# Number Density Calculator

URL: https://speedycalculator.com/calculators/physics/number-density/
Last updated: 2026-10-30

> The number density calculator finds the number of atoms per unit volume and the free-electron density of a metal or other substance from its density and molar mass.

## Inputs

- Substance
- Density ρ (g/cm³, kg/m³)
- Molar mass M (g/mol)
- Free electrons per atom

## Results

- Number density of atoms
- Free-electron density
- Atoms per cm³

## Formula

```
n = ρ·N_A / M,  n_e = n × (free electrons per atom)
```

## Explanation

Number density counts particles per cubic metre. For metals, the free-electron density sets electrical conductivity and the Fermi energy: copper has about 8.5 × 10²⁸ conduction electrons per m³.

### How to use the Number Density Calculator

- **Substance**
- **Density ρ**
- **Molar mass M (g/mol)**
- **Free electrons per atom**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Choose a substance or enter its density, molar mass and free electrons per atom.

### Inputs explained

FieldWhat to enterSubstanceChoose one: Aluminium, Copper, Gold, Silver …Density ρUnits: g/cm³, kg/m³Molar mass M (g/mol)A numberFree electrons per atomA number

### What each result means

- **Number density of atoms** = [rho]×1e6×6.02214076e23÷[M]
- **Free-electron density** = [Number density of atoms]×[fe]
- **Atoms per cm³** = [Number density of atoms]÷1e6

### Example values

How the main result changes when **Density ρ** changes and everything else stays at the example values above:

Density ρ (g/cm³)Number density of atoms (m⁻³)4.488.4912e286.728.4912e288.968.4912e2811.28.4912e2813.448.4912e2817.928.4912e28

## Worked example

Inputs — Substance: Copper.

Results — **Number density of atoms = 8.4912e28 m⁻³**; Free-electron density = 8.4912e28 m⁻³; Atoms per cm³ = 8.4912e22 cm⁻³.

## Key terms

- **Number density**: Particles per unit volume.

## FAQ

**What is the free-electron density of copper?**
About 8.5 × 10²⁸ m⁻³.

**Why multiply by free electrons?**
Each atom contributes its valence electrons to the conduction band.

**What is Avogadro's number?**
6.02214076 × 10²³ per mole, exact by definition.

**Where is number density used?**
Drift velocity, plasma physics, gas kinetics.


## Sources

- [OpenStax – University Physics Volume 1](https://openstax.org/details/books/university-physics-volume-1)
- [The Engineering ToolBox](https://www.engineeringtoolbox.com/)
- [NIST – Fundamental physical constants](https://physics.nist.gov/cuu/Constants/)