Equilibrium and the Driving Forces for Electrochemical Processes

Home , Chemical thermodynamics, Electrode potential, Extent of reaction, Nernst equation

Part I Electrochemical Thermodynamics and Potentials: Equilibrium and the Driving Forces for Electrochemical Processes

Prof. Shannon Boettcher Department of Chemistry University of Oregon

, , aA + bB cC + dD ∆ = ∆ ∆ 𝑜𝑜 𝑜𝑜 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐺𝐺𝑟𝑟𝑟𝑟𝑟𝑟 𝐺𝐺𝑓𝑓 − 𝐺𝐺𝑓𝑓 What criteria do⇆ we use to determine if the reaction goes forward or • Gibbs free energy can be used to calculate the backwards? maximum reversible work that may be performed by system at a constant temperature and pressure.

• Gibbs energy is minimized when a system reaches chemical equilibrium at constant pressure and temperature. 2 Chemical Thermodynamics

If not at standard state, then: = ∆ + ln

𝑜𝑜 ap is the activity of product p 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 ∆𝐺𝐺 𝐺𝐺 𝑅𝑅𝑅𝑅 𝑄𝑄 ar is the activity of reactant r 𝑣𝑣𝑝𝑝 vi is the stoichiometric number of i = = 𝑐𝑐𝑝𝑝 𝑣𝑣𝑝𝑝 𝑝𝑝 𝑝𝑝 0 γi is the activity coefficient of i 𝑝𝑝 𝑝𝑝 ∏ γ 𝑝𝑝 ∏ 𝑎𝑎 𝑐𝑐 c is the concentration of i 𝑣𝑣𝑟𝑟 𝑣𝑣𝑟𝑟 i 𝑄𝑄 𝑟𝑟 𝑟𝑟 𝑟𝑟 0 ∏ 𝑎𝑎 𝑐𝑐 ci is the standard state concentration of i ∏𝑟𝑟 γ𝑟𝑟 0 𝑐𝑐𝑟𝑟 • Activity coefficients are “fudge” factors that all for the use of ideal thermodynamic equations with non-ideal solutions.

• For dilute solutions, γi goes to 1 3 Chemical Thermodynamics

Consider that the Gibbs energy can be written as a sum of enthalpy and entropy terms:

irreversible heat 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 maximum ∆reversible𝐺𝐺 ∆total𝐻𝐻 heat− 𝑇𝑇∆𝑆𝑆 released/gained by the work released/gained reaction by reaction

4 Electrochemical Thermodynamics

Consider the following thought experiment Zn + 2AgCl Zn2+ + 2Ag + 2Cl- do chemical reaction in do electrochemical reaction in do electrochemical reaction in B C in second calorimeter A calorimeter: calorimeter:⇆ calorimeter, resistor - e- e Q Zn + 2AgCl → r Zn2+ + 2Ag + 2Cl- Zn Ag/AgCl Zn Ag/AgCl Zn2+ - Cl 2+ Zn Qc Cl-

Qc = = -233 kJ/mol Qc = = -233 kJ/mol = lim = -43 kJ/mol ∆𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 ∆𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 all species at standard state, extent of reaction small = lim 𝑐𝑐 𝑇𝑇∆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑅𝑅→∞ 𝑄𝑄= -190 kJ/mol 𝑟𝑟𝑟𝑟𝑟𝑟 𝑅𝑅 ∆𝐺𝐺 𝑅𝑅→∞ 𝑄𝑄 Qc + Qr = = -233 kJ/mol Example from Bard and Faulkner 5 ∆𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 Electrochemical Thermodynamics

Gibbs energy change is the maximum reversible work the system can do. nF is the number of charges per mole of reaction,

= Ecell is the voltage… charge × voltage = energy

∆𝐺𝐺 −𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 = + ln 0 = + ln −𝑛𝑛𝐹𝐹𝐹𝐹 = −𝑛𝑛𝐹𝐹𝐸𝐸 ln𝑅𝑅𝑅𝑅 𝑄𝑄 0 0 𝑅𝑅𝑅𝑅log ∆𝐺𝐺 ∆𝐺𝐺 𝑅𝑅𝑅𝑅 𝑄𝑄 𝐸𝐸= 𝐸𝐸 − 𝑄𝑄 Remember Q includes ALL species in the 𝑛𝑛𝑛𝑛log balanced reaction, including ions, solids, 0 𝑅𝑅𝑅𝑅 𝑄𝑄 𝐸𝐸 𝐸𝐸 − etc. 2.302𝑛𝑛𝐹𝐹 𝑒𝑒 = log 0 . 𝑅𝑅𝑅𝑅 … and at 298.15 K, 𝐸𝐸 = 𝐸𝐸 − log 𝑄𝑄 𝑛𝑛𝐹𝐹 6 0 0 0592 V 𝐸𝐸 𝐸𝐸 − 𝑛𝑛 𝑄𝑄 Electrochemical Thermodynamics

– Consider a electrochemical O + ne R reaction: ⇌ activity of R = ln activity of O 0 𝑅𝑅𝑅𝑅 𝑎𝑎𝑅𝑅 𝐸𝐸 𝐸𝐸 − 𝑂𝑂 = 𝑛𝑛𝑛𝑛ln 𝑎𝑎 0 𝑅𝑅𝑅𝑅 γ𝑅𝑅𝐶𝐶𝑅𝑅 𝐸𝐸 𝐸𝐸 − 𝑂𝑂 𝑂𝑂 = ln𝑛𝑛𝑛𝑛 γ 𝐶𝐶 ln 0 𝑅𝑅𝑅𝑅 γ𝑅𝑅 𝑅𝑅𝑅𝑅 𝐶𝐶𝑅𝑅 𝐸𝐸 𝐸𝐸 − − 𝑛𝑛𝑛𝑛 γ𝑂𝑂 𝑛𝑛𝑛𝑛 𝐶𝐶𝑂𝑂

7 Electrochemical Thermodynamics

= ln ln 0 𝑅𝑅𝑅𝑅 γ𝑅𝑅 𝑅𝑅𝑅𝑅 𝐶𝐶𝑅𝑅 𝐸𝐸 𝐸𝐸 − E0' − O + ne– R 𝑛𝑛𝑛𝑛 γ𝑂𝑂 𝑛𝑛𝑛𝑛 𝐶𝐶𝑂𝑂 What happened⇌ to the = ln “electrons” in our Nernst equation? 0′ 𝑅𝑅𝑅𝑅 𝐶𝐶𝑅𝑅 𝐸𝐸 𝐸𝐸 − the formal𝑛𝑛 𝑛𝑛potentia𝐶𝐶l𝑂𝑂… this depends on the identity and concentration of all species present in solution

8 Nernst Equation and Reference Electrodes

In this form we actually mean: ( . ) = ( . ) ln ′ 0 𝑅𝑅 Which is (when the reference is the hydrogen electrode):𝑅𝑅𝑅𝑅 𝐶𝐶 𝐸𝐸 𝑣𝑣𝑣𝑣 𝑟𝑟𝑟𝑟𝑟𝑟 𝐸𝐸 𝑣𝑣𝑣𝑣 𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑂𝑂 n+ + 𝑛𝑛𝑛𝑛 𝐶𝐶 O + (n/2)H2 R + nH

⇌ = ln at standard state, = =1. /𝑛𝑛 + + 𝑜𝑜 H 𝑅𝑅 H 𝐻𝐻2 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅 𝑎𝑎 𝑎𝑎 𝑎𝑎 𝑎𝑎 𝐸𝐸 𝐸𝐸 − 2 𝑛𝑛 2 𝑛𝑛𝑛𝑛 𝑎𝑎𝐻𝐻 𝑎𝑎𝑂𝑂 Key point: anytime you write the Nernst equation for a “half” reaction you are in fact using a short hand to represent the full reaction including the reference electrode/reaction 9 Equilibrium and the Chemical Potential

Gibbs energy is minimized when a system reaches chemical aA + bB cC + dD equilibrium at constant pressure and temperature.

How does the Gibbs energy change with amount of a substance? ⇆ mixture or pure figs. from substance solution Mark Lonergan

(amount in mol) (amount in mol)

= = 𝐺𝐺 𝜕𝜕𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚 10 𝜇𝜇 𝜇𝜇𝑖𝑖 𝑛𝑛 𝜕𝜕𝑛𝑛 Chemical potential

= j is the species , , α is the phase (e.g. metal, electrolyte, ionomer, etc.) α these are held constant 𝜇𝜇𝑗𝑗 𝜕𝜕 𝐺𝐺⁄𝜕𝜕 𝑛𝑛𝑗𝑗 𝑇𝑇 𝑃𝑃 𝑛𝑛𝑖𝑖≠𝑗𝑗 = + ln

α o α chemical potential increases with concentration… 𝜇𝜇𝑗𝑗 𝜇𝜇𝑗𝑗 𝑅𝑅𝑅𝑅 𝑎𝑎𝑗𝑗 (arbitrary) standard activity term state reference

although we call it a “potential” units are energy, J

11 Chemical Reactions and μ

Q 12 Electrochemical potential

When we quantify for a charged particle, it is 𝑗𝑗 useful to define a new𝜕𝜕 𝐺𝐺 quantity,⁄𝜕𝜕 𝑛𝑛 , that partitions the “quantum mechanical” internal free energy𝜇𝜇̅ and the “classical” electrostatic energy

fig. from Mark Lonergan  φ =−⋅∫ Ed path in electrochemistry, electrochemical potential determines equilibrium (other sources of free energy could be included, e.g. gravitational, but these are not generally important) 13 Properties of the electrochemical potential

14 Other Potentials in Electrochemistry

term symbol unit brief definition significance / example of use defines criteria for equilibrium; differences in drive the electrochemical J/mol partial molar Gibbs free energy of a given species j in phase transport, transfer, and reactivity of both charged andα uncharged potential 𝑗𝑗 𝛂𝛂 species 𝜇𝜇̅ 𝛍𝛍�𝐣𝐣 α partial molar free energy of a given species j in phase neglecting differences in describe driving force for reactions between chemical potential J/mol electrostatic contributionsa uncharged speciesα and the direction of diffusive transport 𝛂𝛂 α 𝜇𝜇𝑗𝑗 𝛍𝛍𝐣𝐣 electric work needed to move a test charge to a specific point in space defines direction of electron transport in metals; gradient gives electric potential φ V from a reference point (often at infinite distance) divided by the value of electric field; used to calculate electric potential energy the charge free-energy change divided by the electron charge associated with moving an electron (and any associated ions/solvent indicates oxidizing or reducing power of an electrode; related to electrode potential V movement/rearrangement) from a reference state (often a reference the Fermi level of electrons in electrode 𝐄𝐄𝐰𝐰𝐰𝐰 electrode) to the working electrode free-energy change divided by the electron charge associated with indicates oxidizing or reducing power of electrons involved in moving an electron (and any associated ions/solvent electrochemical redox equilibria; related to “Fermi level” of the solution potential V movement/rearrangement) from a reference state (often a reference electrons in solution and equivalent to the solution reduction 𝐄𝐄𝐬𝐬𝐬𝐬𝐬𝐬 electrode) into the bulk of a solution via a redox reaction potential generally, the difference between the applied electrode potential and F gives the heat released, above that required by overpotential V the electrode potential when in equilibrium with the target thermodynamics, per mole of electrons to drive an electrochemical reaction electrochemical𝜂𝜂 � process at a given rate; F = 96485 C·mol-1 𝛈𝛈 a that is, no ‘long-range’ electrostatic interactions due to uncompensated charge, as would be described by the Poisson equation in classical electrostatics. The electrostatic terms that describe electron-nucleus and electron-electron interactions and dictate Coulombic potentials in the Schrödinger equation are included.

15 What does a voltmeter measure?

Voltage, but what is voltage?  φ =−⋅ ∫ Ed ? Not in this case α β path

If the resistor/wire is the same material, then: = = +α β = ∆𝜇𝜇̅𝑒𝑒 𝜇𝜇̅𝑒𝑒 − 𝜇𝜇̅𝑒𝑒 α α β β 𝑒𝑒 𝑒𝑒 𝜇𝜇 − 𝐹𝐹𝜙𝜙 − 𝜇𝜇 𝐹𝐹𝜙𝜙 −𝐹𝐹∆𝜙𝜙 16 Changes in total free energy drive transport

= 𝑗𝑗 𝑗𝑗 𝒋𝒋 𝐶𝐶 𝐷𝐷 𝑗𝑗 -2𝐉𝐉-1 − 𝛻𝛻𝜇𝜇̅ flux (mol cm s ) 𝑅𝑅𝑅𝑅 gradient in electrochemical potential

= + = + ln α α α α o α j 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 The electrochemical potential𝜇𝜇̅ is𝜇𝜇 usually𝑧𝑧 the𝐹𝐹 𝜙𝜙proper measure of free𝜇𝜇 energy𝜇𝜇 in electrochemical𝑅𝑅𝑅𝑅 𝑎𝑎 systems, though other terms might be added in special cases

u is mobility i 17 Changes in total free energy drive transport

In one dimension, the gradient of leads to the drift diffusion equation.

𝜇𝜇̅𝑗𝑗

ui is mobility

18 Consider the electrode potential

• units of V • given by the difference in , per charge, in the working electrode, relative to in a second electrode 𝑒𝑒 • second electrode is usually𝜇𝜇 ̅reversible electrochemical half reaction (i.e. a 𝑒𝑒 𝜇𝜇reference̅ electrode): ( ) (vs. ) = we re − 𝜇𝜇̅𝑒𝑒 − 𝜇𝜇̅𝑒𝑒 𝐸𝐸we 𝐸𝐸re • and are each themselves defined relative𝐹𝐹 to an arbitrary reference (that cancel in the difference). The cell voltage is usually written = 𝐸𝐸we 𝐸𝐸reor ( ) 𝐸𝐸cell 𝐸𝐸we − 𝐸𝐸re 𝐸𝐸cathode − 𝐸𝐸anode

19 Summary of Key Points

• Measurements of “potential differences” are necessarily of the total free-energy difference. Decomposing into differences in activity, electric potential, and other terms requires a model and assumptions. • Transport of any species is governed by the spatial gradient in the electrochemical potential. • At equilibrium, the electrochemical potential of any given species must be the same throughout the system • For any chemical reaction, the sum of the electrochemical potentials of the reactants must equal those of the products. Processes with very slow kinetics are typically ignored. • The use of the word ‘potential’ alone should be avoided; the type of should be clear.

20 Part II

Electrochemical Thermodynamics and Potentials: Applications

Prof. Shannon Boettcher Department of Chemistry University of Oregon

21 Review of Key Points

22 Equilibration at a metal/redox-electrolyte solution interface consider O + R

𝑛𝑛𝑒𝑒𝑚𝑚 ⇌

We define even though How do these equilibrate? = there are “practically”s no s s 𝜇𝜇̅𝑒𝑒 m 𝑅𝑅 𝑂𝑂 s electrons in the solution 𝑒𝑒 𝜇𝜇̅ − 𝜇𝜇̅ 𝑒𝑒 shows no surface charge𝜇𝜇̅ initially, in reality E≡σ=0 is𝜇𝜇 the̅ applied potential where this surface charge is balanced by electronic charge𝑛𝑛 such that the net charge is zero. 23 The solution potential

expand via definition of electrochemical potential

= = + ln + ln s s s 𝑜𝑜 𝑜𝑜 −𝜇𝜇�𝑒𝑒 𝜇𝜇�𝑅𝑅− 𝜇𝜇�𝑂𝑂 𝜇𝜇𝑅𝑅 𝑅𝑅𝑅𝑅 s 𝑧𝑧𝑅𝑅 s 𝜇𝜇𝑂𝑂 𝑅𝑅𝑅𝑅 s 𝑧𝑧𝑂𝑂 s 𝐹𝐹 − 𝑛𝑛𝑛𝑛 − 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑎𝑎𝑅𝑅 𝑛𝑛 𝜙𝜙 − 𝑛𝑛𝑛𝑛 − 𝑛𝑛𝑛𝑛 𝑎𝑎𝑂𝑂 − 𝑛𝑛 𝜙𝜙 = / ln = s 𝑜𝑜 𝑅𝑅 s 𝑂𝑂 𝑅𝑅 𝑅𝑅𝑅𝑅 𝑎𝑎 s sol 𝐸𝐸 − �𝑎𝑎𝑂𝑂 − 𝜙𝜙 𝐸𝐸 Nernst equation𝑛𝑛𝑛𝑛 need to account for different electrostatic potentials!

the addition of the term that depends on the electric potential reference states and cancels when measured versus a reference electrode− at𝜙𝜙 the same s 24 𝜙𝜙 Equilibration at a metal/redox-electrolyte solution interface • initial difference in drives charge transfer across the interface, leading to an interfacial e electric potential drop𝜇𝜇̅ that affects until it equals m 𝑒𝑒 s 𝜇𝜇̅ 𝑒𝑒 • amount𝜇𝜇 ̅of charge transferred depends on the capacitance of the electrode • small compared to the number of electrons in the metal and redox species in the electrolyte (so that the bulk activity and thus μ for all the species is practically unchanged)

• concentration of the compensating ions given by the Poisson-Boltzmann distribution

• for all species are constant with position

𝜇𝜇̅𝑗𝑗 25 Membrane (Donnan) potentials consider two solutions with different What happens if we connect them with a cation-selective membrane like Nafion? concentrations of KCl = + ln + = + ln + = 𝛽𝛽 𝛽𝛽 α+ o+ 𝛼𝛼+ 𝛼𝛼 o+ + 𝛽𝛽 + 𝜇𝜇̅K 𝜇𝜇K 𝑅𝑅𝑅𝑅 𝑎𝑎K 𝐹𝐹𝜙𝜙 𝜇𝜇K 𝑅𝑅𝑅𝑅 𝑎𝑎K 𝐹𝐹𝜙𝜙 𝜇𝜇̅K = ln 𝛽𝛽 + 𝛼𝛼 𝛽𝛽 𝑅𝑅𝑅𝑅 𝑎𝑎K 𝜙𝜙 − 𝜙𝜙 � 𝛼𝛼+ 𝐹𝐹 𝑎𝑎K

membrane block Cl- transport .

α − β − 26 𝜇𝜇̅Cl ≠ 𝜇𝜇̅Cl Measuring membrane potentials

How do we measure electrostatic potential changes in electrochemical cells?

With a voltmeter? But a voltmeter doesn’t measure electrostatic potential!? Consider what you measure with two reference electrodes:

Ag/AgCl Ag/AgCl 𝛼𝛼 𝛽𝛽 𝜙𝜙 𝜙𝜙

membrane

, , , , = + – and

Ag− Ag s − AgCl s −re1 s −re2 s−re1 s−re2 Cl Cl Cl e e 𝜇𝜇̅ 𝑒𝑒 𝜇𝜇̅ AgThe two𝜇𝜇̅ reference𝜇𝜇̅ AgCl electrodes make𝜇𝜇̅ a “battery”≠ 𝜇𝜇 ̅and are not a𝜇𝜇 equilibrium.̅ ≠ 𝜇𝜇̅ 27 Fuel cells and batteries

first consider “open circuit” negligible net current is flowing through the external circuit (e.g. during measurement with a high-impedance voltmeter) 28 Fuel cells and batteries: open circuit

No. 4H (Nafion) + 4 Pt, 2H (g) Anode: + − 1 2 , = 𝑒𝑒 𝑎𝑎0 k⇌J m ol Why? 2 g Open circuit: If zero net current is Pt−a s+ 𝜇𝜇̅ 𝑒𝑒 𝜇𝜇̅ H2 − 𝜇𝜇̅H ≈ ⁄ flowing does that mean at the O (g) + 4H (Nafion) + 4 (Pt, ) 2H O Pt anode and cathode are the Cathode: e + − same? 𝜇𝜇̅ 2 1 1 𝑒𝑒 𝑐𝑐 ⇌ 2 , = 119 k J m ol 2 4 Pt−c s g s+ out of equilibrium - O and H 𝑒𝑒 H2O O2 H 2 2 𝜇𝜇̅ ,𝜇𝜇 − , 𝜇𝜇̅ − 𝜇𝜇̅ ≈ − ⁄ cannot mix across the membrane = = , and react; electrons cannot Pt−c Pt−a rxn 𝑒𝑒 𝑒𝑒 ∆𝐺𝐺 cell oc exchange 𝜇𝜇̅ − 𝜇𝜇̅ Ecell,oc = 1.23 V −𝐹𝐹𝐸𝐸 𝑛𝑛 29 Fuel cells and batteries: during operation

= + ln +

Nafion+ o+ Nafion+ Nafion H K H under current flow there must be gradients 𝜇𝜇̅ 𝜇𝜇 𝑅𝑅𝑅𝑅 𝑎𝑎 𝐹𝐹𝜙𝜙 What drives the flow of H+? in of all species that transport – electrons, ions, water. 𝜇𝜇� Nafion+ Nafion 𝛻𝛻𝜇𝜇̅H ≈ 𝐹𝐹𝛻𝛻𝛻𝛻 What drives the interfacial electrochemical reactions for ORR and HOR? in terms of electrostatic potentials: = = (for a given reaction) 𝜂𝜂 ∆𝜙𝜙 − ∆𝜙𝜙eq 𝜂𝜂 𝐸𝐸app − 𝐸𝐸rev Electroosmotic effects and concentrated electrolytes

Transport of solvent and electrolyte ions are coupled. Electric potential leads to solvent movement too. ion flux carrying diffusion of water water • κ: ionic conductivity (S/m) • ξ: electroosmotic drag coefficient (unitless) Flux of water:

• φ2: electric potential in electrolyte (V) • α: dimensionless diffusion coefficient (unitless)

• μ0: chemical potential of water (J/mol) Proton current: Ohm’s Law

Gradient in water chemical potential can drive ion transport

see Newman for complete treatment 31 Part III Electrochemical Thermodynamics and Potentials: Double Layer Structure and Adsorption

Prof. Shannon Boettcher Department of Chemistry University of Oregon

32 Double Layer Structure - Basics IHP OHP • The inner Helmholtz plane (IHP) passes through the center of the specifically adsorbed ions. • typically anions, that can shed hydration sphere e.g. sulfate.

• The outer Helmholtz plane (OHP) passes through the center of solvated ions at the distance of their closest approach.

• A layer of orientated “low-entropy” water covers the surface. C d C d • if i = εr,iε0/ i, where i and i are the capacitance and - thickness of layer i, respectively, then εr,I ~ 6 at metals.

• Double layer structure is critical in influencing electrode kinetics, as we will see later.

fig. from S. Ardo 33 σM = -σS Interface Thermodynamics

Consider a ideally polarizable electrode (no faradaic charge transfer)

The interface is a “phase” with finite thickness where define excess concentrations concentrations differ from bulk values.

total differential of electrochemical Gibbs energy of reference phases (no interface)

total differential of Gibbs energy of interface region

change in G with interface area A = surface tension

at equilibrium must be the same everywhere for any species how much free energy it takes to 𝜇𝜇� create new interface

34 Following Bard and Faulkner Ch. 13 Gibbs adsorption isotherm

Gibbs adsorption isotherm differential “excess” free energy of interface

The Euler theorem allows one to define linear homogenous function in terms of derivatives and variables. surface excess concentration

total differential compare to above, then

35 Gibbs adsorption isotherm

Now following Schmickler and Santos for a more general form: Define a reference phase, usually the solvent, and remove from sum in Gibbs-Duhem eqn Gibbs absorption isotherm

absolute surface excess of species

in solution bulk, Gibbs-Duhem Define “relative surface excess” with respect to eqn. (at constant T and P) holds (http://staff.um.edu.mt/jgri1/teaching/che2372/notes/05/02/01/gibbs_duhem.html) the bulk reference phase (i.e. solvent, 0):

places a compositional constraint upon any changes in the electrochemical potential in a mixture

we cannot measure absolute surface excess, only relative to the solvent reference

36 Gibbs adsorption isotherm rewrite Gibbs absorption isotherm positive charge neutral metal in electrode

negative charge everything in the solution in electrode

Expand electrochemical potentials:

(A)

balanced by excess charge in the electrolyte

surface charge density; eo here is fundamental charge 37 Gibbs adsorption isotherm

Mz+ + e- M in eq. with constant φ in metal decompose electrochemical potential of solution species ⇆ (B) (C)

use (A), (B) and (C) to simplify expression for differential surface tension:

This is the electrocapillary equation.

from the electrostatic uncharged solution energy terms species except solvent

38 Gibbs adsorption isotherm Divergence due to increase of the work function by anion adsorption.

Notice there is a maximum in surface energy with potential. Why? can in principle measure surface Lippmann Equation excess with conc. dependence • At the potential of zero charge (PZC), σ = 0 and there is no net charge on the metal. • Moving from the PZC, charge accumulates and tends to repel, counteracting surface tension Interfacial tension of a mercury electrode at 0.1 M electrolyte.

39 Interfacial differential capacitance

The differential capacitance of an interface is given by the second derivative of the interface tension, because:

i.e. the capacitance measures how much = charge is stored as the electrode potential is changed by modulating ∆φ 𝑑𝑑𝜎𝜎 𝐶𝐶 𝑑𝑑𝑑𝑑 This is extremely useful, because we can measure interfacial capacitance directly using impedance spectroscopy

40 Interfacial differential capacitance

higher work function = more electronegative = more positive PZC

PZC

S. Trasatti, Advances in Electrochemistry and Electrochemical Engineering

D. C. Grahame, Chem. Rev., 41, 441 (1947) • A minimum in Cd exists at the pzc. • Cd increases with salt concentration at all potentials, and the "dip" near the pzc disappears. 41 42 Models of the Electrical Double Layer

• Helmholtz - + - + + surface charge on a parallel - plate capacitor - + - + - + - + - + - + predicts constant Cd, which is not + what is observed experimentally. - - + d 43 Gouy-Chapman theory and Boltzmann factors

lamina can be regarded as energy states with equivalent degeneracies – concentrations related by Boltzmann factor

bulk ion “bulk” electrostatic concentration potential

charge density

e is fundamental charge

from Bard and Faulkner 44 Poisson-Boltzmann Equation

Poisson equation: integral of charge density is electric field, integral of electric field is electric potential

apply for 1:1 electrolyte; note: e.g. NaCl or

CaSO4 thus: integrate

from Bard and Faulkner 45 Gouy-Chapman potential distribution

if: separate and integrate

then: resulting in and:

κ = 1/Ld

Ld = Debye screening length

from Bard and Faulkner 46 Capacitance from Gouy-Chapman

Any problems with this?

How is this different than experiment?

from Bard and Faulkner 47 Stern’s Modification Helmholtz parallel plate capacitor

• Ions in Gouy-Chapman are point charges with no restriction on concentration • unrealistic at high ion density • With no physical size, + no distance of closest Gouy-Chapman / Poisson Boltzmann ion approach • no limit to rise in differential capacitance

non-specific adsorption

from Bard and Faulkner 48 Capacitance in the Gouy-Chapman-Stern (GCS) model

Closer to experimental data. What else could be happening?

from Bard and Faulkner 49 Specific adsorption and the PZC

PZC depends on concentration

no specific adsorption Specifically - specific absorbed Br must adsorption be compensated by K+

> 1

from Bard and Faulkner 50 Adsorption phenomena and isotherms

Langmuir Isotherm

W is number of ways of selecting M out of N sites

Helmholtz free energy adsorption entropy (constant V) energy ideal solution

at equilibrium (electro)chemical potentials in adsorbed layer and electrolyte must be the same

M are the filled sites N are the total sites

from Schmickler and Santos 51 Adsorption phenomena and isotherms

Langmuir adsorption ignores interactions between adsorbates. Frumpkin isotherms Frumpkin:

repel positive if repel, negative if attract attract

Langmuir (no interaction)

from Schmickler and Santos 52 Why are these concepts important?

• Changes in ɸ from equilibrium are • How close do ions responsible for affecting electrochemical approach surface? • Is ion transfer or reaction thermodynamics∇ that drive ion and electron transfer electron transfer rate limiting? • In a mean field picture, the location of • What fraction of the electroactive species in the double affects applied potential + affects the transition the driving force for charge transfer M state energy? • New ideas in electrocatalysis involve e- situations where mean-field approach breaks down