Chemical ActivityFaradaic reactions in equilibriumStandard PotentialCell Open Circuit Voltage (OCV)The Equilibrium ConstantII. Equilibrium Thermodynamics Lecture 8: The Nernst Equation Notes by 0,7 6WXGHQW (and MZB 2012) Chemical Activity n general, we can define the chemical activity of species i as follows, δG µi = = kBT log(ai) + zieΦ (per particle) (1)δNiNj ,T,P,Φ lternatively, we can consider the chemical activity per mole, in which case: µi = RT log(ai) + ziF Φ (per mole) (2) R = universal gas constant = kBNA (3) J = 8.31 (N23 A = 6 10 particles)K mol×F = Faraday’s constant = NAe = 96, 487 C (4) kBT RT = = thermal voltage =at25 ◦C 26mV (5) e F ince we focus on microscopic mechanisms, we will typically define physical roperties per particle. As shown in our example above, for a dilute solution, the activity is imply the concentration. In a concentrated solution, there are corrections elated to all other contributions to the chemical potential, which can be xpressed via an activity coefficient γi, whose precise definition depends on he way of measuring concentration (per volume, per mass, per site, etc.) nd is typically scaled such that γi = 1 in the limit of infinite dilution: c m x pai = γici = γimi = γixi = γipi = . . . (6) 1 1IASpsretaLecture 8: Nernst equation 10.626 (2011) Bazant where ci = species concentration (e.g. M = molar = moles/liter), mi = molality in solution (mol/kg solvent), xi = mole fraction or filling fraction of available sites (dimensionless), pi = gas partial pressure (atm), etc. The activity coefficient is also related to the excess chemical potential ex µi = kBT ln γi (7)which contains all the contributions to chemical potential in a concentrated solution, which are not present in the limit of infinite dilution. Example: In the previous lecture, we used statistical mechanics to derive an explicit expression for the electrochemical potential (free energy change for adding an ion of charge ze) in the case of a solid solution or lattice gas of charged particles: x µ − zeφ = h(x) + kB T ln = kBT ln x + µex = kBT ln(γx) (8)1 − x where x is the mole fraction, h(x) is the enthalpy per site, and the second term comes from the entropy of mixing on a lattice. In this case, we have eh(x)/k Tex µ = h(x) − BkBT ln(1 − x), and γ = (9)1 − x Note that the second term in the excess chemical potential and the activity coefficient both grow to infinity as x → 1 due to crowding effects (reduced entropy of vacancies in the lattice). Increased activity means larger chemical potential, and thus greater tendency to have reactions or transport that lower the concentration by removing particles. At the simplest level, this is what stops battery discharge when a reactant is fully consumed. Later we will apply this model to Li-ion batteries, e.g. for lithium intercalation in cobalt oxide, LixCoO2. In electrostatics, the potential φ is just a theoretical construct. Only differences in potential (voltage drops) or gradients (electric fields E = −\φ) have physical meaning. As a result, the choice of a reference potential φ0 is arbitrary and made for convenience only. Similarly, in electrochemistry, only differences in electrochemical po-tential Δ = µ0i − µi, which drive charge transfer reactions or ion trans-port, have physical meaning. This motivates setting a reference state with µ0i= kT log ha0ii+ zieφ0Reference states are usually defined in terms of a standard concentration (1 M for all species), room temperature (25 ◦C) and 2Lecture 8: Nernst equation 10.626 (2011) Bazant electrode(usuallymetal)electrolyte"half cell"+ziφeatmospheric pressure (1atm). Just as only potential differences µ0i − µiare meaningful, only activity ratios ai0 are.a i aµi − µ0i = k iT log = zie(φ − φ0) (10)a0i Note: the reference activity is normally dropped, and it is understood that a = 1 in the standard reference state. 2 Faradaic reactions in equilibrium Any Faradaic “half-cell” charge-transfer reaction at an electrode can be ex-pressed in the following standard form, producing n electrons: zsiMi i→ ne − (11)i where si = stoichiometric coefficients (12) Mi = chemical symbols (13) zi = charge numbers (14) n = number of electrons transferred (15) Charge conservation requires sizi = −n (16) i The forward direction of the Faradaic reaction represents oxidation, or increase of the charge state of the molecules (on the left hand side) while liberating electrons. In a galvanic cell, electrons flow spontaneously from the anode to the cathode, so the forward direction of the reaction is also called the anodic reaction. At the anode, si > 0 for reactants, and si < 0 for 3  Lecture 8: Nernst equation 10.626 (2011) Bazant products, while the signs are reversed for a cathodic reaction which consumes electrodes and leads to reduction of the charge state of the molecules. For example, the hydrogen oxidation reaction can be written as H2 − 2H+ → 2e− (sH2 = 1, sH+ = −2, zH+ = 1, n = 2) In equilibrium, the total electrochemical potential of each side of the reaction must be equal, which implies Xsiµi = nµ− e(17)i For an electrode, the electrochemical potential of the electron is the Fermi energy of the highest occupied electronic quantum state. The energy per charge is the potential of the electron, φe, so we can write µ =− e −eφe (18)The electrostatic potential of the electron, φe, is different from that of all the ions, φ, in the electrolyte. The electrostatic potential of the electron minus that of the ions is the Nerst potential or interfacial voltage drop: Δφ = φe − φ (19) We are now ready to derive a general thermodynamic formula, the Nernst Equation, relating the interfacial voltage Δφ to that of a reference state in equilibrium, Δφ0 . For the reference state we have, X0 s0iµi = −ne(φe) (20)i 4_ _ Lecture 8: Nernst equation 10.626 (2011) Bazant