Measure the equilibrium open-circuit voltage at several controlled temperatures, then differentiate voltage with respect to temperature. At fixed state of charge (SOC), the equilibrium voltage temperature coefficient is

[ \left(\frac{\partial E_0}{\partial T}\right)_p

\frac{\Delta S}{zF} ]

where (E_0) is equilibrium cell voltage, (\Delta S) is reaction entropy, (z) is the number of transferred electrons, and (F=96485\ \mathrm{C,mol^{-1}}). The reversible heat is then (q_{\mathrm{rev}}=T\Delta S) absorbed by the cell, or (-T\Delta S) when reported as heat released to the surroundings.

Core takeaway: Build an equilibrium voltage-versus-temperature map at constant SOC, using sufficient relaxation to eliminate polarization and concentration gradients. Its slope gives the entropy change and temperature coefficient; multiplying the entropy change by temperature gives the reversible Peltier heat, whose direction reverses with current.

What the Measurement Determines

The temperature coefficient of equilibrium voltage

For a reversible cell reaction,

[ \Delta G=-zF E_0 ]

and the thermodynamic relation

[ \Delta S=-\left(\frac{\partial \Delta G}{\partial T}\right)_p ]

gives

[ \boxed{ \left(\frac{\partial E_0}{\partial T}\right)_p

\frac{\Delta S}{zF} } ]

The quantity (\partial E_0/\partial T), usually reported in (\mathrm{V,K^{-1}}) or (\mathrm{mV,K^{-1}}), is the temperature coefficient of the equilibrium cell voltage.

The reversible heat

The reversible heat associated with the reaction is

[ \boxed{ q_{\mathrm{rev,cell}}=T\Delta S } ]

This is the heat absorbed by the electrochemical system under the stated thermodynamic sign convention. Using the measured voltage slope,

[ \boxed{ q_{\mathrm{rev,cell}}

TzF \left(\frac{\partial E_0}{\partial T}\right)_p } ]

The corresponding heat released to the surroundings is

[ \boxed{ q_{\mathrm{released}}

-T\Delta S

-TzF \left(\frac{\partial E_0}{\partial T}\right)_p } ]

This distinction matters because “reversible heat generation” is used inconsistently. Researchers should state explicitly whether positive heat means heat absorbed by the cell or heat generated and released by the cell.

How to Perform the Laboratory Test

Establish a defined SOC

Charge or discharge the cell to the target SOC using a controlled protocol, then stop the current.

The measurement must be repeated across the SOC range of interest because both the voltage temperature coefficient and reversible heat can vary substantially with electrode composition, phase transitions, and lithium concentration.

Allow the cell to reach equilibrium

Hold the cell at open circuit until voltage and temperature have sufficiently stabilized.

This relaxation step is essential. If measurements are taken while concentration gradients, charge-transfer polarization, or thermal gradients remain, the recorded voltage is not the equilibrium voltage required for a thermodynamic derivative.

Use high-impedance voltage measurement so the instrument does not draw a meaningful current from the cell.

Apply controlled temperature steps

Place the cell in a temperature chamber or a well-characterized thermal fixture. At the same SOC, measure the relaxed open-circuit voltage at multiple temperatures, for example:

[ (T_1,E_0(T_1)),\quad (T_2,E_0(T_2)),\quad (T_3,E_0(T_3)) ]

Use small, controlled temperature changes and allow the cell to equilibrate at every step.

The cell temperature—not merely the chamber set point—must be measured. Large cells may require multiple sensors or extended dwell times because internal temperature gradients can distort the result.

Repeat across SOC

Construct a map of

[ E_0=f(\mathrm{SOC},T) ]

Then evaluate the temperature slope at each SOC:

[ \left(\frac{\partial E_0}{\partial T}\right)_{\mathrm{SOC},p} \approx \frac{E_0(T+\Delta T)-E_0(T-\Delta T)} {2\Delta T} ]

A central difference generally reduces bias relative to using only one-sided temperature changes. Regression through several temperature points is preferable when the voltage data contain noise.

Convert the Voltage Slope to Entropy and Heat

Calculate the reaction entropy

Once the slope is known,

[ \boxed{ \Delta S

zF \left(\frac{\partial E_0}{\partial T}\right)_p } ]

For a typical single-electron battery reaction, (z=1), but the appropriate electron-transfer number must be used for the reaction being analyzed.

Calculate reversible heat per mole of reaction

At temperature (T),

[ \boxed{ q_{\mathrm{rev,molar}}

T\Delta S

TzF \left(\frac{\partial E_0}{\partial T}\right)_p } ]

This gives the reversible heat associated with the selected reaction basis. It can be reported per mole of reaction, per mole of active material, per ampere-hour, or using another clearly stated basis.

Calculate reversible heat rate during cycling

If the current is (I), the reversible heat rate is proportional to the molar reaction rate:

[ \boxed{ \dot Q_{\mathrm{rev}}

\frac{T\Delta S}{zF},I } ]

The exact sign depends on the chosen current convention. If positive current is defined as discharge, use that convention consistently when interpreting whether the cell absorbs or releases heat.

Most importantly, reversible heat changes sign when the current direction is reversed. This behavior distinguishes it from ordinary resistive heating, which remains positive for both charge and discharge.

Separate Reversible Heat from Irreversible Heating

Identify the total thermal response

During operation, the cell’s heat rate may include:

• Reversible entropy heat, associated with (T\Delta S).
• Ohmic heating, associated approximately with (I^2R).
• Polarization and overpotential losses at the electrodes.
• Side-reaction and parasitic-reaction heat.
• Heat transferred to or from the fixture and environment.

A temperature rise during cycling is therefore not, by itself, a measurement of the Peltier contribution.

Use current reversal carefully

A current-reversal experiment can help separate terms because the reversible contribution changes sign with current, while Joule heating does not.

However, reversal must be performed under controlled SOC and sufficiently short timescales—or with appropriate correction—because changing current also changes polarization, concentration gradients, and the cell’s thermal state.

Use calorimetry when absolute heat rate is required

A calorimeter or highly characterized thermal enclosure can measure the cell’s total heat flow directly. Combining calorimetry with current, voltage, SOC, and resistance measurements allows researchers to compare the measured thermal response with the thermodynamic reversible-heat prediction.

The OCV-versus-temperature method is usually the direct route to (\Delta S); calorimetry provides an independent validation of the resulting heat-rate model.

Determine Related Thermodynamic Quantities

Obtain the Gibbs free-energy change

From the equilibrium voltage,

[ \boxed{ \Delta G=-zF E_0 } ]

The sign follows the convention that a spontaneous galvanic reaction has negative (\Delta G) and positive discharge voltage.

Obtain the enthalpy change

Using

[ \Delta H=\Delta G+T\Delta S ]

researchers can estimate the reaction enthalpy when both equilibrium voltage and temperature coefficient are known.

Equivalently, with the heat-to-surroundings convention used in the primary reference,

[ \Delta G-\Delta H=-T\Delta S ]

The apparent discrepancy between these forms is only a sign convention for whether heat is defined as entering the cell or leaving it.

Understanding the Trade-offs

Equilibrium measurements are slow

Long relaxation periods improve thermodynamic accuracy but increase test duration and may allow self-discharge or side reactions to alter SOC.

The protocol should define a practical equilibrium criterion, such as a voltage-slope threshold, while recognizing that a finite relaxation time cannot guarantee perfect thermodynamic equilibrium.

Temperature changes can alter SOC indirectly

Temperature affects side reactions, self-discharge, phase equilibria, and concentration distributions. If the SOC changes during a temperature step, the measured voltage change includes both temperature and composition effects.

Use symmetric temperature steps, monitor charge balance, and minimize dwell times consistent with equilibration.

The voltage slope may be nonlinear

The relationship between (E_0) and (T) may not be linear over a broad temperature range, particularly near phase transitions or strong compositional changes.

Use local slopes or piecewise fitting rather than applying one coefficient across the entire temperature range.

Terminal-voltage coefficients are not equilibrium coefficients

The temperature coefficient measured during a high-current pulse includes resistance, charge-transfer kinetics, diffusion, and polarization effects.

For reversible heat calculations, use relaxed open-circuit voltage, not loaded terminal voltage.

Reversible heat is not always the dominant heat source

At high current, irreversible (I^2R) and polarization heating can exceed the reversible contribution. Near equilibrium or at low current, the entropy term may be comparatively more important.

Thermal models should retain both reversible and irreversible terms rather than treating one as a substitute for the other.

Making the Right Choice for Your Goal

Use a controlled equilibrium-voltage experiment as the primary method, and add calorimetry or current-reversal testing when independent heat-flow validation is required.

• If your primary focus is the temperature coefficient: Measure relaxed open-circuit voltage at several temperatures for each SOC, then fit (E_0(T)) and calculate (\partial E_0/\partial T).
• If your primary focus is reversible heat per reaction: Convert the voltage slope using (q_{\mathrm{rev}}=TzF(\partial E_0/\partial T)), while stating whether the sign represents heat absorbed or released.
• If your primary focus is heat generation during cycling: Combine the entropy term with ohmic and polarization heating, using synchronized current, voltage, temperature, resistance, and SOC data.
• If your primary focus is model validation: Compare the thermodynamic prediction with calorimetric heat-flow measurements under controlled charge and discharge conditions.
• If your primary focus is high-confidence thermodynamic data: Use high-impedance instrumentation, small temperature steps, stable thermal control, and relaxation criteria that demonstrate practical equilibrium.

A carefully controlled (E_0)-versus-(T) measurement converts a difficult thermal phenomenon into a measurable thermodynamic quantity and provides a reliable foundation for electrothermal cell models.

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