Number Density Calculator — Particles per m³
Number density (symbol n) is the number of particles (atoms, molecules, photons) per unit volume. Calculate it from a simple count over volume, from a material's mass density and molar mass (using Avogadro's constant), or from gas pressure and temperature using the ideal-gas law.
Input method
m³
Particles per cubic metre
- 1
Particle count N
602,200,000,000,000,000,000,000 - 2
Volume V
1 m³ - 3
Number density n = N ÷ V
602,200,000,000,000,000,000,000 ÷ 1 = 602,200,000,000,000,000,000,000
How does this calculator work?
Number density n is particles per m³. Calculate it as n = N/V (count over volume), n = ρN_A/M (from mass density and molar mass), or n = P/(k_BT) for ideal gases. At 1 atm and 0 °C, air has n ≈ 2.69 × 10²⁵ m⁻³. Solid aluminium has n ≈ 6.0 × 10²⁸ m⁻³.
Formula
How this is calculated
Number density n (SI unit: m⁻³) describes how tightly packed particles are in a given volume. The simplest definition is n = N/V — divide the total count of particles by the volume. This is used in engineering and transport problems where N and V are directly known.
For a pure solid or liquid, the particle count can be derived from the mass density ρ (kg/m³) and the molar mass M (kg/mol): n = ρ N_A / M, where N_A = 6.022 × 10²³ mol⁻¹ is Avogadro's constant. This is the most common route for metals and minerals — for example, solid aluminium (ρ = 2700 kg/m³, M = 0.027 kg/mol) has n ≈ 6.0 × 10²⁸ atoms/m³.
For an ideal gas the number density follows directly from the ideal-gas law PV = NkT rearranged to n = P / (k_B T), where k_B = 1.381 × 10⁻²³ J/K is the Boltzmann constant. At standard conditions (101 325 Pa, 273 K), n ≈ 2.69 × 10²⁵ m⁻³ — the Loschmidt constant. Number density is closely related to molar concentration: divide by Avogadro's number to get mol/m³, or further by 1000 for mol/L (molarity).
Frequently asked questions
Number density n counts individual particles per m³ (n = N/V). Molar concentration (molarity C) counts moles per litre — C = n / N_A / 1000. They convey the same information at different scales. Physicists usually work with n; chemists usually work with C. To convert: C (mol/L) = n (m⁻³) ÷ 6.022 × 10²⁶.
The Loschmidt constant (n_0 ≈ 2.687 × 10²⁵ m⁻³) is the number density of an ideal gas at standard conditions (0 °C, 1 atm). It is the gas-phase equivalent of Avogadro's number and historically the first way atomic-scale particle counts were determined experimentally (Josef Loschmidt, 1865).
Mass density ρ (kg/m³) is total mass per volume and depends on the size of each particle. Number density n (m⁻³) counts particles regardless of their mass. They are related by n = ρ N_A / M — heavier particles (large M) give lower number density for the same mass density, explaining why lead has fewer atoms per m³ than aluminium despite being denser.
Also known as
TG we-Calculate Editorial Team. (2026). Number Density Calculator — Particles per m³ [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/number-density-calculator
TG we-Calculate Editorial Team. "Number Density Calculator — Particles per m³." TG we-Calculate. 2026. https://we-calculate.com/calculator/number-density-calculator.
TG we-Calculate Editorial Team, "Number Density Calculator — Particles per m³," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/number-density-calculator
@misc{wecalculate_number_density_calculator, title = {Number Density Calculator — Particles per m³}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/number-density-calculator}}, year = {2026}, note = {TG we-Calculate} }
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