How batteries work Chapter 3 Cell voltage

Chapter 3

Cell voltage: electrode potentials, the electrical double layer and the electrolyte stability window

Why does a lithium-ion cell give over 3.6 V and a torch cell 1.5 V?

Where we are. Chapter 1 measured 1.10 V between zinc and copper and showed that it sits in two jumps, one at each metal|liquid boundary. Chapter 2 named the parts. Now the physics of a jump: what a potential means for the electrons in a metal, what happens at the boundary, and why the two electrodes of your cell sit where they do on a ladder.

Figure 3.1 Bar chart of the nominal voltages of common battery systems, primary and rechargeable, from 1.2 volts for zinc-air and the nickel cells to 4.0 volts for lithium-ion; a dot marks whether the electrolyte is water-based or nonaqueous.
Different chemistries, different voltages. Try this: hover or tap a bar to see the two electrodes and the electrolyte inside; then find the highest bar with a green dot.
SourceData: Winter and Brodd 2004, Table 2, as printed in 2004; the aqueous versus nonaqueous marking reads the electrolyte column of the same table.

The same four parts, in different materials, give very different cells. The table behind this chart lists the systems on the market in 2004: zinc and manganese dioxide in the 1.5 V cells of a torch, lead and lead oxide in the 2.0 V cells of a car battery, and graphite with lithium cobalt oxide in the 4.0 V cells of a phone.1

Two patterns are already visible: every system with a water-based electrolyte stays at or below 2 V, and every system above 2 V is built on lithium with a nonaqueous electrolyte.1 Where the voltage comes from, and why water sets a ceiling, is the next question.

Figure 3.2 An electrode with its filled electron states up to a Fermi level, beside a species in solution with an empty orbital above and a filled orbital below. Sliding the electrode potential moves the Fermi level: more negative raises the electrons until they can jump into the empty orbital, a reduction; more positive lowers them until electrons from the filled orbital jump onto the metal, an oxidation.
electronorbitals of the species
Potential as electron energy. Moving an electrode’s potential moves the energy of its electrons, one electron-volt per volt, the other way. Try this: slide slowly negative until the first electron jumps, note the potential, then slide positive until one jumps back.
SourceAfter Bard, Faulkner and White 2022, section 1.1.4 and Figure 1.1.3 (reduction when the electrode’s electrons reach a vacant orbital, oxidation when a filled orbital finds a lower energy on the electrode; 1 eV per volt) and section 2.2.5 (the Fermi energy as the energy of transferable electrons). The positions of the two orbitals, −0.6 and +0.6 V from the couple’s standard potential, are illustrative.

In chapter 0 a potential was energy per unit charge. For an electrode that has a sharp meaning: to observe or control the potential of an electrode is to observe or control the energy of its transferable electrons.2 Drive an electrode to a more negative potential and the energy of its electrons is raised; for each volt, each electron gains one electron-volt, 96.5 kJ per mole of electrons, comparable to the energies of chemical bonds.2

The energy of those transferable electrons has a name: the Fermi energy, or Fermi level. In a metal the electron states form a continuum, filled from the bottom up; the Fermi level marks the division between filled and empty, and because empty states lie just above it, an electron can move with almost no activation, which is what makes a metal conduct.2

Raise the Fermi level above an empty orbital of a species in the electrolyte and electrons flow from the electrode into the solution, a reduction current; lower it below a filled orbital and electrons flow from the solution onto the electrode, an oxidation current.2

This is the same fact as the potential ladder further down: a more negative electrode holds its electrons at a higher energy, so it is the stronger reducing agent. Lithium, at the bottom of every ladder, holds them highest of all.

Electrode potential is electron energy in disguise: more negative means higher-energy electrons, one electron-volt per volt.

Figure 3.3 A metal in contact with an electrolyte. Excess charge on the metal sits in a layer less than a nanometre thick at its surface; ions of the opposite sign gather in the liquid, first a compact layer one solvent molecule out and then a diffuse layer a few nanometres deep; a strip below shows the potential flat in the metal, dropping steeply across the compact layer, tailing off through the diffuse layer and flat again in the bulk.
negative chargepositive chargesolvent shellpotential
Inside one jump: the electrical double layer. Try this: press “Replay the formation” and watch the charge separate and the jump grow with it; then slide the charge to positive and watch the ions in the liquid swap sign.
SourceStructure and scales from Bard, Faulkner and White 2022, sections 1.6.2 and 1.6.3 (electrode charge in a layer under 1 nm, diffuse layer under about 10 nm above 0.01 M, capacitance 10 to 40 µF/cm2) and Figure 1.6.3b (the potential profile); equal and opposite charges from section 2.2.2; field up to 107 V/cm from 2.2.3; the Helmholtz capacitor from 14.3.1. Double-layer time constant about 10−8 s from Winter and Brodd 2004, section 4.2. Ion positions are random; the profile shape is schematic.

Now zoom into one of the jumps of figure 1.1. Put a metal into an electrolyte and the two phases exchange a little charge until their electron energies match; equality of the Fermi energies of metal and solution is the condition for equilibrium, and electrons flow from the phase with the higher Fermi energy to the one with the lower until it is reached.2 The amount of charge that moves is tiny, so the composition of the solution does not noticeably change.2

Here is how that field is built, step by step. Dip a piece of zinc into a solution of its own ions. A zinc ion sitting in the metal and a zinc ion dissolved in the liquid are not in the same state: their electrochemical potentials differ, and a species moves across a boundary in the direction that lowers its electrochemical potential until it is the same on both sides.2

So a few Zn2+ ions leave the metal and enter the liquid, and each leaves its two electrons behind. Now the metal is slightly negative and the liquid touching it slightly positive. Those separated charges are the field: the excess electrons sit on the metal’s surface, the excess cations gather in the liquid right against it, and between two sheets of opposite charge a molecular distance apart runs a strong field, pointing from the liquid into the metal. Every ion that crosses makes that field a little stronger, and the field pulls back on the next ion that tries; the crossing stops when the electrical pull exactly balances the chemical push, which is the equilibrium of equal electrochemical potentials.2

At a copper electrode in copper sulfate the exchange runs the other way, ions from the liquid plating onto the metal and leaving it positive. Where the exchange is of electrons rather than ions, as at a platinum wire in a solution of iron ions, the rule is the same: electrons cross from the phase with the higher Fermi energy to the one with the lower until the two are equal.2

In every case the amount of charge that crosses is tiny, far too small to change the composition of the liquid, yet it sets up a potential difference of a volt or so across a few nanometres.2

That charge cannot spread through the metal: a conductor at rest keeps its excess charge on its surface (chapter 0), and on an electrode it lies in a layer thinner than a nanometre.2 The liquid answers with an equal and opposite charge right at the boundary, an excess of cations or of anions.2

The whole array, charged metal surface, oriented solvent molecules, a compact layer of ions one solvent shell out and a diffuse layer beyond, is the electrical double layer.2 Above 0.01 M the diffuse part is less than about 10 nm thick.2

Why does the field stay inside that thin layer and not reach out into the liquid? Inside the metal it is zero for the reason chapter 0 gave: any field in a conductor at rest would move the free charges until it was gone.2 On the liquid side the Coulomb field of the metal’s surface charge is almost entirely cancelled by the polarization of the electrolyte: ions of the opposite sign crowd in and solvent molecules turn, and a few nanometres out the field has been screened to nothing, so the bulk of the liquid sits at one potential.2

That is why the strip under figure 1.1 is flat through the solution at open circuit. The liquid can even hold a potential step with no excess charge at all: water molecules touching a metal tend to line up, and a layer of aligned dipoles is itself a charge separation.2

Draw current and one thing changes. A gentle field appears through the bulk liquid, the slope of the potential that drives the ions, and it is small because the electrolyte conducts well; this is the ohmic drop.2 The two interfacial fields remain, shifted a little from their rest values because each reaction needs an extra push, an overpotential, to run at a finite rate.2

So the field in a working cell has three parts: a strong, nanometre-thin one at each electrode, built by the chemistry itself out of a minute charge separation and held in place by the electrolyte’s screening, and a weak one across the liquid that exists only while current flows.

Two sheets of opposite charge a molecular distance apart: the potential steps across them, and the field inside the step can reach 107 V/cm, strong enough to distort the molecules that sit in it.2 The arrangement behaves like a capacitor, q = C·E, with 10 to 40 µF for every square centimetre of electrode, and it forms and relaxes in about 10−8 s.1,2 A cell’s voltage is the sum of such steps, taken along the path from one electrode to the other.2

A cell’s voltage lives in two double layers, each a few nanometres thick, built by a tiny charge separation that the chemistry itself creates.

Figure 3.4 A ladder of electrode potentials against lithium, negative-electrode materials on the left and positive-electrode materials on the right, the band in which carbonate electrolytes are stable in the middle, and, when switched on, the window in which water is stable, 1.23 volts wide, sitting between 3.05 and 4.27 volts versus lithium at pH 0 and sliding down with pH.
The potential ladder. Pick a negative and a positive electrode and read the gap. Your choice is remembered and carried into the next modules. Try this: guess which pair gives the most voltage before you choose; then switch on the water window and see how many rungs a water-based cell can never reach.
SourceRung values from Goodenough and Park 2013 (graphite about 0.2, alloys 0.2 to 0.8, lithium titanate 1.5, TiS2 about 2.2, LiFePO4 3.5, LiCoO2 about 4.0, LiNi0.5Mn1.5O4 about 4.75 V) and Tarascon and Armand 2001 (sulfur 2.4 V, a lithium-sulfur conversion electrode, not an insertion host); carbonate window from Goodenough and Park 2013. Water window computed from Bard, Faulkner and White 2022: Li+/Li at −3.045 V and O2/H2O at 1.229 V vs NHE (Table C.1), lines E = 0 − 0.059 pH and E = 1.229 − 0.059 pH (eqs. 2.1.66 and 2.1.67), so 3.045 to 4.274 V vs Li at pH 0. Thermodynamic limits; kinetics widen the practical window to about 2 V (Winter and Brodd 2004). Values are approximate operating potentials; real cells give sloping curves, the subject of the next module to come.

Chemists carry the same idea one step further. The energy of a charged species in a phase has two parts: a chemical part, from its surroundings on the scale of a molecule, and an electrical part, its charge times the local potential. Their sum is the electrochemical potential, μ̄ = μ + zFφ, and a species is at equilibrium between two phases when its electrochemical potential is the same in both.2

Apply that to the electrons in the two electrodes of a cell and the open-circuit voltage falls out: it is the difference between the electrochemical potentials of the electrons in the two electrodes, divided by the electron charge, VOC = (μA − μC)/e.2,3

So picture each material as a rung on a ladder of electron energy, drawn as a potential: the cell voltage is the gap between the two rungs you choose. Battery scientists measure the rungs against lithium metal, which is why every value here reads “versus Li/Li+”. Lithium sits at the bottom. In Tarascon and Armand’s words it is “the most electropositive” metal, the one with the most negative standard potential, −3.04 V against the standard hydrogen electrode, and it is also the lightest metal, at 6.94 g/mol and 0.53 g/cm3.4

The gap has a ceiling, and the ceiling is the electrolyte. Water is thermodynamically stable over only 1.23 V, between the potential at which it gives off hydrogen and the potential at which it gives off oxygen;2 slow kinetics stretch that to about 2 V in practice,1 which is why cells with an aqueous electrolyte and a long shelf life stop near 1.5 V.3

But where that 1.23 V sits matters as much as its width: at pH 0 it runs from 3.05 to 4.27 V versus lithium, and it slides 59 mV lower for every pH unit.2 Every rung below about 3 V, which is every negative electrode on the ladder and most of the positive ones, is simply out of reach for water. The organic carbonate electrolytes of lithium-ion cells are stable from about 1.1 V to about 4.3 V versus lithium,3 about 4.6 V in practice,1 and there is the answer to the hook: not a stronger reaction, but an electrolyte that tolerates a wider gap, in the right place.

Two reservoirs at different heights: the pressure at the pipe between them depends on the height difference, not on how wide the reservoirs are. The picture breaks as soon as water flows: the height stays, but the flow costs pressure. The next module to come is about that cost.

Voltage is a property of the pair of materials, and the electrolyte sets how far apart they may be.

Figure 3.5 Electron energy levels of the negative electrode, the electrolyte window between its lowest empty and highest filled levels, and the positive electrode. An electron leaves the negative electrode, travels along a wire drawn above the cell and drops onto the positive electrode, releasing the cell voltage times its charge; a lithium ion crosses the electrolyte inside at the same time.
electronour ionelectrolyte window
The same gap as a picture of electron energy. The figure follows your cell from figure 3.4. Try this: choose lithium titanate in figure 3.4 and watch the warning on the left vanish; then choose the 4.75 V spinel and watch one appear on the right.
SourceAfter Goodenough and Park 2013, Figure 2a and 2c, and their eq. 6: carbonate electrolyte with the LUMO about 1.1 eV and the HOMO about 4.3 eV below the lithium level. The wire is added so that the electron’s path is drawn correctly: through the external circuit, never through the electrolyte (Goodenough and Park 2013, Electrochemical Cells).

The ceiling has a mechanism. An electrolyte is a collection of molecules, and each has a highest filled and a lowest empty energy level; the gap between them is the electrolyte window. A negative electrode whose electrons sit above the empty level will reduce the electrolyte, and a positive electrode whose empty states sit below the filled level will oxidize it, unless a passivating layer forms at the surface and blocks the reaction.3

Graphite sits about 0.2 V above lithium, outside the carbonate window, so a layer must form on it during the first charge; lithium titanate at 1.5 V sits inside the window and needs none.3 A later module follows that layer as it forms.

Notice the path of the electron in the figure. It gives up its e·VOC only by travelling round the external circuit, from the negative electrode to the positive one; it never crosses the electrolyte, which has no free electrons (chapter 1). Inside the cell it is our ion that crosses, keeping every region neutral.2,3

Figure 3.6 Energy bands of a sulfide and an oxide host below the lithium level. The top of the sulfur band lies about 2.5 electron-volts below lithium and the top of the oxygen band about 4.0 electron-volts below; the transition-metal redox couple is pinned at those tops, so the sulfide gives about 2.5 volts and the oxide about 4 volts. The electrolyte's filled level at 4.3 electron-volts is drawn as a dashed line.
Why oxide hosts give 4 V and sulfide hosts about 2.5 V. The redox couple of the transition metal can be lowered only as far as the top of the anion band. Try this: step through the four callouts and check that the oxide’s 4 V sits just above the electrolyte’s 4.3 eV line.
SourceNumbers and the pinning argument from Goodenough and Park 2013 (text on the anion-p bands and Figure 2c); the band as a continuum of electron levels from Bard, Faulkner and White 2022, section 20.1.1. Schematic: band widths are not to scale.

Why do oxide positive electrodes reach 4 V when the sulfide of the first rechargeable lithium cell reached only 2.2 V? In a solid the electron levels of the many atoms merge into bands, continuous ranges of energy, filled from the bottom up.2 The redox couple of the host’s transition metal cannot be pushed below the top of the host’s filled anion band. In the layered sulfides that top sits about 2.5 eV below the lithium level; in the layered oxides it sits about 4.0 eV below, which is why oxide hosts are used in today’s lithium-ion cells.3 Polyanions such as phosphate pull the band lower still through what chemists call the inductive effect, which is how LiFePO4 reaches 3.5 V.3,4

Figure 3.7 A bar whose length is the electrical energy available per mole of reaction, n F E, on a scale from 0 to 500 kilojoules per mole, with the Daniell cell marked at 213.
The same gap in the language of thermodynamics. Try this: double the voltage and double the electrons, and watch which one the energy bar follows; then set n = 2, E = 1.10 V and find the Daniell mark.
SourceComputed from ΔG = −nFE (Bard, Faulkner and White 2022, eq. 2.1.25; Winter and Brodd 2004, eq. 3) with F = 96 485.3 C/mol; the Daniell value uses E = 1.103 V from Table C.1.

In the language of thermodynamics the same gap reads ΔG = −nFE: the free energy of a reaction, the energy it can deliver as work, equals the charge it moves per mole, nF, times the cell voltage, with the minus sign saying that a spontaneous reaction has a positive voltage.1,2

So the voltage is the reaction’s free energy per unit of charge, and nF, the total charge available, is its capacity factor.1 The Faraday constant is the charge carried by one mole of electrons, 96 485.3 coulombs.1,2

Worked example. The Daniell cell of chapter 1, Zn + Cu2+ → Zn2+ + Cu, moves two electrons per zinc atom at 1.103 V. ΔG = −nFE = −2 × 96 485 C/mol × 1.103 V = −213 kJ/mol per electron: 1.103 eV; per zinc atom: 2.21 eV That 213 kJ per mole of zinc is what the lamp of figure 1.1 can get, at most, from 65 g of zinc; chapter 4 turns “per mole” into “per gram” and “per hour”.

Where does ΔG itself come from? Writing ΔG = ΔH − TΔS shows that the open-circuit voltage, V = −(ΔH − TΔS)/nF, depends on temperature through the entropy term.5 For an insertion electrode the pieces have a physical reading. With ΔH = ΔU + PΔV, the terms PΔV and TΔS are the change in volume and the change in configurational entropy as lithium enters or leaves the host. Both are small next to the change in internal energy ΔU, so calculations made for zero kelvin take ΔG ≈ ΔU, and the average voltage between two lithium contents x1 and x2 of a host MO2 follows from three total energies, those of Lix2MO2, of Lix1MO2 and of lithium metal, written Gx2, Gx1 and GLi:5

V¯=−Gx2−Gx1−nGLinF

Here n = x2 − x1 is the amount of lithium moved. This is how voltages are predicted from first principles, after Aydinol and co-workers. At any one lithium content the voltage is the difference in the chemical potential of lithium between the two electrodes, V(x)=−[μC(x)−μA(x)]/zF, and with no current flowing it is the open-circuit voltage.5

You can now explain why a more negative electrode is the stronger reducing agent, where a cell’s voltage physically sits, why lithium cells have the highest voltages, why water caps a cell near 1.5 V and where on the lithium scale that cap lies, why oxide hosts beat sulfides, and what ΔG = −nFE means.

You have chosen a voltage. How much charge can your cell hold?

Go deeper: the electrochemical potential, from two metals in contact to the cell voltage

For electrons in a metal the electrochemical potential is μ̄e = μ°e − Fφ, with no activity term because the electron concentration never changes appreciably.2 Touch a copper wire to a zinc electrode and electrons move between them until their electrochemical potentials are equal, which leaves a potential difference φZn − φCu = (μ°eZn − μ°eCu)/F between the two metals: a contact potential, one of the steps in figure 1.1’s profile.2 Write the equilibrium of the whole cell reaction in electrochemical potentials and the terms in φ cancel everywhere except at the two copper leads, giving −nFE = n(μ̄eCu′ − μ̄eCu) and, on expanding the chemical potentials, the Nernst equation.2 Per electron, the same statement is Goodenough and Park’s VOC = (μA − μC)/e.3

A potential is always a difference, so every value needs a reference. Textbook tables use the standard hydrogen electrode, against which lithium sits at −3.045 V;2 battery papers use lithium metal itself as the zero, which is why the same LiCoO2 electrode reads about 4.0 V “versus Li/Li+” and about 1 V versus hydrogen.3,4 In a lithium-ion cell the negative electrode is not lithium metal but a host that holds lithium near lithium’s potential, so the cell voltage is a little less than the positive electrode’s ladder value: graphite costs about 0.2 V.3

The mathematics of this chapter

μ̄j = μj + zjFφthe electrochemical potential of species j: its chemical potential plus its charge times the local potential (per mole)

μj = μ°j + RT ln ajthe chemical potential in terms of the activity aj (roughly, concentration relative to a standard state)

μ̄jα = μ̄jβequilibrium of species j between phases α and β; the condition that stops the charge transfer described in figure 3.3

μ̄e = μ°e − Fφ, EF = μ̄e / NAelectrons in a metal (no activity term: their concentration never changes appreciably); the Fermi energy is the electrochemical potential of the electrons, per electron (figure 3.2). Making φ more negative by 1 V raises EF by 1 eV.

φZn − φCu = (μ°eZn − μ°eCu) / Ftwo metals in contact exchange electrons until μ̄e is equal: the contact potential

−nFE = n(μ̄eCu′ − μ̄eCu)write the equilibrium of the whole cell reaction in electrochemical potentials; the φ terms cancel everywhere except at the two copper leads, which is why only the whole-cell voltage is measurable

E = E° − (RT/nF) ln (aproducts / areactants)expanding the chemical potentials gives the Nernst equation; at 25 °C, RT/F = 25.7 mV, so a tenfold change in an activity ratio shifts a one-electron E by 59 mV

VOC = (μA − μC) / ethe same statement per electron, in Goodenough and Park’s notation: the open-circuit voltage is the difference in electron energy of the two electrodes (figures 3.4 and 3.5)

E = E° − 0.059 (q/n) pHa half-reaction with q protons per n electrons shifts 59 mV per pH unit; for water, E = 0 − 0.059 pH (hydrogen) and E = 1.229 − 0.059 pH (oxygen), so the window is 1.229 V wide at every pH and sits at 3.045 to 4.274 V versus lithium at pH 0 (figure 3.4)

ΔG = −nFE, ΔS = nF(∂E/∂T)P, ΔH = ΔG + TΔSthe thermodynamics of the cell reaction (figure 3.7); the temperature coefficient of the voltage gives the entropy change, which a later module turns into reversible heat

q = C·E, CH = εε0/dthe double layer as a capacitor (figure 3.3); in the Helmholtz picture two sheets of charge a distance d apart in a medium of dielectric constant ε; measured values 10 to 40 µF/cm2

References for this chapter

Numbers in the text refer to this list. All references of the course, with the chapters that use each one.

  1. 1 M. Winter, R. J. Brodd, "What Are Batteries, Fuel Cells, and Supercapacitors?", Chemical Reviews 104, 2004, 4245-4269. doi:10.1021/cr020730k
  2. 2 A. J. Bard, L. R. Faulkner, H. S. White, "Electrochemical Methods: Fundamentals and Applications", Wiley, 3rd edition, 2022.
  3. 3 J. B. Goodenough, K.-S. Park, "The Li-Ion Rechargeable Battery: A Perspective", Journal of the American Chemical Society 135, 2013, 1167-1176. doi:10.1021/ja3091438
  4. 4 J.-M. Tarascon, M. Armand, "Issues and challenges facing rechargeable lithium batteries", Nature 414, 2001, 359-367. doi:10.1038/35104644
  5. 5 S. A. Arote, "Fundamentals and perspectives of lithium-ion batteries", chapter 1 of Lithium-ion and Lithium–Sulfur Batteries: Fundamentals to Performance, IOP Publishing, 2022, pp. 1-1 to 1-26. doi:10.1088/978-0-7503-4881-2ch1

Cite this page

Using this page in teaching, a thesis or an article? Cite it as follows; the date is the last update of the course.

Shyma Sajeevan, A. (2026). Cell voltage: electrode potentials, the electrical double layer and the electrolyte stability window. How batteries work. https://asajeevan.github.io/batteries/cell-voltage/

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