II. Equilibrium Thermodynamics Lecture 9: Fuel Cells and Lead-Acid Batteries MIT Student (and MZB) We’re going to calculate the open circuit voltage of two types of elec-trochemical system: polymer electrolyte membrane (PEM) fuel cells and lead-acid batteries. To do this, we’re going to make use of two equations from the last lecture. The first is the Nernst equation, which describes the potential difference between the electrode and electrolyte in the half-cell reaction � siMzi i → ne− . (1) i The potential difference, which is always defined to be the potential of the electrode minus the potential of the electrolyte, is Δφ =Δφ◦ − kB T ln(� a si i ). (2) ne i Δφ◦ is the potential difference of this half-cell reaction in a particular refer-ence state. The values of the activity ai, are then scaled so that the activity of species i in the reference state is 1. Reference states are set to be 1 in convenient units, such as 1 atmosphere of pressure for gases. The second equation is for open circuit voltage, and comes from combin-ing the Nernst equation at each electrode. The total open circuit voltage of a cell is VO =Δφc − Δφa sj =Δφ◦ − Δφ◦ − kB T ln ��cathode aj . (3)c a sine anode ai This describes the open circuit voltage in terms of the half-cell reactions at the cathode and anode, but we can also describe it in terms of the full cell 1� � Lecture 8: Fuel cells and lead-acid batteries 10.626 (2011) Bazant reaction, siRi → sj Pj . (4) i j Since the reaction at the anode is going forwards in order to produce elec-trons, si for reactants in the half-cell reaction is the same as si in the full cell reaction, while the value of si for products is defined to be negative in the half-cell reaction but positive in the full cell reaction. The situation is the opposite for the cathode, because the half-cell reaction proceeds backwards, in order to accept electrons. This means that we can rewrite equation 3 in the form �sjkBT products ajVO = V ◦ − ln �, (5)sine areactants i using the stoichiometric coefficients defined for the full cell reaction. Here we have made the definition V ◦ =Δφ◦ − Δφ◦ , (6)c a the open circuit voltage of the reaction under reference conditions. 1 Polymer Electrolyte Membrane Fuel Cells 1.1 Hydrogen concentration cell A hydrogen concentration cell has a porous electrode situated in hydrogen gas for both the anode and the cathode, the difference being the pressure of the hydrogen. The two electrodes are separated by a polymer membrane that permits the flow of hydrogen ions but not hydrogen gas. The reactions are anode: H2 → 2H+ +2e− (7) cathode: 2H+ +2e− → H2 (8) net: H2(a) → H2(c). (9) Uθ = 0 for each reaction, so V ◦ = 0. The open circuit voltage is kB TpHa 2VO = ln c . (10)2e pH2 Here we have assumed that activity is proportional to pressure: we choose our reference state so that activity is equal to pressure measured in atmo-spheres. In this particular case we could measure pressure in other units, 2Lecture 8: Fuel cells and lead-acid batteries 10.626 (2011) Bazant e− PEM a H+ cp pH2 H2 Porous Porous anode cathode Figure 1: Hydrogen concentration cell because the units cancel when we take the ratio. To achieve a positive open a ccircuit voltage we need p >pc . As pH2 → p , open circuit voltage H2 H2 H2 drops to zero. This kind of cell will never give large voltage, because kBT/e is only 25 mV and logarithms never become particularly large. For example, a cell run between a pressure equal to that at the centre of the sun (around 1011 atm, mostly hydrogen) and the earth’s atmosphere (where hydrogen is present at around 1 ppm, i.e. a partial pressure of 10−6 atm) would only produce approximately 0.5 V. 1.2 Standard PEM fuel cell A standard PEM fuel cell has hydrogen at the anode and oxygen (converted to water) at the cathode. The reactions are anode: H2 → 2H+ +2e− Uaθ = 0 (11) cathode: 1O2 +2H+ +2e− H2O2 → cUθ =1.229 V (12) 1net: H2 + 2O2 → H2O V◦ =1.229 V. (13) 3Lecture 8: Fuel cells and lead-acid batteries 10.626 (2011) Bazant e− H2 O2 PEM H+ H2O Porous Porous anode cathode Figure 2: PEM fuel cell The open circuit voltage is kB T aH2OVO = V ◦ − ln 1/2 . (14)2eaH2 aO2 Liquid water is a reference state, so as long as water remains in contact with the cathode, aH2O will be 1. The activity of a gas is its pressure measured in atmospheres, so VO = V ◦ + kBT ln(pH2 p 1/2). (15)2e O2 2 Lead-Acid Batteries The open circuit voltage of a battery is more complicated than that of a fuel cell. This is because it must depend on the state of charge of the battery, which determines how much of each reactant and product is present, and hence their activities. One example of a battery is the lead-acid battery, used in cars. The anode is lead metal and the cathode is lead oxide, with an electrolyte of sulfuric acid, approximately 6 M (one third H2SO4 by mass). This is very acidic (pH around 0), making battery acid potentially very 4Lecture 8: Fuel cells and lead-acid batteries 10.626 (2011) Bazant dangerous. This battery has reactions anode: Pb + SO24 − → PbSO4 +2e− Uaθ = −0.356 V (16) cathode: PbO2 +SO24 − +4H+ +2e− → PbSO4 +2H2O Ucθ =1.685 V (17) net: Pb + 2H2SO4 + PbO2 → 2PbSO4 +2H2O V ◦ =2.041 V. (18) The open circuit voltage is 2 2 VO = V◦ − kBT ln aPbSO 4 4 aH2O 2 . (19)2eaPbaPbO2 aH+ a SO42− The activity of water and of the solids is 1, so the open circuit voltage will depend on the activities of the hydrogen and sulfate ions. We’ll particularly focus on the activity of the hydrogen ion, because it can be involved in several other reactions including corrosion and electrolysis. To explore the effect of hydrogen concentration, we can set the activity of sulfate equal to 1. If its activity is something else, this will simply give some constant offset to the open circuit voltage. In a real battery, sulfate activity will vary, but this effect will be less important than the variation of hydrogen activity, so we’ll ignore it. The open circuit voltage is then 2kBT VO = V◦ + ln aH+ (20) e 2kBT = V◦ − ln 10(− log10 aH+ ). (21) e 2kBT/eln 10 is 0.12 V, while − log10 aH+ is the definition of pH. We often use − log10 cH+ as pH, where cH+ is the concentration of