How batteries work Chapter 4 Capacity and energy

Chapter 4

Capacity, energy and power: Faraday's law, the C-rate and the Ragone plot

What do mAh, Wh and Wh/kg mean, and how long would your cell run a lamp?

Where we are. Chapter 3 gave your cell a voltage: the gap between two rungs, which is the free energy of its reaction per unit of charge. Now the other factor: how much charge the materials hold. Charge times voltage is energy; energy per second is power.

Figure 4.1 Bar chart comparing the theoretical specific capacity of the chosen material with graphite, lithium iron phosphate and lithium metal, each labelled with the mass it is referred to.
The Faraday calculator. Pick a material, or type your own molar mass and electron count. Try this: pick lithium metal and see why it tempts every battery designer; then pick your positive electrode and multiply by your cell’s voltage.
SourceComputed from Faraday’s law as given in Winter and Brodd 2004, eq. 7, and Bard, Faulkner and White 2022, eq. 1.1.12, with F = 96 485.3 C/mol; molar masses from IUPAC atomic weights. Printed check values: graphite 372 mAh/g on the C6 basis (Tarascon and Armand 2001), LiFePO4 170 mAh/g (Goodenough and Park 2013). Wh/kg refers to the active material of the chosen electrode alone.

Capacity is a count of electrons. Faraday’s law ties the amount of material transformed to the charge that flows, Q = nFN for N moles of reactant, and it has “no known exceptions”.1,2 Turn it around: a gram of a material of molar mass M that gives up n electrons per formula unit holds nF/M coulombs, which the battery world writes in milliampere-hours per gram, Q = nF/(3.6 M). The Faraday constant is the charge of one mole of electrons, 96 485.3 C or 26.8015 Ah.2

Two conventions hide in that formula. Graphite’s 372 mAh/g refers to the mass of carbon, the host as assembled, not to the mass of LiC6: Tarascon and Armand give 372 mAh/g for the end compound LiC6 and about 350 mAh/g in practice.3 Positive-electrode materials are referred to their lithiated mass: LiFePO4 holds 170 mAh/g in theory4 and is used at about 90 % of it,3 while LiCoO2 is limited to about half of its lithium for safety, 140 mAh/g in practice.3

Worked example. Graphite, one lithium per six carbons, referred to the carbon. M(C6) = 6 × 12.011 = 72.07 g/mol, n = 1 Q = 1 × 96 485.3 C/mol ÷ (3.6 × 72.07 g/mol) = 372 mAh/g at 3.8 V: 372 mAh/g × 3.8 V = 1414 mWh/g = 1414 Wh/kg of graphite, graphite alone The same calculation on the mass of LiC6 (79.0 g/mol) gives 339 mAh/g. Both are right; they answer different questions.

Light elements and many electrons per formula unit give high capacity; high voltage turns capacity into energy.

Figure 4.2 A discharge curve of voltage against capacity with the area under it shaded; a higher rate lowers the curve, ends it earlier and shrinks the area; the capacity axis runs from 0 to 100 percent of the low-rate value and a dashed reference curve marks the low-rate discharge.
Energy is the area under the discharge curve. The curve follows your cell’s positive electrode: flat for LiFePO4, sloping for the others. Try this: slide the rate up to 5C and watch the area shrink from both sides, lower and shorter.
SourceEnergy = ∫V dq after Goodenough and Park 2013, eq. 5. The curve is a schematic model: an illustrative open-circuit curve minus ohmic, activation and concentration polarization (Winter and Brodd 2004, section 1.4); parameters are illustrative, not measured. “C/2” means a current that would empty the C/10 capacity in two hours.

Energy is the integral of the voltage over the charge delivered: on a discharge curve, the area under the line, which is also the average voltage times the capacity.4 Per kilogram it is called specific energy (Wh/kg), per litre energy density (Wh/L); how fast it can be delivered is specific power (W/kg) and power density (W/L).2

The capacity itself depends on how hard you pull. At high current the transfer of ions across the interfaces becomes diffusion-limited and the cell reaches less of its charge before the cut-off; that part of the loss is reversible and comes back at a gentler rate.4 The next module to come takes the curve apart.

Worked example. A cell of capacity 2 Ah discharged at an average of 3.6 V. energy = 3.6 V × 2 Ah = 7.2 Wh a 6 W lamp (P = IV, chapter 0) runs for 7.2 Wh ÷ 6 W = 1.2 hours, drawing 6 W / 3.6 V = 1.7 A At that current the curve would sag and the capacity would fall short of 2 Ah; the figure shows by how much, schematically.
Figure 4.3 A cell gauge filled to the state of charge, beside a discharge curve of voltage against depth of discharge. The upper and lower cut-off voltages bound the useful range, with overcharge above and over-discharge below; a dashed line marks 80 percent depth of discharge, a deep discharge, and another the nominal voltage. A marker shows where the cell is on the curve; buttons set the C-rate and the readout gives the time to empty.
State of charge, depth of discharge and the C-rate. Try this: set the state of charge to 20 % and read the depth of discharge; then switch from C/20 to 2C and watch the time left shrink.
SourceDefinitions of capacity, the C-rate and its examples, state of charge, depth of discharge, deep discharge, cut-off voltage, overcharge, over-discharge, shelf life and cycle life after Arote 2022, section 1.3.3 and section 1.4.5. The discharge curve is a schematic; the cut-off voltages are illustrative; the nominal voltage drawn is the average of this curve. The cell is a nominal 2.0 Ah.

The words on a datasheet

A datasheet uses a small vocabulary, and it is worth having it straight. Capacity is the charge a battery stores, in ampere-hours: a 10 Ah cell can supply 10 A for one hour, and since an ampere is a coulomb per second, 1 Ah is 3600 C.5 The C-rate scales the current to the capacity, relative to a one-hour discharge: at 1C a cell empties in an hour, at 2C in 30 minutes, at C/5 in five hours and at C/20 in twenty. The higher the C-rate, the shorter the discharge; some high-performance cells are charged and discharged above 1C with moderate stress.5

The state of charge (SOC) says how much of the capacity is left; the depth of discharge (DOD) says how much has been used, so the two add up to 100 %. A discharge to 80 % DOD is called a deep discharge, and for long cycle life and safety designers keep the depth of discharge small.5 The cut-off voltages bound the useful range: charging past the upper one is overcharge, which decomposes the electrolyte and releases gas, and discharging below the lower one is over-discharge.5

Two words describe time. The cycle life is the number of cycles a battery lasts at a given C-rate; by industry convention it is nearly over when the capacity has fallen to 80 % of its initial value (16.2). Because it depends so strongly on the depth of discharge, a cycle life means little unless the DOD is quoted with it; temperature, rate, cut-off voltages and storage conditions matter too.5 The shelf life is how long a battery can be stored without a load, and self-discharge, the slow loss of charge at open circuit through internal reactions, is what shortens it (chapter 16).5

Worked example. Electric vehicles may draw a continuous 20 to 400 A, from a single cell or from smaller cells in parallel.5 Take a 100 Ah cell. at 1C: 100 A for one hour rated at 5C for 10 s: 5 × 100 A = 500 A charge in that burst: 500 A × 10 s = 5000 C = 5000 ÷ 3600 Ah = 1.4 Ah A burst of 500 A takes only 1.4 % of the cell’s charge; what limits it is power, the subject of figure 4.4.
Figure 4.4 Power against current for a cell of 3.6 volts. A dashed straight line is the power the reaction releases, current times voltage; below it, a curve bends over and returns towards zero: the power delivered to the load, P = IV minus I squared R. The wedge between them, shaded red, is the heat in the cell’s internal resistance. A dot marks the peak of the curve, at a current of V over 2R; sliders set the internal resistance and the current.
power delivered, IV − I2Rheat in the cell, I2R
Power, and what the internal resistance costs. Try this: find the current at which the cell delivers the most power, then halve the resistance and find it again.
SourceP = IV − I2Rint, power density and what sets Rint after Arote 2022, section 1.4.5; the maximum at I = V/2R follows from setting dP/dI = 0. The voltage of 3.6 V and the range of resistance are illustrative, not a measured cell.

Power: how fast

Energy says how much work a battery can do; power says how fast it can do it.5 Power density is the greatest power a battery can deliver per kilogram (W/kg) or per litre (W/L), and it is measured by cycling the battery at different rates.5 The limit is the cell’s internal resistance. Of the power that the reaction releases, current times voltage, a part I2R is turned into heat inside the cell, so the power delivered is5

P=IV−I2Rint

which is simply the current times the terminal voltage, V − IR, of chapter 0. To get the most power, the internal resistance must be kept low. It depends on how well the electrolyte conducts lithium ions, how well the electrode materials conduct electrons, and how fast the insertion and extraction reactions run.5 So power is not only a matter of new materials: optimizing known ones, their conductivity, surface morphology and electrochemical, chemical and structural stability, can raise it too.5 Chapter 5 takes R apart into its pieces.

Worked example. The figure’s cell: V = 3.6 V, R = 30 mΩ. at 20 A: IR = 20 A × 0.030 Ω = 0.6 V, so the terminals give 3.0 V delivered: 20 A × 3.0 V = 60 W; heat: I2R = 400 A2 × 0.030 Ω = 12 W the peak: dP/dI = V − 2IR = 0, so I = V/2R = 60 A and P = V2/4R = 108 W At its peak the cell turns as much power into heat as it delivers: an efficiency of 50 %.
Figure 4.5 A Ragone map: supercapacitors high in power and low in energy, fuel cells high in energy and low in power, batteries in between, and the combustion engine beyond all three because its energy is stored in fuel. specific energy, Wh/kg (log scale, unnumbered) specific power, W/kg (log scale) supercapacitorshigh power, low energy batteriesintermediate in both fuel cellshigh energy, low power combustion engineenergy stored in fuel
The Ragone map: energy against power. Regions only; the axes carry no numbers because the source figure is itself simplified. Try this: find where a thin-film battery would sit, then read why the engine is off on its own.
SourceSchematic after Winter and Brodd 2004, Figure 3 and section 1.1. The combustion engine is included there for comparison only: it is not an electrochemical system and stores its energy in a fuel, not in electrodes.

To compare the two abilities, energy and power, engineers use the Ragone plot. On it, supercapacitors are high-power systems, fuel cells high-energy systems, and batteries sit between them, overlapping both: a battery with thin-film electrodes reaches the power of a supercapacitor.2 No single electrochemical system matches the combustion engine on both axes at once, which is why the review suggests combining them: batteries or supercapacitors for the power, fuel cells for the energy.2

One caution before you trust any Wh/kg figure: the theoretical energy of a chemistry, reckoned on the mass of everything that reacts, is never what a cell delivers. Practical values are significantly lower, and Winter and Brodd list among the causes the inert parts a cell needs (conductive diluents, current collectors, containers), the internal resistances of electrodes, electrolyte and contacts, and incomplete use of the active masses.2

Chapter 2 showed the inert parts; chapter 0 showed why resistance costs voltage, and therefore energy, whenever current flows. How large the gap is depends on the chemistry and on how the cell is built, so this page quotes delivered values only where a source gives them.

To give the numbers a feel, Winter and Brodd note that a specific energy of 30 Wh/kg means one kilogram of lead-acid battery can power a 60 W lamp for only half an hour.2 The first commercial lithium-ion cell of 1991 gave over 3.6 V and 120 to 150 Wh/kg, two to three times the usual nickel-cadmium cell of the day.3

Worked example. Theoretical against delivered, for that 1991 cell: graphite against LiCoO2, counting all the lithium, so one electron per LiCoO2 (97.87 g/mol) and one LiC6 host (72.07 g of carbon) per electron. active mass per mole of electrons = 97.87 + 72.07 = 169.9 g; Q = 26 801 mAh ÷ 169.9 g = 158 mAh/g theoretical energy at 3.8 V = 158 mAh/g × 3.8 V ≈ 600 Wh/kg of active material delivered by the 1991 cell: 120 to 150 Wh/kg, a fifth to a quarter of that The rest of the mass was copper, aluminium, separator, electrolyte, can and carbon, and part of the lithium in LiCoO2 was never used.2–4
Figure 4.6 Energy drawn as area on a plot of voltage against charge: an outlined rectangle for the charge put in, as wide as the charge and as high as the average charge voltage, and a filled rectangle for the charge taken out at the lower average discharge voltage. The difference, shaded, is the energy lost; sliders set the coulombic efficiency and the two average voltages.
energy put in on chargeenergy taken out on dischargeenergy lost
Three efficiencies, drawn as areas. Try this: set the coulombic efficiency to 100 % and see that energy is still lost; then close the gap between the two voltages.
SourceDefinitions of coulombic, voltage and energy efficiency, the charge factor, nominal voltage and capacity, and the round-trip efficiency after Arote 2022, sections 1.3.3, 1.4.5 and 1.5. The product rule follows from E = ∫V dQ with voltages averaged over the charge. The default values are illustrative, not a measured cell.

Three efficiencies

A cell never gives back all it took. Three ratios measure how much it keeps.5 The coulombic efficiency is the ratio of the charge taken out on discharge to the charge put in on charge, over one cycle; for a good cell it is close to 100 %, and its reciprocal is called the charge factor. The voltage efficiency is the ratio of the average discharge voltage to the average charge voltage. The energy efficiency is the ratio of the energy delivered on discharge to the energy taken on charge, and it depends strongly on the charge and discharge rates.5 Because energy is charge times voltage, E = ∫V dQ, the energy efficiency is the product of the other two when each average voltage is taken over the charge.

The two factors have different causes. The coulombic efficiency counts charge lost to side reactions such as the growth of the SEI (chapter 16); the voltage efficiency counts the overpotentials and the hysteresis of chapter 5. One textbook puts the round-trip efficiency of lithium-ion batteries at about 95 %, higher than that of other batteries.5 Data sheets add two reference values: the nominal voltage, the average voltage over a whole discharge at 0.2 C, and the nominal capacity, the capacity of that discharge.5

Worked example. The default setting of the figure. voltage efficiency = 3.65 V ÷ 3.80 V = 96.1 % energy efficiency = 0.995 × 0.961 = 95.6 %; charge factor = 1 ÷ 0.995 = 1.005 Almost all of the loss here is voltage, not charge: a cell with a perfect coulombic efficiency would still waste about 4 % of its energy as heat.
Figure 4.7 The mass of battery that stores the same energy, drawn as one-kilogram blocks: about 6.7 kilograms of lithium-ion, 10 of nickel-metal hydride and 40 of lead-acid for one kilowatt-hour. Below, nine chips list the advantages of lithium-ion cells (cell voltage, no memory effect, low self-discharge, cycle life, round-trip efficiency) and their drawbacks (protection circuit, battery management cost, sensitivity to heat, ageing on the shelf).
Why lithium-ion won, and what it costs. Try this: switch to 150 Wh and count the lead-acid blocks; then open the four red chips.
SourceSpecific energies of 150, 100 and 25 Wh/kg, the 6 kg comparison, and the advantages and drawbacks after Arote 2022, section 1.5. These are one textbook’s round figures for typical batteries, not measurements of particular cells. Blocks drawn to scale by mass, in the colour of each chemistry’s main electrode material.

Why lithium-ion won

Why did lithium-ion displace the older rechargeable cells? Mostly because of mass. In one textbook’s round numbers, a typical lithium-ion battery stores 150 Wh/kg, nickel-metal hydride perhaps 100 and lead-acid only 25, so a lead-acid battery must weigh 6 kg to store the energy of 1 kg of lithium-ion; lithium-ion also has roughly twice the energy density of a typical nickel-cadmium battery.5 Winter and Brodd give lead-acid 30 Wh/kg (figure 4.5); the point is the ratio, not the last digit.2

The rest of the case is a list. A working voltage of about 3.6 V, two to three times that of nickel-cadmium, nickel-metal hydride or lead-acid cells, held steady under load; low self-discharge; no memory effect, so no need to discharge fully before recharging; no regular maintenance, unlike lead-acid; a cycle life ten times that of lead-acid; and a round-trip efficiency of about 95 %.5 Behind the list is lithium itself: much lighter than the other metals used in batteries, and quick to give up its outer electron.5

Every advantage has a price. Lithium-ion cells need a protective circuit and a cover against short circuits; high-power uses need an advanced battery management system, which adds cost; they are sensitive to temperature and can explode when overheated; and they age markedly, losing performance even when unused.5 Chapter 17 and chapter 16 take these up.

You can now explain where a capacity in mAh/g comes from and which mass it refers to, why energy is an area, what the Ragone plot shows, and why a cell delivers less than its chemistry promises.

Now let us watch your cell do it.

Go deeper: the mass basis, and 372 against 339

Run Faraday’s law on the lithiated compound LiC6 (79.0 g/mol) and you get 339 mAh/g; run it on the carbon host C6 (72.07 g/mol) and you get 372 mAh/g. Both are correct; they answer different questions. The convention in the field is to refer negative-electrode hosts to their delithiated mass and positive-electrode materials to their lithiated mass, which is how each is weighed when the cell is built.3,4 Whenever you compare capacities, check the basis first.

The mathematics of this chapter

Q = nFN, m = ItM / (nF)Faraday’s law: the charge for N moles of reactant that give n electrons each; turned round, the mass transformed by a current I for a time t

Qspec = nF / (3.6 M) mAh/gthe theoretical specific capacity of a material of molar mass M (g/mol); 3.6 converts coulombs per gram to milliampere-hours per gram (figure 4.1). Graphite as C6: 372; LiFePO4: 170; lithium metal: 3861.

energy = ∫0Δt IV dt = ∫0Q V(q) dq = 〈V〉 Qthe energy delivered is the area under the discharge curve, or the average voltage times the capacity (figure 4.2)

P = IV, specific energy = energy / mass, specific power = P / massthe two axes of the Ragone plot (figure 4.5), in Wh/kg and W/kg

ηCoulombic = 100 × Qdis / Qchthe share of the charge put in on charging that comes back on discharge; a later module follows it cycle by cycle

ηenergy = 100 × ∫Vdis dq / ∫Vch dqthe share of the energy that comes back; lower than the Coulombic efficiency because the charge branch sits above the discharge branch

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 A. J. Bard, L. R. Faulkner, H. S. White, "Electrochemical Methods: Fundamentals and Applications", Wiley, 3rd edition, 2022.
  2. 2 M. Winter, R. J. Brodd, "What Are Batteries, Fuel Cells, and Supercapacitors?", Chemical Reviews 104, 2004, 4245-4269. doi:10.1021/cr020730k
  3. 3 J.-M. Tarascon, M. Armand, "Issues and challenges facing rechargeable lithium batteries", Nature 414, 2001, 359-367. doi:10.1038/35104644
  4. 4 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
  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). Capacity, energy and power: Faraday's law, the C-rate and the Ragone plot. How batteries work. https://asajeevan.github.io/batteries/capacity-energy/

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