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Metal Free-Electron Density from Mass and Molar Mass

Estimate free-electron number density from material density, molar mass and electrons per atom, with ten material presets, custom inputs and inverse controls.

Formula & worked example
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Calculator inputs

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How to use this calculator

Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

With 8960 kg/m³, 63.546 g/mol and one modeled free electron per atom, the estimate is about 8.49123 ×10²⁸ carriers/m³.

n = z ρ N_A / M_kg_per_mol, with N_A = 6.02214076 ×10²³ mol⁻¹ exactly. The displayed coefficient is n divided by 10²⁸ in the selected volume unit.

Worked example

Enter these known values and leave the other values blank.

Material mass density: 8960 kg/m³
Molar mass: 63.546 g/mol
Modeled free electrons per atom: 1

Free-electron number density: 8.491 ×10²⁸ carriers/ m³

Assumptions and limitations

This is a simple metal conduction-electron model: multiply the estimated atom count by the positive integer number of free electrons contributed per atom. It does not count every bound electron or determine electron mobility.
The result control has a ×10²⁸ carriers scale. A displayed 1 with m³ selected means 10²⁸ carriers/m³. Changing the volume unit changes the coefficient, not the physical concentration.
Enter molar mass in g/mol. The formula converts it to kg/mol before combining it with mass density in SI. The Avogadro constant is exact; density and molar-mass measurements are not.
The ten material presets use reviewed Royal Society of Chemistry room-temperature density and listed atomic-mass values, checked October 13, 2026. They are approximate pure-element inputs; temperature, composition and isotopic abundance can change the appropriate values.
A material preset fills density and molar mass and keeps those controls read-only. Choose Custom to supply measured values for the actual material or use the retained inverse controls. The preset does not choose the number of free electrons per atom.
This model retains positive density, molar mass and number density, and positive integer electrons per atom. Zero, negative values and fractional electrons-per-atom results are outside its declared domain. Only the computed electron-count inverse is rounded to 60 decimal places to remove conversion arithmetic noise before the integer check.
This is not a semiconductor doping, alloy electronic-band or mixed-species model. A count of free electrons per atom is an explicit modeling assumption; it cannot establish a measured carrier concentration by itself.

Common questions

Is 8.49 the number of carriers per cubic meter?

With m³ selected, a coefficient of 8.49 means approximately 8.49 ×10²⁸ carriers per cubic meter. The scale printed next to the control is part of the quantity.

Does choosing copper automatically mean one free electron per atom?

No. The material selection supplies approximate density and molar mass. Enter the positive integer carrier assumption separately; one is a common simplified copper model.

Can I use a density measured at another temperature?

Choose Custom and enter the measured density and appropriate molar mass. The preset values are room-temperature reference inputs rather than universal constants.

References

Use reviewed Royal Society of Chemistry room-temperature densities and listed atomic masses for the ten material presets, correcting five captured cells in both the table and keyed scope. These are approximate material inputs, not exact or temperature-independent constants. Replace the four imperial density conversions consistently with exact avoirdupois pound and international length definitions. Use exact SI Avogadro constant 6.02214076e23 throughout all four relationships, retaining the 10^28 carrier scale, positive integer electrons-per-atom model and both unit preferences; the retained positive-domain checks reject each singular dividing denominator. Round only the computed electrons-per-atom inverse to 60 decimal places to remove conversion arithmetic noise before the retained integer-domain check.

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