VisualChem The interactive chemistry textbook
中文

Lesson 7 · 30 min

Batteries: chemistry that stores charge

Voltage comes from Gibbs energy and capacity from Faraday’s law: compute the theoretical capacities of graphite, LiCoO₂ and LiFePO₄ from their formulas, see how a lead–acid battery consumes its acid, and explain the flat discharge plateau of LiFePO₄ with a lattice-gas model.

In this lesson

  • Compute the theoretical specific capacity of an electrode material from its formula and the Faraday constant, and estimate specific energy from capacity and voltage
  • Explain the voltage of a lead–acid battery from standard potentials and compute the sulfuric acid a discharge consumes with Faraday’s law
  • Explain the insertion reactions of a lithium-ion cell, and why a positive electrode with two coexisting phases gives a flat voltage plateau

The batteries in phones, electric cars and grid storage are all galvanic cells that can be run back and forth: on discharge the negative electrode is oxidised and the positive one reduced, and electrons flow through the outer circuit doing work; on charge an applied voltage drives the reactions backwards — an electrolysis.

Chemists judge a battery by three numbers: its voltage (set by the Gibbs energy difference of the electrode reactions), its capacity (how many electrons each gram of electrode can exchange, set by Faraday’s law) and their product, the energy. The 2019 Nobel Prize in Chemistry went to J. B. Goodenough, M. S. Whittingham and A. Yoshino for developing the lithium-ion battery; this lesson starts from those three numbers.

Predict first

Per gram of electrode, which stores more charge: graphite (one Li per six carbons, becoming LiC₆) or lithium cobalt oxide, LiCoO₂ (one Li per formula unit)?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

Capacity: Faraday’s law again

Battery capacity is usually quoted in milliampere-hours: 1 mAh = 3.6 C. A mole of formula units exchanging z electrons stores zF coulombs; divide by the molar mass and you have the theoretical specific capacity per gram. It depends only on the formula — the ceiling for that material.

The voltage comes from the Gibbs energy difference of the electrode reactions: ΔG = −zFE. Capacity times voltage is the stored energy (1 mAh × 1 V = 3.6 J; 1 Wh = 3600 J). Both electrodes weigh something, so the capacity of a pair combines like resistors in parallel: 1/q = 1/q₋ + 1/q₊.

q=zF3.6 M mAh g−1,1qcell=1q−+1q+,w=E qcellq = \frac{zF}{3.6\,M}\ \mathrm{mAh\,g^{-1}},\qquad \frac{1}{q_\text{cell}} = \frac{1}{q_-} + \frac{1}{q_+},\qquad w = E\,q_\text{cell}
M in g/mol; w in Wh/kg (= mWh/g), counting only the active materials.
qq
theoretical specific capacity
zz
electrons exchanged per formula unit
EE
cell voltage
ww
theoretical specific energy
Capacities from the formula and the Faraday constant · potentials are literature averages

Build a lithium-ion cell

Negative electrode
Positive electrode
Potential and capacity of each material
3.35 V LiLiC₆Li₄Ti₅O₁₂LiFePO₄LiCoO₂LiCoO₂ ½100200500100020005000theoretical capacity / (mAh/g)01234E / V (against Li⁺/Li)
The cell (sketch)
negativepositiveseparator and electrolyte

Voltage ≈ 3.35 V; negative 372 mAh/g, positive 170 mAh/g, both together 117 mAh/g; theoretical specific energy of the active materials 391 Wh/kg.

A real cell also carries current collectors, separator, electrolyte and casing, so a whole cell stores clearly less energy per kilogram than its active materials alone.

Theoretical specific capacity = zF/(3.6 M) mAh/g, with z electrons (lithium ions) per formula unit and molar mass M. Graphite is counted per C₆ (one Li per six carbons); LiCoO₂ gives up only about half of its lithium in practical cells, so it is listed twice. Average potentials are representative literature values: graphite about 0.1 V (Dahn 1991), Li₄Ti₅O₁₂ 1.55 V (Ohzuku et al. 1995), LiFePO₄ about 3.45 V (Padhi et al. 1997), LiCoO₂ about 3.9 V (Mizushima et al. 1980). Lithium positions in the cell sketch come from a deterministic hash.
Swap the electrodes: graphite with LiFePO₄ gives about 3.35 V and 391 Wh/kg (active materials only); a lithium titanate negative drops the voltage to 1.9 V and the energy to 164 Wh/kg, but its volume hardly changes as lithium goes in and out, so it lasts for a very large number of cycles. Drag the state of charge to watch the lithium ions move between the electrodes.

Worked example

The theoretical capacity of LiFePO₄

On discharge each formula unit of LiFePO₄ takes up one Li⁺ and one electron (Fe³⁺ → Fe²⁺). Find its theoretical specific capacity.

The lead–acid battery: the acid is a reactant

The lead–acid battery that starts a car has a negative electrode of spongy lead and a positive one of PbO₂, both in sulfuric acid. On discharge both electrodes turn into PbSO₄:

Pb+SOX4X2−→PbSOX4+2 eX−PbOX2+SOX4X2−+4 HX++2 eX−→PbSOX4+2 HX2O\ce{Pb + SO4^2- -> PbSO4 + 2e-}\qquad\ce{PbO2 + SO4^2- + 4H+ + 2e- -> PbSO4 + 2H2O}

From the standard potentials, E° = 1.6913 − (−0.3588) = 2.050 V, and six cells in series make 12 V. The overall reaction Pb + PbO₂ + 2H₂SO₄ → 2PbSO₄ + 2H₂O shows that every electron transferred uses up one H₂SO₄ and makes one H₂O — the deeper the discharge, the more dilute and less dense the acid. A hydrometer reading of the electrolyte density therefore tells the state of charge.

Try the calculation

One cell of a lead–acid battery delivers 60 Ah. How many grams of H₂SO₄ (M = 98.08 g/mol) does it consume?

g

The lithium-ion cell: lithium in a rocking chair

There is no lithium metal in a lithium-ion cell. On charge, lithium ions leave the positive electrode (between the layers of LiCoO₂, for example), cross the electrolyte and slip between the layers of graphite; on discharge they go back — hence the name “rocking-chair battery”. The host lattices stay essentially intact and only the lithium content x changes: an insertion (intercalation) reaction.

The shape of the discharge curve reveals the phases in the electrode. If lithium spreads evenly through the host (one phase), the voltage falls smoothly as x grows; if inserted ions prefer to cluster, the material splits into a lithium-poor and a lithium-rich phase, and discharge simply grows the rich phase — while the two coexist the chemical potential of lithium cannot change, so the voltage is a plateau. That is the phase rule: two components, two phases, fixed temperature and pressure, no freedom left. It is why LiFePO₄ discharges at such a flat voltage.

Half-cell against lithium · lattice-gas (regular-solution) model

The discharge curve of an insertion electrode

Lithium sites in the host lattice
Open-circuit voltage against lithium content
00.250.50.751x (LiₓHost)3.13.23.33.43.53.63.73.8E / V (against Li⁺/Li)
  • equilibrium curve
  • if forced to stay one phase

x = 0.50: E = 3.450 V. Two phases: a lithium-poor phase (x = 0.012) coexists with a lithium-rich one (x = 0.988), and the rich domains grow as the cell discharges.

While two phases coexist, lithium has the same chemical potential in both and it does not change with x (the phase rule: at fixed temperature and pressure a two-phase, two-component system has no freedom left), so the voltage is a plateau.

Model: a lattice gas on the lithium sites, E = E° − (RT/F)[ln(x/(1 − x)) + (Ω/RT)(1 − 2x)]. For Ω > 2RT the uniform phase is unstable; the common-tangent (equal-area) construction gives the plateau, whose ends satisfy ln[x/(1 − x)] + (Ω/RT)(1 − 2x) = 0. E° and Ω are parameters set by hand: the defaults make the curve resemble LiFePO₄ (a plateau near 3.45 V), while Ω = 0 with E° = 3.9 V gives a sloping curve like LiCoO₂; real materials add ordering, staging and other details. Lithium positions in the lattice sketch come from a deterministic hash and are only illustrative.
Lower Ω slowly from 4.5 RT to 0: once it drops below 2RT the plateau disappears and the curve becomes the smooth slope of a single phase. The grey dashed line is the voltage if the material were forced to stay one phase; it falls and then rises across the two-phase region — such states are unstable.

Concept check

Why does a LiFePO₄ cell hold an almost constant voltage for most of its discharge?

Key takeaways

  • Theoretical specific capacity q = zF/(3.6 M) mAh/g: graphite 372, LiCoO₂ 274 (about 137 in practice), LiFePO₄ 170, Li₄Ti₅O₁₂ 175; for a pair of electrodes 1/q = 1/q₋ + 1/q₊, and specific energy = voltage × capacity.
  • The lead–acid cell has E° = 2.050 V; each electron uses one H₂SO₄ — about 3.66 g per ampere-hour — so the electrolyte grows less dense as it discharges.
  • A lithium-ion cell rocks lithium between two insertion electrodes; a single-phase electrode gives a voltage that varies smoothly with x, and a two-phase one such as LiFePO₄ gives a flat plateau, as the phase rule requires.
Progress is stored only in this browser and never uploaded.

Data and references

  1. J. R. Dahn, “Phase diagram of LiₓC₆”, Phys. Rev. B 44, 9170–9177 (1991).
  2. K. Mizushima, P. C. Jones, P. J. Wiseman, J. B. Goodenough, “LiₓCoO₂ (0 < x ≤ 1): a new cathode material for batteries of high energy density”, Mater. Res. Bull. 15, 783–789 (1980).
  3. A. K. Padhi, K. S. Nanjundaswamy, J. B. Goodenough, “Phospho-olivines as positive-electrode materials for rechargeable lithium batteries”, J. Electrochem. Soc. 144, 1188–1194 (1997).
  4. T. Ohzuku, A. Ueda, N. Yamamoto, “Zero-strain insertion material of Li[Li₁/₃Ti₅/₃]O₄ for rechargeable lithium cells”, J. Electrochem. Soc. 142, 1431–1435 (1995).
  5. P. Vanýsek, “Electrochemical Series”, in CRC Handbook of Chemistry and Physics, 91st ed. (2010–2011): PbO₂/PbSO₄ and PbSO₄/Pb potentials.