Potentials in Electrochemistry

Viewpoint

http://pubs.acs.org/journal/aelccp Potentially Confusing: Potentials in Electrochemistry

Cite This: ACS Energy Lett. 2021, 6, 261−266 Read Online

ACCESS Metrics & More Article Recommendations

potential quantifies the capacity of a system to do causes transport. The flux (J) of species j (usually in mol· − − work. A simple example is from mechanics: by lifting cm 2·s 1) is given by: a weight, its potential energy increases. When the A CD weight is dropped, that potential energy is converted into jj J =− ∇μ kinetic energy. Applying the concept of potential to j j̅ − RT (2) electrochemical systems can be surprisingly confusing.1 4 “ ” Table 1 shows some potentials used in electrochemistry and The termsi C andy D are the concentration and diffusion 5 j j z j − their different units. coefficientsj forz the species j (e.g., in units of mol·cm 3 and 2· −1 k { ∇μ To start, let us consider a voltage measured with a cm s , respectively), and ̅j is the spatial gradient in the voltmeter for a given system. Analog voltmeters operate by electrochemical potentialthe underlying driving force for passing a small current through a calibrated resistor and wire transport. R is the gas constant, and T is temperature. coil, generating a small magnetic field, and thus deflecting a Chemical reactions can occur throughout the system and needle attached to a fixed magnet. The current passed progress in their thermodynamically favorable direction to μ through the resistor, which causes needle deflection, is ∂ j̅ minimize free energy. When ∇μ̅= 0 and = 0 for all proportional to the voltage. This voltage, however, is not the j ∂t electric potential difference of the system. Electric potential ϕ species j and everywhere in a system, the system is at is the line integral of the electric field E⃗along a path from a electrochemical equilibrium (i.e., the system cannot lower its  reference point (often at infinite distance from the system) to total free energy via the net movement of any species  a given position: electrons, ions, or molecules to another part of the system or via a chemical reaction). Equation 2 governs transport by drift/migration (movement of charged particles in the ϕ =−∫ E⃗·dS fi ff path (1) presence of an electric eld), di usion (movement due to a concentration gradient), and less-common processes (for ffi Electric fields are generated by (uncompensated) electric example, due to spatially dependent activity coe cients). “ ff ” charge. Thus, q·ϕ is a measure only of the electric Equation 2 also yields the common drift-di usion equation .

Downloaded via UNIV OF OREGON on February 2, 2021 at 19:06:16 (UTC). The electrochemical potential for species j can be component of the work needed to move a test charge q 7 from a reference position to the position of interest in the conceptually decomposed: system. If the charge moves through different phases, i.e., μ αα=+μϕzF α See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. across interfaces of different materials, the electric work does jj̅ j (3) not fully account for the total work. μα Consider two metals with different work functions (i.e., the The term j is the chemical potential of j, relative to a ϕα energy required to remove an electron from the metal to reference state, and zjF is the electrostatic energy per mole α vacuum), e.g., Au and Ti. If Ti and Au are put in contact, of j in phase , relative to a reference state. The term zj is the signed charge number of j (e.g., +1, −1, +2, −2), and F is the initially electrons transfer from Ti to Au, generating an − fi Faraday constant (96 485 C·mol 1). The chemical-potential- interfacial electric eld and thus an electric potential μα ff term j is the partial molar Gibbs free energy, ignoring di erence across the interface. Yet if you connect a voltmeter fi with one lead to Ti and the other to Au, you measure 0 V. A electrostatic contributions. It is de ned as the derivative (at constant temperature, T, and pressure, P, and concentration voltmeter does not only measure the electric potential difference between the two points in the system. A voltmeter measures the difference in the electrochemical Received: November 23, 2020 μ ff Accepted: December 10, 2020 potential of electrons ( ̅e), because it is this di erence that drives the current through the voltmeter needed to make the Published: December 24, 2020 measurement.6 The electrochemical potential for electrons is also the Fermi level. In general, it is the gradient (i.e., derivative with respect to position, usually per cm) in μ̅that

− of all other species, ni≠j) of the Gibbs free energy with eld;

ﬁ respect to the number of particles (in the phase in question):

= 96485 α

F μ =∂(/)Gn ∂ j intjTPn , , i≠j (4)

The term Gint isthechemicalfreeenergyneglecting “ ” ﬀ contributions from long-range electrostatic e ects, and nj is the number of moles of species j. In a mixture, the chemical potential is determined as a function of the activity, drive the transport, transfer, α α j ̅

μ aj , and referenced to the chemical potential at standard state, μo of the electrons in solution and

tic terms that describe electron j . ” μα =+μo α jjRTln aj (5) erences in ﬀ α γα α o,α γα ﬃ cance/example of use The term a = C /C , where is the activity coe cient,

Fermi level j j j j j ﬁ “ a factor that allows the use of ideal thermodynamic equations

usive transport α o,α o,α ﬀ with concentrations Cj and Cj as inputs. The term Cj is a reference concentration, usually taken to be 1 M for soluble μo

describe driving force for reactions between uncharged species species. The term j is determined relative to a reference

α j μo  ° μ state (e.g., H+ 0at25 C in water). The activity term takes into account that the chemical potential increases with 1 − concentration. In physics, the electrochemical potential is

mol fi nes direction of electron transport in metals; gradient gives electric nes criteria for equilibrium; di · erences in often not de ned explicitly, instead the electrostatic energy fi fi ff F gives the heat released, above that required by thermodynamics, per mole used to calculate electric potential energy and reactivity of both charged and uncharged species and the direction of di of electrons in electrode redox equilibria; related to equivalent to the solution reduction potential of electrons to drive an electrochemical process at a given rate; C α · ϕ “ ” de di indicates oxidizing or reducing power of an electrode; related to the Fermiindicates level oxidizing or reducing power of electrons involved in electrochemical de η zjF is included directly in the chemical potential , and an intrinsic or internal chemical potential without long-range a electrostatic effects is defined as a new quantity.7 The fact that gradients in electrochemical potential, not electric potential, drive charge flow explains why a voltmeter cannot measure the electric potential between two metals, i.e., μTi μAu ϕTi ≠ ϕAu Δϕ above ̅e = ̅e but . A voltmeter measures between two points in a system only if they have a common chemical potential. Consider a metal wire acting as a resistor. The concentration of electrons is large, ∼1022 cm−3, and the density of electronic states at the Fermi level is high. Thus,

α even when current flows the chemical potential of electrons is dinger equation are included. ̈ ff neglecting electrostatic contributions invariant across the length of the wire. Thus, for two di erent c point in space from a reference point α α β fi locations and in the same metal in phase j

nition signi αβα αβ β ﬁ Δμμee̅ = ̅ − μ e̅ =− μ eFFF ϕ −+ μ e ϕ =−Δ ϕ in phase

j (6) −Δμ A voltmeter thus measures e̅ =Δϕ between two points in F a common metallic phase. By definition, electrons flow to μ regions of more-negative ̅e (to lower their free energy), but to more-positive ϕ. Under this condition, eq 2 simplifies to a form of Ohm’s law. The foregoing discussion emphasizes that all measurements of potential are necessarily measurements of erence between the applied electrode potential and the electrode potential when

ﬀ electrochemical potential and any inferences about electric nite distance) divided by the value of the charge

ﬁ potential require one or more assumptions. Now let us consider the term electrode potential, Ewe,in the context of these other potentials (the “we” refers to the working electrode). The electrode potential, with units of V, (often at in associated ion/solvent movement/rearrangement)electrode) from to a the reference working state electrode (often aassociated reference ion/solvent movement/rearrangement)electrode) from into a the reference bulk state of (often a a solution reference via a redox reaction in equilibrium with the target electrochemical reaction ﬀ μ is given by the di erence in ̅e, per charge, in the working μ electrode, relative to ̅e in a second electrode that is typically set via a reversible electrochemical half reaction (i.e.,a V electric work needed to move a test charge to a speci V generally, the di J/mol partial molar GibbsJ/mol free energy of partial a molar given free species energy of a given species V free-energyV change divided by the electron charge associated with free-energy moving change an divided electron by (and the any electron charge associated with moving an electron (and any reference electrode, indicated by "re"): electrostatic interactions due to uncompensated charge, as would be described by the Poisson equation in classical electrostatics. The electrosta ”

electron interactions and dictate Coulombic potentials in the Schro we re we sol α j α j ̅ −()μμ− E μ ϕ E η μ − ̅ ̅ EE(vs ) = e e we re F (7) long-range

“ The sign convention is again consistent with electrons ﬂ spontaneously owing toward more-positive Ewe. Practically,

term symbol unit brief de one measures the voltage diﬀerence between the working and reference electrodes. Because E and E are each themselves potential potential we re That is, no electrode chemical potential electric potential solution potential overpotential electrochemical

nucleus and electron ﬁ Table 1. Potentials in Electrochemistry a de ned relative to an arbitrary reference (that cancel in the

262 https://dx.doi.org/10.1021/acsenergylett.0c02443 ACS Energy Lett. 2021, 6, 261−266 ACS Energy Letters http://pubs.acs.org/journal/aelccp Viewpoint

Figure 1. Electrochemical equilibrium at an electrode-electrolyte interface. (a) Initially, the electrochemical potential of the electrons is different in the metal electrode and solution. (b) Once the two phases are brought into contact, electrons are driven by the gradient in electrochemical potential across the interface. This process leads to the addition of positive charge on the metal surface that is balanced by negative ions in the solution (the so-called double layer). The ionic charge in the solution matches the charge on the metal due to overall charge neutrality. This is analogous to processes that occur to form electric potential differences (Δϕ) as space-charge regions at semiconductor interfaces and as Donnan potentials at membrane-electrolyte interfaces. The absolute charge of the metal surface (σm) will E fi also depend on the potential of zero charge ( σ=0) of the speci c electrode material in the given electrolyte. Even in the absence of electron transfer across the interface, most electrode surfaces carry surface charge due to adsorption of electrolyte ions or ionization of σm E σm surface-functional groups that induces changes in . σ=0 is the applied potential where = 0. The equilibration process described in fi σm σm E this gure leads to a change in , although the ultimate sign and magnitude of depends on σ=0 and the redox species. ff − fi di erence), the cell voltage is usually written Ecell = Ewe Ere. Typically, the standard hydrogen electrode (SHE) is de ned The open-circuit (oc) cell voltage is related to the Gibbs free- as the reference state. A Pt electrode in equilibrium with H2 + Pt s s energy change of the overall electrochemical reaction μ μ − μ + and H (at unit activity) thus has ̅e = 0 because ̅H2 ̅H occurring at the working and reference electrodes: = 0 under this (arbitrary) definition. Since electrochemical potentials and reduction potentials −ΔG E = rxn are both referenced to a standard state, we can explore the cell,oc (8) fi nF meaning of Esol. From the de nition of the electrochemical potential: Electrode potentials can also be measured during sponta- s s s o neous electrochemical reactions (like corrosion) or can be −μμμe̅ R̅ − O̅ μR RT sszR controlled by a potentiostat in a conventional three-electrode =− =− +ln aR + ϕ μ FnFnFnF n experiment. While eq 7 focuses on ̅e, it is important to note μ μ o that these ̅e are coupled to ̅j of other species, for example μ RT z RT s 8 O i ssy O i o aR s −−j lnaO z −ϕj =EO/R − ln a ions, via chemical reactions. nF j nF z n j nF O Now let us consider three simple electrochemical systems j z j α k { k where the conceptual framework of partitioning μ̅ into −=ϕs E j sol zy (11) chemical and electric components is useful. z z Example 1: The electrochemical double layer and the The solution potential, Esol (in V), is thus given by the { ϕs solution potential/Fermi level. Consider a metal electrode Nernst equation (with the addition of the - term that in equilibrium with a redox couple (at say 10 mM) with both depends on the electric potential reference state and cancels ϕs the oxidized (O) and reduced (R) form present dissolved in when measured versus a reference electrode at the same ) an inert electrolyte (Figure 1). Intuitively, we know that the and directly related to the solution Fermi level. As a caveat, it “ ” μ is important to remember that, practically, E can be difficult electron energy in the solution must equilibrate with ̅e sol (the Fermi level) in the electrode. But how precisely does to measure because of slow electrochemical kinetics. As an this occur? We can write the electrochemical reaction as example, consider an electrolyte containing water and

dissolved O2; EO2/H2O cannot be measured with a metal OR+ ne F (9) electrode because other undefined electrochemical processes set E as opposed to equilibration with the kinetically slow where e is an electron residing in the metal electrode.At we μα oxygen couple (the species with the largest exchange current equilibrium, the sum of ̅j for products must be equal to that ffi dominates the measured electrode potential). for reactants. This is true for any process that has a su cient Let us now return to the concept of electrochemical rate to be considered at equilibrium over the relevant time ff μ μ μ equilibration. Upon contact, the initial di erence in ̅e drives scale. We can therefore relate ̅e in the metal to ̅for O and charge transfer across the electrode/electrolyte interface, R in solution and define a solution Fermi-level (even though ff “ ” 9 leading to an interfacial electric potential drop that a ects there are practically no free electrons in the solution). μm μs ̅e relative to ̅e until they are equal. The amount of charge s s that must be transferred to reach equilibrium depends on the m μμR̅ − O̅ s μ̅ = ≡ μ̅ capacitance of the electrode but is generally small compared e n e (10) to the number of electrons in the metal and redox species in

263 https://dx.doi.org/10.1021/acsenergylett.0c02443 ACS Energy Lett. 2021, 6, 261−266 ACS Energy Letters http://pubs.acs.org/journal/aelccp Viewpoint

Figure 2. Establishing electrochemical equilibrium across a perfectly cation-selective membrane. (a) The two adjacent phases, varying in − ﬀ μ + KCl concentration, have di erent ̅K before coming into contact. The Cl ions in this example do not equilibrate across the interface + ∇μ + i.e. because the membrane is assumed to completely exclude them. (b) Upon contact, K transports from left to right according to ̅K , , μα μβ − until equilibrium is reached and ̅K+ = ̅K+.AsCl cannot transport, space-charge regions are formed that result in an electric potential a + gradient and a position-dependent K on both sides of the membrane.

Pt μ − Figure 3. Electrochemical potentials in a fuel cell. In equilibrium, ̅e match with the electrochemical potentials of the other reaction partners at the cathodic and anodic triple-phase boundaries (according to stoichiometry). When a fuel cell is operated at a cell potential E cell < 1.23 V the free energy stored in the adjacent 1 atm H2 and O2 compartments is released to do work in the external circuit. Current-density-dependent overpotentials develop at the anode and cathode which are associated with kinetic limitations at the catalyst sites, leading to a free-energy loss, as heat, compared to the theoretical value of 1.23 V. the electrolyte (so that the bulk activity and thus μ for all the ϕβ across the membrane. As the membrane was assumed to − μ − species is practically unchanged). The extra charge on the block Cl transport, there is no way for ̅Cl to equilibrate, α β − μ − ≠ μ − ff metal electrode surface is compensated by ionic charge in and ̅Cl ̅Cl . This electric potential di erence is known as the electrolyte. The concentration of the compensating ions a Donnan potential and found in biological cells and various decreases with distance from the electrode (as given by the electrochemical devices.7 μ Poisson-Boltzmann distribution) such that, at equilibrium, ̅j It is instructive to consider how membrane electric for all species are constant with position, through the double potentials are experimentally determined. Usually two layer, and into the bulk electrolyte. identical reference electrodes are placed on either side of Example 2: Semipermeable membranes and membrane the membrane, and the membrane potential, ϕα − ϕβ,is potentials. Consider a membrane that is selective for the taken as the measured voltage difference. But if voltmeters transport of cations, e.g., an ionomer or a biological ff μ measure di erences in ̅e, and not electric potential membrane. Imagine the membrane separating two compart- ff ff di erences, how can this be? The electrons in the ments containing, for example, aqueous KCl at di erent reference-electrode wire are in equilibrium with electrons concentrations (Figure 2). How do the chemical, electric, and involved in the reference-electrode half reaction, e.g., AgCl + electrochemical potentials change in moving across the − − e (Ag) ⇄ Ag + Cl (aq), such that membrane from one compartment to the other at equilibrium? Let us assume perfect permselectivity, meaning Ag Ag s AgCl μ −−= μμμ+ − only the cations can be moved and equilibrated across the e̅ Ag̅ Cl̅ AgCl̅ (13) membrane (the rate of anion equilibration is negligible). At − equilibrium, the electrochemical potential of K+ must be The activity of Cl is the same in the two reference constant across the entire system. Labeling points far from electrodes, and we neglect junction potentials across the frits the interface in the two compartments α and β yields used to separate each reference electrode from each test

α o α α o β β β solution (which are small if both contain electrolytes at high μ ++=+μϕμϕμRTln a+ +=+ F+ RT ln a++ + F = KK̅ K K KK̅ concentrations). The membrane potential affects the value of s Ag αβRT β α μ − μ − ϕϕ−=lnaa++ / ̅Cl and thus also ̅e between the two Ag wires. A voltage is F KK (12) measured because Cl− ions and electrons in the reference β α s,re1 s,re2 re1 ≠ + fl μ − ≠ μ − μ − ≠ Because aK+ aK+,K ows from one compartment to the electrodes are not in equilibrium ( ̅Cl ̅Cl and ̅e α β re2 μ μ μ − other to reach equilibrium, i.e. when ̅K+ = ̅K+, resulting in a ̅e ). Because the voltmeter provides a high-impedance path charge imbalance and thus electric potential difference ϕα − for electronic current, the rate at which equilibrium is

264 https://dx.doi.org/10.1021/acsenergylett.0c02443 ACS Energy Lett. 2021, 6, 261−266 ACS Energy Letters http://pubs.acs.org/journal/aelccp Viewpoint approached is slow enough to not perturb the measurement can thus be described as migration or drift in the Nafion, and of the membrane potential. losses (overpotentials) are ohmic and generally increase Example 3: A fuel cell. Electrochemical energy-conversion linearly with current. In the Pt-catalyst and electrode devices such as batteries and fuel cells convert between structures, electron current flows, which leads to small μPt chemical and electrical energy. Consider a proton exchange gradients in ̅e (particularly in the catalyst layer where membrane (PEM) fuel cell with a Nafion membrane, Pt electronic connectivity between Pt particles and carbon hydrogen oxidation reaction (HOR) catalyst, and Pt oxygen support is often nonideal). The largest loss in free-energy fi reduction reaction (ORR) catalyst (Figure 3). O2 gas at 1 driving force occurs at the Pt/Na on interfaces because of fl fl atm is owed over the cathode, and H2 gas at 1 atm is owed the slow kinetics of the ORR (and to a lesser extent HOR). over the anode. First, let us consider open circuit when The total free energy extractable from a fuel cell is thus never negligible net current is flowing through the external circuit ideal under current flow; output voltages are always less than (e.g., during measurement with a high-impedance voltmeter). 1.23 V. Because there is negligible flux, the electrochemical potential More generally, an overpotential is the free energy used to + fi μs of H in the Na on, ̅H+, is constant through the phase. drive an electrochemical reaction away from equilibrium. Electric fields exist only at the interfaces between Nafion and Including kinetic, transport, and other contributions, the the Pt catalysts and extend only a few nanometers into overpotential is Nafion (i.e., the Debye screening length), because the H+ are η =−EE mobile and concentrated in Nafion. The chemical potential app rev (19) μs H+ is thus also constant through the membrane (except at where E is the electrode potential applied, for example by a μs app the interfaces where double layers form and H+ is spatially potentiostat, relative to a defined reference, and E is the μs rev varying to give constant ̅H+). reversible potential of the redox reaction (i.e., the potential of Now consider the Pt catalyst/electrodes. If zero net current a hypothetical nonpolarizable electrode driving the desired fl μ is owing, does that mean ̅e values at the Pt anode and reaction with no overpotential under the given conditions) cathode are the same? No. The electrons in the Pt anode (a) relative to the same (arbitrary) reference. We note that are in equilibrium (in the absence of net current flow) with because of mass-transport limitations, Esol can be position- the H (g) and H+ in Nafion, governed by fi 2 dependent while Erev is not (as it is de ned under equilibrium 4H(Nafion)+−+ 4e(Pt,a)F 2H(g) conditions). For this reason, the kinetic component of the 2 (14) η ff overpotential, k, is the di erence between the electrode and We write the electrochemical potential equality that is solution potentials (as defined in Table 1) across the required under no current flow: electrode interface over the relevant distance where electron/ion-transfer takes place (thus avoiding convolution Pt,a 1 g s μ̅ −+= μμ̅ − ̅ ≈ 0kJ/mol with mass-transport and ohmic losses). It is this kinetic e H2 H (15) 2 overpotential that practitioners are generally referring to Pt,a μ − ∼ The value of ̅e is 0 because of the standard states chosen when discussing electrocatalyst performance, although it is o o μ + μ ffi fl ( ̅H = H2 = 0) and is arbitrary. often di cult to completely remove the in uence of other Pt,c μ − overpotential contributions (e.g., from electronic and ionic At the Pt cathode (c), ̅e is governed by the equation +− transport). O22 (g)++ 4H (Nafion) 4e (Pt, c)F 2H O (16) Summary. Fundamentally, there are several key points we In the absence of current, we then have hope the reader will internalize and use when considering any electrochemical system to avoid confusion regarding Pt,c 1 s 1 g s potentials: μe̅ −+= μμμHO̅ − O̅ − H̅ ≈−119 kJ/mol 2 224 (17) • Measurements of potential differences are necessarily of g s ff Both μ̅ and μ̅+ would be (arbitrarily) defined as zero if O the total free-energy di erence. Decomposing this total O2 H 2 ff and H+ are in their standard states, the temperature is 25 °C, into di erences in activity, electric potential, and other and electrostatic potential is defined as zero in the solution. terms (as needed to describe all the relevant The open-circuit cell potential can then be calculated: contributions to the free energy) therefore requires a model and assumptions. Pt,c Pt,a ΔGrxn • Transport of any species is governed by the spatial μ̅ −−− μ̅ = =−FEcell,oc e e n (18) gradient in the electrochemical potential. Generally, fi This yields E = 1.23 V, the thermodynamic voltage for knowledge of the electric eld alone (i.e., the gradient cell,oc in the electric potential) is insufficient to understand water electrolysis under standard conditions. Ecell,oc is also known as the electromotive force or emf of a battery or fuel the transport of charged species except in special cases. • cell. The system thus could be considered out of equilibrium At equilibrium, the electrochemical potential of any given species must be the same throughout the system, in this state because the O2 and H2 on either side of the cell cannot mix across the Nafion membrane and react, and and for any chemical reaction, the sum of the electrons cannot exchange between electrodes at open-circuit. electrochemical potentials of the reactants must equal During operation, when ionic and electronic current flow those of the products. Processes with slow kinetics are, μα fi in practical systems, typically ignored in the thermody- in the fuel cell, there must be gradients in ̅j . In the Na on μs namic equilibrium analysis. membrane there is a gradient in ̅H+, but, as in the case of the metal wire, the high concentration of H+ means that changes • The use of the word “potential” alone should be s μs to aH+ are negligible and the spatial dependence of ̅H+ is avoided unless the type of potential is made clear. In largely due to a gradient in electric potential ϕs. The current electrochemistry, “potential” is usually implied to mean

265 https://dx.doi.org/10.1021/acsenergylett.0c02443 ACS Energy Lett. 2021, 6, 261−266 ACS Energy Letters http://pubs.acs.org/journal/aelccp Viewpoint

the electrochemical potential of electrons. When the electric potential or chemical potential is meant, that should be indicated explicitly. Practitioners should also indicate what species and phase are being referred to. Shannon W. Boettcher orcid.org/0000-0001-8971-9123 Sebastian Z. Oener Mark C. Lonergan orcid.org/0000-0003-4070-5067 Yogesh Surendranath orcid.org/0000-0003-1016-3420 Shane Ardo orcid.org/0000-0001-7162-6826 Carl Brozek orcid.org/0000-0002-8014-7904 Paul A. Kempler ■ AUTHOR INFORMATION

Complete contact information is available at: https://pubs.acs.org/10.1021/acsenergylett.0c02443 Notes Views expressed in this Viewpoint are those of the authors and not necessarily the views of the ACS. The authors declare no competing financial interest. ■ ACKNOWLEDGMENTS The authors thanks Marc Koper and Plamen Atanassov for useful comments regarding this Viewpoint. ■ REFERENCES (1) Newman, J.; Thomas-Alyea, K. E. Electrochemical Systems; Wiley, 2004. (2) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications; 2001. (3) Schmickler, W.; Santos, E. Interfacial Electrochemistry; Springer, 2010. (4) Gileadi, E. Physical Electrochemistry. Fundamentals, Techniques and Applications; Wiley, 2011. (5) Parsons, R. Electrochemical Nomenclature. Pure Appl. Chem. 1974, 37, 499−516. (6) Riess, I. What Does a Voltmeter Measure? Solid State Ionics 1997, 95, 327−328. (7) It is important to further recognize that the electrochemical potential formulation given here is but one model that can be used to describe the total free energy of a given species. Electric potential gradients exist in all molecules because of the nuclei and electrons. The separation of electric energies into “long-range” (zFϕα) and “short-range” (μα) components is particularly useful in electrochemical systems where (long-range) interfacial electric fields can be controlled between and across phases, but it is not otherwise unique. (8) Peper, J. L.; Mayer, J. M. Manifesto on the Thermochemistry of Nanoscale Redox Reactions for Energy Conversion. ACS Energy Lett. 2019, 4, 866−872. (9) If there are other species whose free energy changes during the reaction, for example, ions that intercalate (e.g.,inbattery electrodes) or form ion pairs, these must be explicitly included in the electrochemical reaction (eq 9), and eq 10 must be modified appropriately.

266 https://dx.doi.org/10.1021/acsenergylett.0c02443 ACS Energy Lett. 2021, 6, 261−266