Number Density Calculator
Number density is the count of particles, atoms, or charge carriers packed into a unit of volume. Enter values in any of the three modes below: material density and molar mass for solids, pressure and temperature for ideal gases, or the free-electron model for conductors. Results update instantly with a step-by-step breakdown of the calculation.
What is number density?
Number density (symbol n) is the number of particles in a unit of volume. It is measured in particles per cubic metre (m-3) or, for atomic-scale work, per cubic centimetre (cm-3). Number density appears across nearly every branch of physics and materials science: it describes how many atoms are packed into a solid, how many gas molecules occupy a chamber at a given pressure and temperature, and how many free electrons are available to carry current in a conductor. Unlike mass density (which mixes the mass of each atom with how tightly they are packed), number density counts the actual particles, which makes it the natural quantity for cross-section calculations, kinetic-theory formulas, and transport equations.
Three ways to calculate number density
This calculator offers three modes, each matching a different physical situation. For a solid or liquid, the formula is n = (rho x NA) / M, where rho is the mass density in g/cm3, NA is Avogadro's constant (6.022 x 10^23 mol-1), and M is the molar mass in g/mol. For an ideal gas, the kinetic theory gives n = P / (kB x T), where P is pressure in pascals, kB is the Boltzmann constant (1.381 x 10^-23 J/K), and T is absolute temperature in kelvin. For the free-electron model of a conductor, carrier density adds the number of free electrons per atom (Z) as n = (NA x Z x rho) / M. Each approach is exact within its model assumptions, and the derivation is shown in the "Show your work" panel.
Mean free path and flow regimes (gas mode)
When you use the ideal-gas mode, the calculator also estimates the mean free path (lambda), the average distance a molecule travels between collisions: lambda = 1 / (sqrt(2) x n x pi x d^2), where d is the effective collision diameter. At atmospheric pressure (101 325 Pa) and 20 degC, the mean free path of air is roughly 68 nm, so gas flow is fully in the continuum regime. As pressure falls below about 1 Pa (high vacuum), the mean free path exceeds centimetres and molecules travel ballistically without inter-molecular collisions. This distinction, described by the Knudsen number Kn = lambda / L where L is a characteristic dimension, determines whether to apply Navier-Stokes equations (Kn less than 0.01) or free-molecular theory (Kn greater than 10).
Free-electron carrier density in conductors
In a metal, each atom donates Z valence electrons to a shared electron gas that carries current. The free-electron carrier density n = (NA x Z x rho) / M tells you how many of those electrons exist per cubic metre. Copper, gold, and silver donate one free electron per atom (Z = 1); aluminum donates three (Z = 3). The resulting carrier densities in common metals are on the order of 10^28 to 10^29 m-3, far higher than a doped semiconductor (typically 10^16 to 10^24 m-3). Resistivity is inversely related to carrier density and mean free time between collisions, so metals with higher n and longer scattering times (like copper) conduct electricity most efficiently.
Number density of common materials
| Material | Density (g/cm3) | Molar mass (g/mol) | Number density (10^28 m-3) | Carriers/m3 (10^28) |
|---|---|---|---|---|
| Copper (Cu) | 8.96 | 63.546 | 8.491 | 8.491 |
| Aluminum (Al) | 2.7 | 26.982 | 6.026 | 18.078 |
| Gold (Au) | 19.32 | 196.967 | 5.904 | 5.904 |
| Silver (Ag) | 10.49 | 107.868 | 5.857 | 5.857 |
| Iron (Fe) | 7.87 | 55.845 | 8.488 | 16.977 |
| Nickel (Ni) | 8.908 | 58.693 | 9.141 | 18.281 |
| Tungsten (W) | 19.35 | 183.84 | 6.339 | 12.679 |
| Platinum (Pt) | 21.45 | 195.084 | 6.622 | 6.622 |
| Silicon (Si) | 2.329 | 28.085 | 4.993 | 19.974 |
Calculated from published mass density and atomic molar mass values using n = rho x NA / M.
Frequently asked questions
What is the difference between number density and mass density?
Mass density (rho) is the total mass per unit volume (kg/m3 or g/cm3). Number density (n) is the count of individual particles per unit volume (m-3 or cm-3). They are related by n = (rho x NA) / M. Mass density tells you how heavy a material is; number density tells you how many atoms or molecules are actually present, which is what matters for nuclear cross-sections, kinetic theory, and charge-transport calculations.
How do I calculate the number density of an ideal gas?
Use n = P / (kB x T), where P is the absolute pressure in pascals, kB = 1.381 x 10^-23 J/K is the Boltzmann constant, and T is the absolute temperature in kelvin. At standard conditions (101 325 Pa, 273.15 K) this gives roughly 2.687 x 10^25 m-3, known as the Loschmidt constant. At room temperature (293 K) it is about 2.465 x 10^25 m-3.
What units is number density measured in?
The SI unit is particles per cubic metre (m-3). In nuclear engineering and chemistry, particles per cubic centimetre (cm-3) is common. Neither unit has a special name. Because the numbers are very large or very small, scientific notation is almost always used, for example 8.49 x 10^28 m-3 for copper.
Why does aluminum have a higher carrier density than copper even though it is lighter?
Aluminum donates three free electrons per atom (Z = 3) compared to copper's one (Z = 1). Even though aluminum has a lower mass density and a smaller molar mass, the factor-of-three advantage in Z means its carrier density (about 1.81 x 10^29 m-3) ends up higher than copper's (about 8.49 x 10^28 m-3). Despite more carriers, aluminum's higher resistivity compared to copper is due to shorter electron mean free paths from greater phonon scattering.
How is number density used in nuclear engineering?
In reactor physics, the macroscopic cross-section (Sigma) for neutron interaction is computed as Sigma = n x sigma_micro, where sigma_micro is the microscopic cross-section of a single nucleus in cm2 (barns x 10^-24). The macroscopic cross-section has units of cm-1 and represents the probability per unit path length that a neutron will interact. Accurate number densities for each isotope in a compound or mixture are therefore a starting point for every shielding and criticality calculation.