Thermodynamicsof Fuel Cells:Nernst EquationEocrefers to the open circuit potential of the cell understandard conditions.Whathappenstothevoltageastheconcentrationof reactants&products changes?Thermodynamics:dG=VdP-SdTwhereV&Pare thevolumeandpressureof species of interest.Under isothermal conditions dT=O:hence dG=VdpUsing Ideal Gas Law: PV=mRT and substituting for Vwe get: G =mRr - RTd(ln P")Integratingfromstandard state(superscript'o)conditions:fdG=RTd(ln P")=G=G°+RTInGPConvert Pinto concentration, c, via a form of the Ideal Gas Law P=cRT, noting thatunderisothermalconditionsT=Tgives:G~G°+RT°InForourreactionH,+YO,H,o(notestoichiometry!)wenowget:CHVCOAG=AGo-RT'InwherecCH,O
Thermodynamics of Fuel Cells: Nernst EquaGon • EOC refers to the open circuit potenGal of the cell under standard condi6ons. What happens to the voltage as the concentraGon of reactants & products changes? • Thermodynamics: dG = VdP – SdT where V & P are the volume and pressure of species of interest. • Under isothermal condiGons dT = 0: hence dG = VdP • Using Ideal Gas Law: PV=mRT and subsGtuGng for V we get: • IntegraGng from standard state (superscript ‘0’) condiGons: • Convert P into concentraGon, c, via a form of the Ideal Gas Law P=cRT, noGng that under isothermal condiGons T=To gives: • For our reacGon H2 + ½ O2à H2O (note stoichiometry!) we now get: dG = mRT dP P = RTd(lnPm ) dG = RT d(lnPm ) ⇒ G = G0 + RT ln P P0 " # $ % & ' m " # $ $ % & ' ' P0 P ∫ G0 G ∫ G ≈ G0 + RT 0 ln c c 0 " # $ % & ' m " # $ $ % & ' ' ΔG = ΔG0 − RT 0 ln c ~ H2 c ~ O2 c ~ H2O # $ % % % & ' ( ( ( where c ~ = c c 0
ThermodynamicsofFuelCells:NernstEquationAGRearrange equation 1.1 EocnFandsubinbydividingbothsidesby(-nF)gives:RTCH2VCO,(Eq 1.2)TheNernst EquationEoc=EnFCH20Toavoid confusion for anygiven reaction as written:remember either“plus RT/F In(reactantsoverproducts)"or“minusRT/FIn(products overreactants)":Yourpreference howyou write the Nernst formula, but that way you can'tgo wrong!The Nernst Eguationsimplytells us that the potential of thefuel cell varieslogarithmicallywiththeconcentrationsof reactantsorproducts
• Rearrange equaGon 1.1 • and sub in by dividing both sides by (–nF) gives: Thermodynamics of Fuel Cells: Nernst EquaGon EOC = EOC 0 + RT nF ln c ~ H2 cO2 ~ c ~ H2O ! " # # # $ % & & & The Nernst Equa6on (Eq 1.2) To avoid confusion for any given reacGon as wriben: remember either “plus RT/F ln (reactants over products)” or “minus RT/F ln(products over reactants)”: Your preference how you write the Nernst formula, but that way you can’t go wrong! EOC = − ΔG nF The Nernst EquaGon simply tells us that the potenGal of the fuel cell varies logarithmically with the concentraGons of reactants or products
ThermodynamicsofFuel Cells:Temperaturedependenceof EocWe can now seehow thefuel cell open circuit potential varies with concentration (orpressure)of reactants and products under isothermal (constant T)conditions,butwhathappens ifTchangesunderconstantpressure,P?Weknow:dG=VdP-SdT butnowdP=o (isobaricconditions)Therefore dG=-SdT and integrating both sides from standard conditions(To=298K)gives: G = Go-S(T-TO)Writing this forthe initial'',and final'fstates, then subtracting weget:△G = AG°-S(T-TO)(Eq 1.3)Dividing both sides by (-nF) using equation 1.1 then gives:Note:1)AS<oforourreactionsoEocdecreasesasTincreases.2)Thevariation of Eocwith Tis linear3)BecausenF>>△Sthischangeissmall(10-3VK-1)4)WehaveassumedsisconstantovervaryingT,whichisreasonableprovided the temperature difference(T-T')is nottoo large
• We can now see how the fuel cell open circuit potenGal varies with concentraGon (or pressure) of reactants and products under isothermal (constant T) condiGons, but what happens if T changes under constant pressure, P? • We know: dG = VdP – SdT but now dP = 0 (isobaric condiGons) • Therefore dG= –SdT and integraGng both sides from standard condiGons (To = 298 K) gives: G = Go – S(T – T0) • WriGng this for the iniGal ‘i’, and final ‘f’ states, then subtracGng we get: ΔG = ΔG0 – S(T – T0) • Dividing both sides by (–nF) using equaGon 1.1 then gives: • Note: 1) ΔS<0 for our reacGon so EOC decreases as T increases. 2) The variaGon of EOC with T is linear 3) Because nF>>ΔS this change is small (10-3 V K-1) 4) We have assumed ΔS is constant over varying T, which is reasonable provided the temperature difference (T – T0)is not too large. Thermodynamics of Fuel Cells: Temperature dependence of EOC EOC = EOC 0 + ΔS nF T −T 0 ( ) (Eq 1.3)
Potentialsina Fuel CellSofar,we haven't mentioned the needfora Pt catalyst in thefuel cell.TheEocvalueof1.2Visthetheoretical maximumforperfectcatalyst+perfectelectrolyte"system.Real systemsarenotideal,sothevalueof Eoc(max)isdecreased.This is the nature of electrode kinetics-we will come to this in amoment.Firstconsidercurrentloadingofafuelcell:If no current is drawn from the FC, E=EociDrawing currentfrom the FC"costs"some energy, hence E decreases as i increases by an amount Eloss'Ptparticles supported onthecarbonelectrodegiveaporousstructureoflargearea, A.The potential of the Pt=the potential of the carbon, and we ignoreporosityin a"macrohomogeneous"model (which works,but is theoreticallyunjustified)
PotenGals in a Fuel Cell • So far, we haven’t menGoned the need for a Pt catalyst in the fuel cell. The EOC value of 1.2 V is the theoreGcal maximum for “perfect catalyst + perfect electrolyte” system. Real systems are not ideal, so the value of EOC(max) is decreased. This is the nature of electrode kineGcs – we will come to this in a moment. • First consider current loading of a fuel cell: • If no current is drawn from the FC, E = EOC ; Drawing current from the FC “costs” some energy, hence E decreases as i increases by an amount Eloss. • Pt parGcles supported on the carbon electrode give a porous structure of large area, A. The potenGal of the Pt = the potenGal of the carbon, and we ignore porosity in a “macrohomogeneous” model (which works, but is theoreGcally unjusGfied)
Potentialsina Fuel CellBecauseeachcatalystisnotperfect,thereis anextrapotential"cost",theoverpotential,n(greeksymbol"eta")-moreonthislater.Thus,wecanmodel potential lossesinafuel cell asfollows:Potentialofanodecatalystlayer(ACL),a;potentialofcathodecatalystlayer(CCL),bc=0Overpotentialsforanodeandcathodenaandn'respectively(yellowshadedareas);EjossduetocurrentdrawnMembranepotentialm=flux (currentdensity)of protons xmass transport resistance,jxRmtOverall,potential lostisthesumofthesethreeterms:=VeellEloss=na+ jRmt + ncViaAndtherealcell voltageavailableissimply:ne/0boEcell=Eoc-ElossACLmembraneCCLof=0Distanceacrossthecell
PotenGals in a Fuel Cell • Because each catalyst is not perfect, there is an extra potenGal “cost”, the overpoten:al, η (greek symbol “eta”) – more on this later. • Thus, we can model potenGal losses in a fuel cell as follows: – Poten:al of anode catalyst layer (ACL), ψa; potenGal of cathode catalyst layer (CCL),ψc = 0 – OverpotenGals for anode and cathode ηa and ηc respecGvely (yellow shaded areas); – Eloss due to current drawn – Membrane potenGal ψ’m = flux (current density) of protons x mass transport resistance, j×Rmt • Overall, potenGal lost is the sum of these three terms: Eloss = ηa + jRmt + ηc • And the real cell voltage available is simply: Ecell = EOC – Eloss