Battery Thermodynamics: Gibbs Free Energy, Nernst Equation & Energy Formulas

A battery’s voltage comes from the Gibbs free energy of its cell reaction: ΔG = −nFE. Battery thermodynamics links that free energy to cell voltage (the Nernst equation), to heat generation (entropy) and to how much energy a battery can store (the battery energy formula, Wh = V × Ah, and specific energy). This guide explains each idea with worked examples, including a complete specific energy calculation for a lithium–manganese dioxide cell.

• Battery energy formula: how to calculate the energy stored in a battery
• What is battery thermodynamics? (including metastability)
• Gibbs free energy and cell voltage
• The Nernst equation for a battery
• Entropy, enthalpy and heat in batteries
• Extensive vs. intensive properties
• Specific energy and specific capacity (with a worked example)
• Battery thermodynamics FAQ

Battery Energy Formula: How to Calculate the Energy Stored in a Battery

The energy stored in a battery is its capacity multiplied by its voltage:

Energy (Wh) = Voltage (V) × Capacity (Ah)

To use milliamp-hours, divide by 1,000 first (4,000 mAh = 4 Ah). Divide watt-hours by 1,000 to get kilowatt-hours, and multiply by 3,600 to get joules. The same idea gives the electrical work a battery does as it moves charge: W = QV, with W in joules, Q in coulombs and V in volts. Use the nominal voltage for a quick estimate. The energy you actually get depends on discharge rate and temperature. For the difference between energy (Wh) and power (W), see Energy vs Power.

Nominal voltage and typical specific energy by battery chemistry

Chemistry	Nominal cell voltage	Typical practical specific energy
Lead-acid	2.0 V	30–50 Wh/kg
Nickel–metal hydride (NiMH)	1.2 V	60–120 Wh/kg
Lithium iron phosphate (LFP)	3.2 V	90–160 Wh/kg
Lithium-ion (NMC or LCO)	3.6–3.7 V	150–250 Wh/kg

See the dedicated posts on lead-acid batteries, nickel battery technologies and the lithium battery for chemistry details.

Worked examples: calculating battery energy

1. Phone battery: 4,000 mAh = 4 Ah at 3.85 V gives 4 × 3.85 = 15.4 Wh. If the cell weighs 0.060 kg, its specific energy is 15.4 / 0.060 ≈ 257 Wh/kg.
2. LFP cell: 100 Ah at 3.2 V gives 100 × 3.2 = 320 Wh.
3. EV pack: a 400 V nominal pack rated 200 Ah stores 400 × 200 = 80,000 Wh = 80 kWh.

What Is Battery Thermodynamics?

Thermodynamics describes how energy is stored, released and converted. In a battery, three quantities do most of the work: the Gibbs free energy change (ΔG), which sets the voltage and the maximum electrical work; the enthalpy change (ΔH), the total energy of the reaction; and the entropy change (ΔS), which sets the reversible heat and how voltage changes with temperature.

What does thermodynamically metastable mean?

A thermodynamically metastable state is a local minimum of free energy. It is stable against small disturbances, but it is not the lowest-energy state the system could reach. It persists only because an energy barrier blocks the transformation to the lower-energy state. Many batteries are metastable. A charged battery holds a large amount of free energy that it can release through a controlled reaction. In addition, one or both electrodes often operate at a potential outside the electrolyte’s electrochemical stability window, so the electrolyte tends to oxidize at the cathode or reduce at the anode. In lithium-ion cells this reaction forms a thin passivation layer, the solid–electrolyte interphase (SEI), which slows further decomposition and keeps the cell usable.

Why are thermodynamic methods important for batteries?

Thermodynamic methods help explain the behavior of anodes, cathodes and complete lithium-ion cells during charge, discharge, overcharge and cycle aging. A lithium-ion electrode that reversibly takes up lithium follows the general reaction H + xLi⁺ + xe⁻ ⇌ Li_xH, where H is the host structure and x is the state of lithiation (0 ≤ x ≤ 1). For background, see the basics of electrochemistry.

Gibbs Free Energy and Cell Voltage

The Gibbs free energy change of the cell reaction is tied directly to the open-circuit voltage (OCV):

ΔG(x) = −nFE⁰(x)

Here n is the number of electrons transferred (n = 1 for Li⁺), F is Faraday’s constant (96,485 C/mol) and E⁰(x) is the equilibrium cell potential. A negative ΔG means the reaction is spontaneous and the cell voltage is positive. The maximum electrical work a cell can deliver is −ΔG. For the broader idea of useful work, see availability and exergy and Gibbs free energy.

The cell potential is the cathode potential minus the anode potential: E⁰(x) = E⁰₊(x) − E⁰₋(x). Gibbs free energy also relates to enthalpy and entropy:

ΔG(x) = ΔH(x) − TΔS(x)

Electrode interfaces and half-reactions

Cell reactions occur at the two electrode interfaces. On discharge, reduction takes place at the positive electrode (aA + ne⁻ → cC) and oxidation at the negative electrode (bB → dD + ne⁻). Adding the two half-reactions gives the overall reaction aA + bB → cC + dD, and its standard free energy change is:

ΔG⁰ = −nFE⁰

For example, a Li–MnO₂ cell with a nominal 3.0 V and one electron per reaction has ΔG ≈ −(1)(96,485)(3.0) ≈ −289.5 kJ per mole of lithium. This change in standard free energy is the driving force that lets a battery supply electrical energy to an external circuit.

The Nernst Equation for a Battery

The Nernst equation describes how an electrode’s potential depends on its chemical environment. Under non-standard conditions it gives the actual cell voltage, linking potential to temperature and to the ratio of products to reactants:

E = E⁰ − (RT / nF) ln Q

At 298.15 K (25 °C) this simplifies to E = E⁰ − (0.0592 / n) log₁₀ Q, because RT/F ≈ 25.69 mV.

• E⁰ is the standard cell potential,
• R is the gas constant (8.314 J/mol·K),
• T is the absolute temperature in kelvin,
• n is the number of moles of electrons transferred,
• F is Faraday’s constant (96,485 C/mol),
• Q is the reaction quotient (products over reactants, using activities).

Nernst equation example: Daniell cell

1. Reaction: Zn + Cu²⁺ → Zn²⁺ + Cu, with n = 2 and E⁰ = 1.10 V.
2. Let [Zn²⁺] = 0.10 M and [Cu²⁺] = 1.0 M, so Q = 0.10.
3. E = 1.10 − (0.0592 / 2) log₁₀(0.10) = 1.10 + 0.0296 ≈ 1.13 V.

Lower product concentration raises the voltage slightly, just as the Nernst equation predicts.

Why does open-circuit voltage change with state of charge?

In an intercalation electrode, ΔG depends on composition (x), so the equilibrium potential E⁰(x) slopes as the electrode fills with lithium. In lead-acid cells the acid concentration also changes with state of charge, which shifts the voltage through the Nernst term. This is why OCV can be used as a rough state-of-charge indicator.

Entropy, Enthalpy and Heat in Batteries

Combining ΔG = ΔH − TΔS with ΔG = −nFE gives E = −ΔH/nF + TΔS/nF. Differentiating with respect to temperature shows that the entropy change of the reaction is measurable from voltage:

dE/dT = ΔS / nF

This quantity is the entropic coefficient. For lithium-ion cells it is usually a fraction of a millivolt per kelvin, but it sets the reversible heat, which can warm or cool a cell depending on state of charge. The total heat generation rate (the Bernardi equation, with discharge current I taken as positive) is:

q = I(E_oc − V) − I·T·(dE_oc/dT)

The first term is irreversible heat from resistance and polarization. The second is reversible heat from entropy change. For how these terms feed into pack cooling design, see Battery Thermal Management 101. For the entropy concept itself, see entropy and the second law.

Extensive vs. Intensive Thermodynamic Properties of a Battery

Thermodynamic quantities are either extensive (they scale with the size or amount of material) or intensive (they do not). Telling them apart is essential when you compare cells of different sizes.

Property	Type	Why
Mass	Extensive	Grows with battery size
Capacity (Ah)	Extensive	Total charge the battery can store
Enthalpy, entropy, Gibbs free energy	Extensive	Scale with the amount of reacting material
Temperature	Intensive	Does not depend on battery size
Open-circuit voltage (OCV)	Intensive	Set by the cell chemistry, not its size
Specific energy (Wh/kg)	Intensive	Energy per unit mass
Specific heat (J/kg·K)	Intensive	Heat to warm one kilogram by one kelvin
Molar heat capacity (J/mol·K)	Intensive	Heat to warm one mole by one kelvin. (A cell’s total heat capacity in J/K is extensive.)

Specific Energy and Specific Capacity

How to calculate the specific energy of a battery

Specific energy is the electrical energy stored per unit mass, usually in watt-hours per kilogram (Wh/kg):

Specific energy (Wh/kg) = Specific capacity (Ah/kg) × Cell voltage (V)

It also equals the total energy in Wh divided by the battery mass in kg.

How do you calculate specific capacity?

Specific capacity is the charge stored per unit mass, in ampere-hours per kilogram (Ah/kg):

Specific capacity (Ah/kg) = Energy (Wh) / [Cell voltage (V) × Mass (kg)]

For a theoretical value from the chemistry, use Q/M = nF / (3,600 × M_molar), with F = 96,485 C/mol and the molar mass of the active materials in kg/mol.

Example problem: theoretical specific energy of a Li–MnO₂ cell

Calculate the theoretical specific capacity and specific energy for the one-electron reaction Li + MnO₂ → LiMnO₂.

1. Mass per mole of reaction: 6.94 g (Li) + 86.9 g (MnO₂) = 93.84 g.
2. Charge per mole of reaction: 1 × 96,485 C.
3. Specific capacity: 96,485 / 93.84 ≈ 1,028 C/g = 1,028,000 C/kg. Dividing by 3,600 s/h gives ≈ 285.6 Ah/kg.
4. Specific energy at a nominal 3.0 V: 285.6 × 3.0 ≈ 857 Wh/kg.

This is a theoretical ceiling that counts only the active materials. Real cells deliver much less, as the next section explains.

Theoretical vs. Practical Energy Density

Real batteries store a fraction of their theoretical specific energy for several reasons:

• Inactive materials such as current collectors, separators, electrolyte and the cell casing add mass without storing energy.
• Not all of the active material is used. Part of the lithium capacity is left in the electrode to protect cycle life.
• Voltage sags under load, so the average discharge voltage is below the nominal value.
• At pack level, the battery management system, cooling, wiring and structure add further mass. See energy storage for system-level examples.

Why Is High Energy Density (Specific Energy) Important?

High specific energy gives longer run time for the same weight or volume, or the same run time from a smaller, lighter battery. In portable electronics and electric vehicles, where size and weight are critical, high-energy-density batteries make devices more compact and efficient without sacrificing performance.

How Do You Achieve High Specific Energy?

• High-capacity electrode materials: materials that store more lithium per unit mass raise capacity. Lithium cobalt oxide (LiCoO₂) cathodes and graphite anodes are common in lithium-ion cells.
• Higher cell voltage: energy is capacity times voltage, so a higher-voltage cathode raises specific energy. LiCoO₂ and NMC cells run near 3.6–3.7 V, while lithium iron phosphate runs near 3.2 V. LFP gives up some specific energy in exchange for better safety and longer life.
• Lightweight components: thinner current collectors, separators and casings cut inactive mass.
• Efficient cell and pack design: keeping inactive material low at cell and pack level preserves more of the theoretical energy.

Key Takeaways on Battery Thermodynamics

• Cell voltage and Gibbs free energy are linked by ΔG = −nFE.
• The Nernst equation, E = E⁰ − (RT/nF) ln Q, corrects the voltage for concentration and temperature.
• The entropic coefficient, dE/dT = ΔS/nF, sets the reversible heat in a cell.
• Energy in a battery is voltage times capacity (Wh = V × Ah), and specific energy is that energy per kilogram.
• Theoretical specific energy is a ceiling; practical cells reach a fraction of it.

Battery Thermodynamics FAQ

How do you calculate the energy stored in a battery?

Multiply the battery voltage by its capacity: energy in watt-hours (Wh) = voltage (V) × capacity (Ah). For example, a 4 Ah battery at 3.85 V stores 15.4 Wh. Divide by 1,000 for kilowatt-hours.

What is the Nernst equation in a battery?

The Nernst equation gives the actual cell voltage under non-standard conditions: E = E⁰ − (RT/nF) ln Q, where Q is the reaction quotient. At 25 °C it simplifies to E = E⁰ − (0.0592/n) log₁₀ Q.

Why is ΔG negative for a discharging battery?

A discharging battery runs a spontaneous reaction, and spontaneous reactions release free energy, so ΔG is negative. Because ΔG = −nFE, a negative ΔG corresponds to a positive cell voltage, and the free energy released is the maximum electrical work the cell can deliver.

What does thermodynamically metastable mean?

A metastable state is a local free-energy minimum that is stable against small disturbances but not the lowest-energy state. It persists because an energy barrier blocks the change. Batteries are metastable: a charged cell holds free energy that is released through a controlled reaction.

What is the difference between specific energy and energy density?

Specific energy is energy per unit mass (Wh/kg), also called gravimetric energy density. Energy density usually means energy per unit volume (Wh/L), also called volumetric energy density. The two terms are often used interchangeably, so check the units.

How does temperature affect battery voltage?

Temperature changes the open-circuit voltage through the RT/nF term in the Nernst equation and through the entropic coefficient, dE/dT = ΔS/nF, which is usually a fraction of a millivolt per kelvin in lithium-ion cells. Cold temperatures also raise internal resistance, which lowers the voltage and capacity available under load.

What is the formula for the work done by a battery?

The electrical work is W = QV, where Q is the charge moved in coulombs and V is the voltage. In thermodynamic terms, the maximum work per mole of reaction is −ΔG = nFE.

Related Reading

• Electric Battery Explained: how batteries are built and work
• Battery Thermal Management 101: heat generation and cooling
• Basics of Electrochemistry
• Gibbs free energy explained
• Lithium battery basics
• Lead-acid car batteries
• Energy vs Power: key differences, formula and examples