II. Equilibrium Thermodynamics. Lecture 8: The Nernst Equation

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1 II. Equlbrum Thermodynamcs Lecture 8: The Nernst Equaton Notes by MIT Student, re-wrtten by MZB 2014 February 27, Chemcal Actvty The fundamental thermodynamc quantty controllng transport and reactons s the (electro)chemcal potental of speces, δg µ = (1) δn N j,t,p,φ where G s the Gbbs free ergy (for an open system at constant temperature and pressure). whch we decompose nto three parts: µ = µ Θ + k B T ln(a ) + z eφ(per partcle) (2) Θ the chemcal potental µ n an arbtrarly defd standard state, a correcton k B T ln a where a s the (dmensonless) chemcal actvty relatve to a = 1 n the standard state, and the electrostatc ergy z eφ for a charge z e n the electrostatc potental φ. In chemstry t s more common to def chemcal potental and other thermodynamc quanttes per mole, n whch case: µ = µ Θ + RT ln(a ) + z F Φ (per mole) (3) R = unversal gas constant = k B N A (4) J = 8.31 K mol (N A = partcles) F = Faraday s constant = N A e = 96, 487 C (5) k B T RT = = thermal voltage = at25 C 26mV e F (6) 1

2 Lecture 8: Nernst equaton Snce we focus on mcroscopc mechansms, however, we wll typcally def all thermodynamc propertes per partcle (whch s more common n physcs). As shown n the example below, n the lmt of an nfntely dlute soluton, the actvty s proportonal to the concentraton. In a non-dlute concentrated soluton where partcles nteract wth each other, there are correctons related to all other contrbutons to the chemcal potental, such as short-range forces between molecules (contrbutng to enthalpy h) or temperature-dependent crowdng effects (contrbutng to the entropy). These non-deal contrbutons to the chemcal potental can be expressed va an actvty coeffcent γ, whose precse defnton depends on the way of measurng concentraton (per volume, per mass, per ste, etc.) and s typcally scaled such that γ = 1 n the standard state of a sngle speces (n a dlute soluton): a = γ c c = γ m m = γ x x = γ p p =... (7) where c = c /c ref s the dmensonless speces concentraton wth c ref = 1M = 1 mole/lter = 1 molar, m = m /m ref s the dmensonless molalty n soluton wth m ref = 1 mol/kg solvent = 1 molal, x = N /N ste s the mole fracton or fllng fracton of avalable sold stes (dmensonless), p = p /p ref s the dmensonless gas partal pressure wth p ref = 1 atm, etc. Example: Lattce gas, or deal sold soluton. In the prevous lecture, we used statstcal mechancs to derve an explct expresson for the chemcal potental of an deal sold soluton (or lattce gas) of non-nteractng partcles on a lattce: x µ = k B T ln = k B T ln(γx) (8) 1 x whch comes from the Gbbs free ergy of mxng (only entropy). In ths case, we have 1 γ = (9) 1 x The actvty coeffcent goes to nfnty as x 1 due to crowdng effects (reduced entropy of vacances n the lattce). Increased actvty means larger chemcal potental, and thus greater tendency to have reactons or transport that lower the concentraton by removng partcles. At the smplest level, ths s what stops battery dscharge when a reactant s fully consumed. Later we wll apply ths model to L-on batteres. 2

3 Lecture 8: Nernst equaton φ e +z electrode (usually metal) electrolyte "half cell" 2 Faradac reactons n equlbrum Any Faradac half-cell charge-transfer reacton at an electrode can be expressed n the followng standard form, producng n electrons: where s M z (10) Charge conservaton requres s = stochometrc coeffcents (11) M = chemcal symbols (12) z = charge numbers (13) n = number of electrons transferred (14) s z = n (15) The forward drecton of the Faradac reacton represents oxdaton, or ncrease of the charge state of the molecules (on the left hand sde) whle lberatng electrons. In a galvanc cell, electrons flow spontaously from the anode to the cathode, so the forward drecton of the reacton s also called the anodc reacton. At the anode, s > 0 for reactants, and s < 0 for products, whle the sgns are reversed for a cathodc reacton whch consumes electrodes and leads to reducton of the charge state of the molecules. For example, the hydrogen oxdaton reacton can be wrtten as H 2 2H + 2e (s H2 = 1, s H + = 2, z H + = 1, n = 2) The Faradac reacton can also be wrtten wth only postve coeffcents as R = s M z s M z + = O + (16) s >0 s <0 3

4 Lecture 8: Nernst equaton φ e electrode$$ $$solu/on$ φ (a)$ O z Oe (e) R z Re (e) O z Os (s) R z Rs (s) (b)$ O z O R z R (c)$ R z R O z O Fgure 1: Types of Faradac reactons O + R. (a) Geral mxed on-electron conductor electrode/electrolyte nterface. (b) Redox n soluton. (c) Ion ntercalaton or electrodeposton. where R s the reduced state and O s the oxdzed state. The fgure shows dfferent Faradac reactons at an electrode/electrolyte nterface. Let us assume that ons only exst n the electrolyte (glectng mxed on-electron conductors). For redox reactons (1(b)), e.g. Fe 3+ +e Fe 2+, the reduced state s n the soluton at the same potental, φ R = φ O = φ. For electrodeposton (1(c)), e.g. Cu e Cu, or on ntercalaton as a utral polaron (on/electron par), e.g. CoO 2 +L + + e LCoO 2, the reduced state s uncharged, z R = 0, so we can also set φ R = φ, even though t s n the electrode. For ths broad class of Faradac reactons, we can thus let φ be the electrostatc potental of each on, and φ e be the potental of the electrons n the electrode. The Nernst potental or nterfacal voltage drop (electrode mnus electrolyte) s the Δφ = φ e φ (17) and we wll now derve ts value n equlbrum. The total electrochemcal potental of each sde of the reacton must be 4

5 Lecture 8: Nernst equaton equal n equlbrum, s µ = nµ e (18) where the electrochemcal potental of the electron µ e s the Ferm ergy of the hghest occuped electronc quantum state. Insertng the defntons above, µ = µ Θ + k B T ln a + z eφ (19) µ e Θ = µ + k B T ln a e eφ e (20) e and usng charge conservaton s z = n (21) we arrve at the Nernst equaton: _ s k B T a Δφ = Δφ 0 ln (22) where Θ e s µ Δφ Θ = µ (23) s the standard potental, or nterfacal voltage when all reactants are n ther standard states. The Nernst equaton shows how the nterfacal voltage of the half-cell reacton vares wth reactant actvtes away from the standard state. We can also wrte a n e Θ k B T a O a Δφ = Δφ Θ + ln e (24) a R n where 3 a R = s a, a O = s a (25) s >0 s <0 Redox Equlbrum Constants For a geral chemcal reacton, the equlbrum constant K s the rato of the actvtes of the products to those of the reactants, n chemcal equlbrum. If K > 1 (or K < 1), the forward (or backward) reacton s thermodynamcally favored under standard condtons and leads to more (or 5

6 Lecture 8: Nernst equaton less) products compared to reactants. In the case of the geral oxdaton reacton, R O +, the equlbrum constant s n = K 1 = a Oa e (26) K ox red a R whose nverse K red s for the backward reducton reacton. (The term redox s used to refer to reversble charge transfer reactons, and combnatons of oxdaton and reducton half reactons are called redox couples.) The physcal sgnfcance of the Nernst equaton becomes more clear n terms of the redox equlbrum constants, ( ( φ K ox = K 1 eq red = exp φ Θ ) ) (27) k B T Now we see that by ncreasng the Nernst equlbrum potental, Δφ eq, we ncrease K ox so as to favor the forward oxdaton reacton that produces electrons. Ths makes sense because electrons are drawn to the more postve potental of the electrode as ts voltage relatve to the soluton s ncreased. The opposte trend holds for the standard Nernst potental. If we ncrease Δφ Θ, then K ox decreases so as to favor the reducton reacton that consumes electrons. In galvanc cells, therefore, half-cell Faradac reactons wth larger Δφ Θ act as cathodes (consumng electrons) when coupled wth those havng smaller Δφ Θ, whch act as anodes (producng electrons). 4 Standard Cell Potental In order to measure the potental of an electrode φ e, we must form a complete electrochemcal cell wth two electrodes, and two half-cell Faradac reactons, snce only dfferences n potental are well defd. As such, the standard cell potental, E Θ = Δφ Θ Δφ Θ ref = φ e φ ref (28) s defd relatve to a standard reference electrode. An electrode reacton wth E Θ < 0 acts as an anode (gatve battery termnal, producng electrons), and o wth E Θ > 0 as the cathode (postve termnal, consumng electrons), when coupled to the standard reference electrode. Comment on notaton: In electrochemstry, t s common to denote cell voltage (potental dfference) wth E to denote electro-motve force or emf. Ths can cause confuson wth the electrc feld ampltude, also denoted E (n all felds of scence), whch s the local gradent of the potental! 6

7 Lecture 8: Nernst equaton Faradac reactons n aqueous electrolytes L metal L + (aq) + e P b + SO4 2 P bso 4 + 2e H 2 2H + 2e 2H 2 O O 2 + 4H + + 4e P bso 4 + 2H 2 O P bo 2 + SO H + + 2e E Θ (V) To avod confuson, we use φ to denote the electrostatc potental, E = Vφ for the electrc feld, Δφ for potental dfferences (e.g. across nterfaces), V for cell voltage (Δφ between two electrodes), and V Θ for ts reference value under standard condtons. However, for consstency wth electrochemstry tables, we use E Θ for the standard cell potental, relatve to a gven reference. For aqueous electrolytes, the usual reference electrode s the standard (or normal) hydrogen electrode (SHE), whch conssts of H 2 gas at 1 atm partal pressure at room temperature undergong fast, reversble oxdaton, typcally at a platnum electrode, n acdc soluton (a H + = 1, ph=0). As shown n the fgure, o can measure the potental of another electrode relatve to SHE by preparng two aqueous solutons concted by a lowresstance salt brdge, whch allows ons to pass and equlbrate across the two chambers, whle a very small current s drawn to measure the open crcut voltage of the cell. If the test system s at standard condtons (1atm, 25 C) then we wrte E Θ = standard cell potental relatve to SHE. Some examples are shown n the table: lthum oxdaton, lead-acd battery reactons, and water electrolyss. 5 Open Crcut Voltage In equlbrum, the open crcut voltage (OCV) of a two-electrode electrochemcal cell can be obtad as the dfference of the two half-cell voltages. Under standard condtons, ths s just gven by the dfference of standard potentals, whch can be looked up n a table or measured expermentally. For example, we can create a galvanc cell wth lthum metal oxdaton (electro-dssoluton) occurrng at the anode, and oxygen gas reducton to produce water at the cathode, as shown n the fgure. The open crcut voltage under standard condtons would be V O = ( 3.045) = 4.3V. 7

8 Lecture 8: Nernst equaton PEM L water H + + L O 2 H O 2 More gerally, we use the Nernst equaton to descrbe how the OCV changes wth the actvtes of the reactants away from standard condtons. The cell voltage s gerally the dfference V = φ c φ a (29) between the potentals of the cathode and anode, or the postve and gatve termnals, respectvely, n the case of a galvanc cell. Under open crcut condtons n equlbrum, the electrolyte potental s constant, so we can wrte V O = Δφ c Δφ a (30) and use the Nernst equaton, _ s kt _ j a j j (anode reacton) V = V Θ O + ln s (31) (cathode reacton) where the standard OCV wth all reactants n the standard states s a V Θ = Δφ Θ Δφ Θ = E Θ E Θ c a c a (32) By charge conservaton, t cell reacton nvolves only utral reactants and products, so we can also wrte the OCV as ( ) V Θ kt acell reactants V O = + ln (33) If we wrte out the redox reactons, a cell products R a O a + (anode half cell) O c + R c (cathode half cell) (34) R a + O c O a + R c (full cell) then Θ kt a R a a O c VO = V + ln a Oa a Rc (35) 8

9 Lecture 8: Nernst equaton 6 Cell Equlbrum Constant products The equlbrum constant of the t cell reacton, K = a reactants, s measured under short crcut condtons at equlbrum, V O = 0, when there s no spontaous current flow, and thus no chemcal drvng force between the anode and the cathode. The Nernst equaton relates the cell equlbrum constant to the standard cell voltage, V K = e Θ /kt = Ka ox Kred c a (36) whch s also related to the rato of redox equlbrum constants of the two electrodes. Because ths value can be very large or small, t s convent to refer to the pk value defd by n V Θ pk = log 10 K = (37) log 10 e although snce the hstorcal conventon s to use base ten, we must keep track of the factor (log 10 e) 1 = From the defnton of cell voltage as the Gbbs free ergy change of the system per charge passng through the external crcut, V = ΔG/( ), we also recover the fundamental relaton ΔG Θ /kt K = e (38) where ΔG Θ s the free ergy of reacton (of products mnus reactants) for the t cell reacton under standard condtons. Example: Short-crcuted hydrogen-oxygen fuel cell. The t reacton s water producton: H 2 + O 2 H 2 O. V Θ = 1.229V (39) n = 2 (40) k B T = 26mV (room temperature) e (41) K e 94.5 = » 1, pk = 41» 1 (42) We see that wth only o volt standard potental, the equlbrum constant s astronomcally large, almost as large as Avogadro s number squared (N 23 A = partcles per mole), mplyng that essentally every sngle molecule of H 2 and O 2 s converted to H 2 O n equlbrum, f no other 9

10 Lecture 8: Nernst equaton reactons or ktc lmtatons can occur. Ths s the typcal stuaton when an electrochemcal cell s short-crcuted. Because Faradac reactons nvolve breakng and reformng stable chemcal bonds (whch are not destroyed by thermal fluctuatons), the free ergy of reacton s usually much larger than the thermal voltage, leadng to a very large equlbrum constant for the t reacton. Away from short crcut condtons, the OCV can be expressed n terms of the cell equlbrum constant, kt Ka reactants V O = ln (43) a products or n terms of the redox equlbrum constants, kt VO = ln (44) Koxa a Ra a Oc K c red a Oa a Rc where t s clear that V O = 0 n equlbrum. 10

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