For oxygen evolution, Faradaic efficiency (FE) is the oxygen you actually measure divided by the oxygen that the passed electrical charge should produce. Calculate charge first, convert the oxygen measurement to moles over the same time interval, then apply FE(O2) = 100 × n(O2,measured) / [Q/(4F)]. Here Q = ∫I(t)dt, F = 96,485 C mol−1, and the factor of four comes from the four electrons required to make one molecule of O2 from water.
The calculation is simple; obtaining a defensible oxygen amount is the demanding part. A concentration, sensor percentage or chromatographic peak is not yet a mole quantity. You must apply calibration, flow or sampled volume, pressure, temperature and gas-composition corrections, and match the oxygen sampling interval to the charge interval.
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The equations and what each term means
1. Determine the charge passed
Use the measured current trace rather than a nominal instrument setting whenever possible:
Q = ∫ I(t) dt
For genuinely constant current, this reduces to Q = I × t. Express current in amperes and elapsed time in seconds so that charge is in coulombs. If current changes during a run, integrate the recorded trace over exactly the interval for which oxygen was quantified. Including charging transients, pauses or portions of the experiment that are absent from the gas measurement creates a mismatch.
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2. Calculate the theoretical oxygen amount
Water oxidation to oxygen requires four electrons per mole of oxygen. Faraday’s law therefore gives:
n(O2,theoretical) = Q/(4F)
Use F = 96,485 C mol−1. The result is moles of oxygen. Multiplying by 1,000 gives millimoles; multiplying by 1,000,000 gives micromoles.
3. Compare measured and theoretical oxygen
Once the instrument output has been converted to moles over the same interval:
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FE(O2) = 100 × n(O2,measured) / n(O2,theoretical)
Substituting Faraday’s law produces the equivalent form:
FE(O2) = 100 × 4F × n(O2,measured) / Q
FE is a percentage, not an oxygen concentration. A sensor reading of 5% oxygen, for example, cannot be inserted directly into the equation without the total flow or sampled gas volume that turns that fraction into moles.
Convert each oxygen measurement into moles
Collected gas volume
If oxygen is collected in a vessel, use the ideal-gas relationship:
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n(O2) = p(O2)V/(RT)
p(O2) is oxygen’s partial pressure, V is the oxygen-containing gas volume, R is the gas constant in units consistent with pressure and volume, and T is absolute temperature in kelvin. Keep pressure absolute, not gauge pressure, and convert volume to the units required by your chosen value of R.
Total pressure is not automatically oxygen pressure. For a gas mixture, use the oxygen mole fraction: p(O2) = y(O2)P. For gas collected over water, subtract water-vapor pressure from the total pressure before applying the oxygen fraction. If the collection is wet but the oxygen analysis is reported on a dry basis, make the pressure and composition bases agree before calculating moles.
Subtract any oxygen that was already present in the vessel or headspace when the interval began. Leaks, dissolved oxygen released during the run and gas that remains trapped as bubbles can all make the recovered volume differ from the oxygen generated at the electrode.
Oxygen fraction in a carrier-gas stream
A calibrated oxygen sensor or analyzer reports a fraction or concentration. Combine it with the carrier-gas flow to obtain an oxygen molar flow. If y(O2) is the oxygen mole fraction and ṅgas is the total molar flow:
ṅ(O2) = y(O2) × ṅgas
For a volumetric flow, convert to molar flow with the ideal-gas law using the actual pressure and temperature at the flow measurement point. Integrate the oxygen molar flow over the same time window as the electrical charge:
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An inert carrier such as argon does not itself supply oxygen, but the oxygen fraction still needs a clearly defined wet or dry basis. Water vapor can dilute a wet measurement, while a dry-basis analyzer has already removed that contribution mathematically. Do not combine a dry oxygen percentage with a wet volumetric flow without correction.
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A published 2022 Nature Communications method used an in-situ calibrated oxygen microsensor with argon carrier gas. It waited for a stable oxygen concentration before calculating efficiency and compared measured oxygen flow with the theoretical rate I/(4F). Stability before calculation matters: a concentration during headspace filling is not the steady oxygen output of the electrode.
Online gas chromatography
For online GC, first calibrate oxygen peak area or another signal against standards of known oxygen concentration. Convert each injection’s calibrated concentration to an amount using the sampled volume or the constant carrier flow and sampling interval. Then compare that amount with the charge passed during the corresponding interval.
A 2022 Nature Communications procedure used an oxygen calibration gas, helium carrier and constant flow. Its calculation excluded the initial headspace-equilibration injection from the average. That treatment prevents a transient, not-yet-representative sample from being presented as steady-state oxygen production.
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- Define the interval. Record the exact start and stop times for oxygen collection, sensor integration or GC sampling. The electrical data must cover those same times.
- Measure and integrate current. Export the potentiostat or electrochemical workstation trace and calculate
Q = ∫I(t)dt. For constant current, multiply amperes by seconds. - Choose one gas basis. State whether pressure, flow and oxygen concentration are wet or dry, and use that basis consistently.
- Calibrate the oxygen measurement. Calibrate a sensor with appropriate reference gases, calibrate GC response with known standards, or measure collected-gas pressure, temperature and composition with traceable instruments.
- Convert output to moles. Apply
pV/(RT)to collected gas, multiply oxygen fraction by molar flow for a stream, or convert each GC result using its sampling flow and interval. - Correct backgrounds and transients. Remove initial headspace oxygen, account for water vapor, and exclude equilibration data when the setup has not reached a stable condition.
- Calculate theoretical oxygen. Divide charge by
4F. - Calculate FE. Divide measured moles by theoretical moles and multiply by 100.
- Audit the result. Check units, pressure basis, time alignment, calibration, leaks, bubbles and competing electrochemical reactions before interpreting an unusual percentage.
Worked example
Suppose an electrolyzer operates at a constant 10.0 mA for 30.0 min, and the oxygen measurement reports 0.0400 mmol over precisely those 30 minutes.
- Convert current and time:
I = 0.0100 Aandt = 30.0 × 60 = 1,800 s. - Calculate charge:
Q = 0.0100 A × 1,800 s = 18.0 C. - Calculate theoretical oxygen:
n(O2) = 18.0/(4 × 96,485) mol = 4.66 × 10−5 mol = 0.0466 mmol. - Calculate FE:
FE = 100 × 0.0400/0.0466 = 85.8%.
The 85.8% value is an illustrative calculation, not a performance result for a particular catalyst. Its validity depends on the oxygen amount being calibrated and collected for exactly the same interval as the 18.0 C of charge.
Choosing an oxygen-measurement route
| Method | Measured output | Time resolution | Required setup | Typical failure modes to check | Selectivity considerations |
|---|---|---|---|---|---|
| Collected gas | Total gas volume, with oxygen partial pressure determined from composition | Batch or end-point collection over a defined interval | Calibrated volume plus pressure and temperature measurement; oxygen composition if other gases are present | Leaks, incomplete recovery, bubbles, initial headspace oxygen, water-vapor pressure and incorrect gas fraction | Needs composition analysis to distinguish oxygen from other collected gases |
| Carrier-gas sensor or analyzer | Oxygen fraction or concentration multiplied by total gas flow | Continuous tracking once the signal is stable | Calibrated sensor, inert carrier, known flow, pressure and temperature data, and suitable headspace or cell plumbing | Sensor calibration drift, unstable concentration, wet/dry basis mismatch, flow error and transport delay | Directly reports oxygen when calibrated, but does not by itself account for oxygen trapped in bubbles or the cell |
| Online gas chromatography | Calibrated chromatographic signal converted to oxygen amount per injection | Periodic injections; each interval is separately quantifiable | Oxygen standards, carrier gas, controlled flow, sampling schedule and GC calibration | Calibration error, sampling mismatch, headspace equilibration and treating a transient injection as steady state | Separates oxygen from other gases when the chromatographic method resolves them |
| Rotating ring-disk electrode in a suitable configuration | Electrochemically detected oxygen reaching the ring | Potentially time-resolved during operation | Controlled hydrodynamics and a calibrated collection-efficiency relationship | Collection-efficiency uncertainty, gas bubbles and transport effects | Can help distinguish oxygen electrochemically, but interpretation depends strongly on collection and bubble behavior |
No route is universally best. A collection vessel can be simple but loses time resolution and may miss bubbles. A sensor provides continuous data but requires stable flow and calibration. GC adds chemical separation and interval-by-interval analysis, at the cost of standards, sampling control and equilibration checks.
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Common calculation errors and how to correct them
Using current instead of charge
Current is a rate. The theoretical oxygen amount depends on total charge, so multiply constant current by elapsed seconds or integrate a varying current trace. A 10 mA reading without a time interval is insufficient to calculate oxygen production.
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One mole of O2 from water corresponds to four moles of electrons. Dividing by F instead of 4F overestimates theoretical oxygen by a factor of four and distorts FE.
Comparing different time windows
Oxygen accumulated for 20 minutes cannot be compared with charge passed for 30 minutes. Align collection, sensor integration or GC injections with the same electrical start and stop times, or calculate FE separately for matching sub-intervals.
Treating concentration as an amount
Percent oxygen or a concentration in parts per million is not moles. Multiply by total molar flow for a stream or by sampled gas volume for a discrete sample, then integrate or sum over time.
Mixing gas-law units
Use one consistent set of units for pressure, volume and the gas constant. Convert Celsius to kelvin and gauge pressure to absolute pressure. Unit errors can produce apparently plausible but incorrect FE values.
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Gas collected over water contains water vapor. Subtract its partial pressure before calculating dry oxygen. Conversely, do not apply a dry-gas correction to a flow or concentration already reported on a wet basis.
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Ignoring background oxygen
Initial headspace oxygen, dissolved oxygen and oxygen entering through a leak can inflate the measured amount. Establish a baseline, quantify the initial inventory where relevant and subtract only a defensible background rather than assuming the first reading is generated electrochemically.
Assuming all anodic current is oxygen evolution
Total anodic current can include catalyst or support oxidation and other competing reactions. As the EcoMat recommended-practice review states, “The current density observed in the OER region comprises the cumulative currents of all redox reactions transpiring at that potential.” Oxygen FE therefore measures the fraction of charge that results in detected O2, not simply the current at a nominal OER potential.
Counting every bubble as recovered oxygen
Bubbles can remain attached to the electrode, dissolve, bypass the collection path or release later. Inspect cell geometry, mixing and gas routing, and state whether the method measures evolved, transported or recovered oxygen.
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How to interpret values below or above 100%
FE below 100%
A value below 100% can reflect genuine competing oxidation, incomplete oxygen recovery, oxygen dissolution, sensor or GC calibration error, leaks, transport delay or a mismatch between gas and charge intervals. Do not assign the deficit to catalyst selectivity until recovery and measurement checks are complete.
Apparent FE above 100%
More measured oxygen than the charge predicts is a diagnostic signal, not evidence that the electrochemical reaction exceeded Faraday’s law. Check oxygen calibration, baseline and initial headspace, oxygen fraction, pressure, water-vapor correction, flow measurement, current integration, leaks and interval alignment. Chemical oxygen release from a stored oxidized species can also add oxygen that was not generated by the measured charge.
Potential-dependent results
Oxygen selectivity can change with potential. An Energy & Environmental Science report from 2022 found oxygen efficiency near 100% above 1.40 VRHE and around 20% at 1.33 VRHE in its own acidic-water-oxidation system. Those figures are system-specific observations, not a universal operating range or benchmark for every catalyst and cell.
No single “typical” oxygen FE applies across electrolytes, electrodes, reactors and measurement methods. Report the potential, electrolyte, current or current density, gas-analysis route, calibration basis and recovery arrangement with the percentage.
Uncertainty and quality checks
FE inherits uncertainty from both numerator and denominator. Important contributors include current measurement and integration, elapsed time, gas volume, pressure, temperature, flow, oxygen calibration, background subtraction and recovery. If the measured oxygen amount is obtained by several steps, propagate the uncertainty of those steps rather than reporting more significant figures than the least certain input supports.
- Run an instrument blank or no-current baseline to identify background oxygen.
- Verify flow with a calibrated standard and record the pressure and temperature at the flow meter.
- Check sensor response with reference gases before and after a run, or bracket GC measurements with calibration standards.
- Allow the cell and headspace to reach the stated operating condition before using steady-state data.
- Compare duplicate runs and inspect the current trace for pauses, spikes or dropout.
- Confirm that the reported oxygen is on the same wet or dry basis as the gas-flow calculation.
What to report so another person can reproduce FE
- Electrolysis start and stop times, applied potential or current, and the current trace or integrated charge.
- The Faraday constant used and the four-electron OER stoichiometry.
- Oxygen measurement type, calibration gas or standard, instrument sampling interval and response-stability criterion.
- Gas flow, pressure, temperature, oxygen fraction basis and whether values are wet or dry.
- Collection volume, headspace treatment, water-vapor correction and background subtraction where applicable.
- How bubbles, dissolved oxygen and transport delays were handled.
- The measured oxygen amount, theoretical oxygen amount, resulting FE and uncertainty or replicate spread.
- Any competing anodic chemistry that could consume current without producing oxygen.
With those details, a reader can reproduce the conversion from instrument output to moles and verify that the oxygen interval and electrical charge are commensurate.
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