g) where b = b/bø µ = µideal + RTln(g) • µideal (= µø + RTln(b) ) is the chemical potential of an ideal dilute solution of molality bhere b = b/bø

7. Mean Activity coefficients • Consider a solution of two monovalenent cations (M+) and anions (X-) • Deviation from ideality contained in term RTln(g+ g-) • Define (g+ -) = (g+ g-)0.5 (geometric mean) • Reflects fact you can’t really separate deviation from non-ideality • (g+ -) is the mean activity coefficient for monovalent ions • Then, µ+ = µ+ideal + RTln(g+-)and µ- = µ-ideal + RTln(g+-) • Generally for compound MpXq that dissolves into p cations and q anions, by same process define mean activity coefficient as (g+ -) = (g+ g-)1/swhere s = p + q • The chemical potential, µi, becomes µi = µi ideal + RTln(g+-) • And G becomes G = p µ+ + q µ- • Again non-ideality is shared

8. Estimating (g+ -) - Debye-Hückel Theory • Coulomb interactions imply oppositely charged ions attract each other • In solutions, near an ion counter ions are found (ionic atmosphere) • Coulomb potential(f) drops as 1/r: fi = Zi/r Zia ionic charge • G (& µ) of ion lowered by electrostatic interactions • Since µi = µi ideal + RTln(g+-), lowering is associated with RTln(g+-) • ln(g+-) can be calculated by modeling these interactions • Debye-Hückel Limiting Law (proof Justification 10.2): • zi is charged number on ions • Must sum all ions in solution • Sign of charge included, e.g, zNa+ = +1; zSO42- = -2 • You’ll be using this in lab (Expt. 7, 9) • Works well at dilute solutions (b < 1 mmol/kg) • Extended Debye-Hückel Limiting Law (1 mmol/kg <b < 0.1 mol/kg): • B dimensionless const., adjustable empirical parameter • b>0.1 mol/kg (e.g. sea water): Model dependence of g of solvent on solute and use Gibbs-Duhem equation (SnJdµJ) = 0 to estimate g of solute

9. Estimating (g+ -) - Debye-Hückel Theory Limiting Law vs. Ionic Strength

10. Electrochemical Cells • Electrochemical cell - two electrodes in contact with an electrolyte • Electrolyte is an ionic conductor (solution, liquid, or solid) • Electrode compartment = electrode + electrolyte • If electrolytes are different compartments may be connect with salt bridge • Electrolyte solution in agar • Galvanic cell - an electrochemical cell that produces electricity • Electrolytic cell - an electrochemical cell in which a non-spontaneous reaction is driven by an external source of current

11. Types of Electrodes • Metal/metal ion (a) • Designation:M(s)|M+(aq) • Redox couple: M+ /M • Half reaction: M+(aq) + 1e-® M(s) • Gas (b) • Designation*: Pt(s)|X2(g)|X+(aq) or Pt(s)|X2(g)|X-(aq) • Redox couple: X+ /X2 or X2 / X- • Half reaction: X+(aq) + 1e-® 1/2X2(g) or 1/2X2(g) + 1e-® X-(aq) • Metal/insoluble salt(c) • Designation:M(s)|MX(s)|X-(aq) • Redox couple: MX /M,X- • Half reaction: MX(s) + 1e-® M(s) + X-(aq) • Redox (d) • Designation*: Pt(s)| M+(aq), M2+(aq) • Redox couple: M+/M2+ • Half reaction: M2+(aq) + 1e-® M+(aq) *Inert metal (Pt) source or sink of e-

12. Half-Reactions • Recall definition of redox: • Redox reaction is one involving transfer of electrons • OILRIG • Oxidation is loss of electrons • Reduction is gain of electrons • Reducing agent (reductant) is electron donor • Oxidizing agent (oxidant) is electron acceptor • Any redox reaction can be expressed as the difference of two reduction half reactions (sum of oxidation and reduction half reaction) Cu2+(aq) + 2e-® Cu(s) Zn2+(aq) + 2e-® Zn(s) (copper - zinc): Cu2+(aq) + Zn(s) ® Cu(s) + Zn2+(aq) • Redox couples are the reduced and oxidizing species in a redox reaction • Written Ox/Red for half reaction Ox + ne- ® Red • Example above: Cu2+/Cu; Zn2+/Zn

13. Half Reactions • Reaction quotient for half-reaction (Q) • Like reaction quotient for overall reaction (activity of product over activity reactant raised to appropriate power for stoichiometry) except electrons omitted • Cu2+(aq) + 2e-® Cu(s) Q = 1/aCu {Metal in standard state aM = 1} • O2(g) + 4H+(aq) + 4e- ® 2H2O (l) O2 assumed to be ideal gas • Redox couples in an electrochemical cell separated in space • Oxidation in one compartment, reduction in another • Oxidation compartment • Red1® Ox1 + ne- • Electrode at which this occurs is the cathode • Reduction compartment • Ox2 + ne- ® Red2 • Electrode at which this occurs is the anode

14. Electrolytic Cell Galvanic Cell Half Reactions • Galvanic cell (produces electricity) • Cathode at higher potential than anode • Species being reduced withdraws electrons from cathode giving it a relative (+) charge • Species being oxidized deposits electrons in anode giving it a relative (-) charge • Electrolytic cell(electricity supplied) • Oxidation still occurs at anode • Oxidation doesn’t occur spontaneously • Electrons come from the species in that compartment • Anode relatively positive to cathode • Cathode • Supply of electrons drives reduction

15. Daniel Cell Types of Cells • Commonest cell has single electrolyte in contact with both electrodes • Daniel cell - electrode compartments separated • Different electrolyte in each compartment • Electrolyte concentration cell - same electrolyte, different concentration • Electrode concentration cell - electrodes have different concentration • Gas cells at different pressures • Amalgams at different cocentrations • Additional potential difference across interface of two electrolytes - liquid junction potential • Present in electrolyte concentration cells • Due to differing mobility of ions of different sizes across interface • Can be reduced with salt bridge • Potential is then independent of concentration of electrolyte solution

16. Cell Reactions • Notation - overall cell reaction denoted by cell diagram • Phase boundary denoted by single vertical line (|) • Liquid junction denoted by single vertical dotted line (:) Daniel cell • Interface with no junction potential double verticla line (||) Salt bridge • Cell Reactions: reaction occurring in the cell with the right-hand as the cathode (spontaneous reduction) • To write cell reaction: 1. Write r.h.s. as reduction 2. Write l.h.s. as reduction 3. Subtract 2 from 1 • Cell Potential - electrical work that can be done through the transfer of electrons in a cell • Depends on the potential difference between electrodes • If overall reaction at equilibrium cell potential is zero • If reaction spontaneous, w is negative. At constant p and T we,max = DG • Work is maximum only if cell is operating reversibly • G is related to composition, work is reversible @ constant composition, i.e., no current • Under these conditions the potential difference is the electromotive force (emf) of the cell, E

17. Cell Reactions: Relation between E and DrG • For a reversible cell, the Gibbs energy of reaction, DrG, is proportional to the cell potential, DrG = -nFE where n is the number of electrons transferred and F is Faraday’s constant  Proof • When potential is high,DrG is negative and cell reaction is spontaneous • The emf (driving power of cell) is related to the extent of reaction

18. Nernst Equation: Relationship of emf to Activity • Recall that the Gibbs energy of reaction is related to composition: DrG = DrGø + RTlnQ where Q is reaction quotient(anJproducts/anJreactants) • Since DrG = -nFE E = -(DrGø/nF) - (RT/nF)lnQ • Define -(DrGø/nF) as Eø, the standard emf of the cell • So, E = Eø - (RT/nF)lnQ(Nernst Equation) • Eø is the emf when all reactants and products are in their standard states • aproducts = 1 and areactants = 1, so Q=1 and ln(Q) = 0 • Nernst Equation indicates a plot of E vs. ln Q will have • Slope = -(RT/nF) • Intercept = Eø You’ll see this in Lab #6 • At 25°C, RT/F = 0.0257 mV, Nernst equation becomes E = Eø - (0.0257 V/nF)lnQ

19. Nernst Equation: Concentration Cells M|M+(aq, L)|| M+(aq, R)|M • In above cell the only difference is the concentration of electrolyte in each cell • Left cell - molality is L; right cell - molality is R • Cell reaction: M+(aq, R) ® M+(aq,L) • nis 1 • Eø is 0 because when R=L the two compartments are identical and no driving force • Nernst Equation: E = Eø - (RT/nF)lnQ • In this case: E = - (RT/F)ln(aL/aR) = - (RT/F)ln(bL/bR) • If R>L, ln(bL/bR)<0, E>0, concentration will be lowered by reduction in right compartment • If L>R, ln(bL/bR)>0, E<0, concentration will be lowered by reduction in left compartment • This has biological application - nerve firing involves change in permeability of cell membrane to Na+ This changes nerve cell potential. (see text)

20. Nernst Equation: Equilibrium Cells • At equilibrium, by definition no work can be done • E = 0 • Concentrations are fixed by the equilibrium constant(K) • K=Q • Nernst Equation: E = Eø - (RT/nF)lnQ • In this case: E = 0 so Eø = (RT/nF)lnQ = (RT/nF)lnK • Rearranging, lnK = Eø/ (RT/nF) = nFEø/ RT or K = exp(nFEø/ RT) • This means cell potentials can be used to determine equilibrium constants

21. Standard Potentials • Since you can’t measure the potential of a single electrode, one pair has been assigned, by convention a potential of 0 • Standard hydrogen electrode (SHE): • Other potentials determined by constructing cells in which SHE is left hand electrode: • Silver Chloride|Silver Pt(s)|H2(g)|H+(aq)||Cl-(aq)|AgCl(s)|Ag Eø(AgCl, Ag, Cl-)=+0.22V Reaction: AgCl(s) + 1/2H2(g) ® Ag(s) + H+(aq) + Cl-(aq) • Because all potentials are relative to the hydrogen electrode, the reaction is listed without the contribution of the SHE, AgCl(s) ® Ag(s) + Cl-(aq) • Numerical factors • If std emf reaction is multiplied by numerical factor, DrG increases by that factor • Standard potential does not increase! • Recall Eø = -(DrGø/nF) If DrGø(new) = n x DrGø, n(new) = n x n Eø (new) = -(DrGø(new) / n(new) F) =-(n x DrGø /nn(new) F)= -(DrGø/nF) = Eø

22. Cell emf & Standard Potentials • Cell emf of any cell can be calculated from table of standard potentials • emf of cell is just difference in standard potential • Procedure: 1) Write new cell diagram 2) Eø = Eø(right) - Eø(left) • Sc(s)|Sc3+(aq)||Al3+(aq)|Al(s) • Al3+(aq) + 3e- ® Al(s) Eø = 1.69 V • Sc3+(aq) + 3e- ® Sc(s) Eø = -2.09 V • Al3+(aq) + Sc(s) ® Sc3+(aq) +Al • Eø = 1.69 V - (-2.09 V) = 3.78V • Recall: Eø = -(DrGø/nF) • If Eø > 0, DrGø < 0 and K>1 • Example above, at 25°C, K = exp(nFEø/ RT) = exp (3 x 3.78V/0.0257V) = exp(441) = 4.27 x 10191 K = [Sc3+]/[Al3+]

23. Measuring Standard Potentials Harned Cell • From Nernst Equation E = E°(AgCl/Ag,Cl-) - (RT/F)ln Q E = E°- (RT/F)ln ((aH+ aCl-)/(fH2/pØ)0.5 = E°- (RT/F){ln (aH+ aCl-) -ln(fH2/pØ)0.5 } Let fH2= pØ. E = E°- (RT/F)ln (aH+ aCl-) But a =bg± and bH+= bCl-) so E = E°- (RT/F)ln (b2g± ) = E°- (RT/F)ln (b2) - (RT/F)ln (g±2) = E°- (2RT/F)ln(b) - (2RT/F)ln(g±) • Debye-Hückel Limiting Law log(g±) a -I0.5solog(g±) a -b0.5orln(g±) a -b0.5 • E = E°- (2RT/F)ln(b) + Cb0.5 or E + (2RT/F)ln(b) = E° + Cb0.5 • This meansa plot of {E + (2RT/F)ln(b)} vs. b0.5 has Eø as intercept • Measuring activity coefficients • Since E = E°- (2RT/F)ln(b) - (2RT/F)ln(g±) , ln(g±) = {(Eø - E)/(2RT/F)}- ln(b) • Knowing Eø and measuring E at known molality allows you to calculate activity coefficient

24. More Reducing Applications of Standard Potential • Electrochemical series • Because Eø = -(DrGø/nF), if Eø > 0, DrGø < 0 • Since Eø = Eø2 - Eø1 , the reaction is spontaneous as written • Red1 has tendency to reduce Ox2 , if Eø1 <Eø2 • More directly in the electromotive series elements arranged such that low on the chart reduces high • Calcium reduces platinum (Eø = 4.05 V) • Platinum reduces gold (Eø = 0.51) • Tin reduced lead (Eø = 0.011 V) • Sodium reduces magnesium (Eø = 0.34 V) More Oxidzing

25. pH and pKa Glass Electrode • For hydrogen electrode (1/2 reaction above), Eø = 0 • If fH2= pø, Q = 1/aH+ and E = (RT/F) ln(aH+) • E = Eø - (RT/nF) lnQ • Converting ln to log (ln =2.303log), E = (RT/F) 2.303log(aH+) • Define pH=-logaH so E = -2.303(RT/F)pH • At 25°C, E= -59.16mVpH • Measurement • Direct method: hydrogen electrode + saturated calomel reference electrode (Hg2Cl2) • At 25°C, pH = (E + E(calomel))/ (-59.16mV ) • Indirect method: • Replace hydrogen electrode with glass electrode sensitive to hydrogen activity (but not permeable to H+ • E(glass) a pH, E(glass) = 0 when pH = 7 • pKA • Since we learned pH = pKa when concentration of acid and conjugate base are equal pKa can be measured directly from pH measurement • Ion-Selective electrodes - related to glass electrode except potentials sensitive to other species (see Box 10.2, p 278)

26. Electrochemical Cells and Thermodynamic Functions • Since the standard emf of a cell is related to the Gibbs energy, electrochemical measurements can be used to obtain other thermodynamic functions • Complementary to calorimetric measurements • Esp. useful for ions in solutions (aqueous, molten salts, etc.) • Starting point: DrGø = -nFEø and thermodynamic relationships we saw earlier • DrGø can be used to calculate Eø directly (or reverse) • Look at Example 10.4 - important caution about the relationship between numerical factors and Eø • Entropy (S) • Thermodynamic relationship: (∂G/∂T)p =-S DrGø = -nFEø At constant p, (dGø/dT)=-nF(dEø/dT) -DS=-nF(dEø/dT) or (dEø/dT) =DS/nF • Enthalpy (S) • Thermodynamic relationship: DrHø = DrGø + T DrSø DrHø = -nFEø + T(nF(dEø/dT)) DrHø =-nF(Eø -TdEø/dT)