Number Density Calculator

Number density (n) is the number of particles — atoms, molecules, ions, or electrons — per unit volume, usually written as particles per cubic meter (m⁻³) or per cubic centimeter (cm⁻³). For example, air at standard temperature and pressure holds about 2.687e+25 molecules per cubic meter (2.687e+19 per cm³), a value known as Loschmidt's number. Mass density tells you how much mass a volume holds; number density tells you how many particles it holds.

For an ideal gas, n = p / (kB·T), where p is the pressure in pascals, kB = 1.380649×10⁻²³ J/K is the Boltzmann constant (exact), and T is the absolute temperature in kelvin. Worked example: air at 101325 Pa and 273.15 K gives n = 101325 / (1.380649×10⁻²³ × 273.15) = 2.687e+25 m⁻³. The formula assumes an ideal gas — dense gases and liquids deviate from it.

Loschmidt's number (n0) is the number density of an ideal gas at standard temperature and pressure (273.15 K, 101325 Pa): n0 = 2.687e+25 molecules/m³, or 2.687e+19 molecules/cm³. Named after Johann Josef Loschmidt, it also equals p0·NA/(R·T0), the same quantity written with the gas constant R and Avogadro's number NA. This calculator shows every ideal-gas result relative to n0 as a bonus readout.

Use n = Z·ρ·NA / M, where ρ is the mass density in kg/m³, NA = 6.02214076×10²³ /mol is Avogadro's number (exact), M is the molar mass in g/mol, and Z is the number of free carriers per particle (Z = 1 for molecules; Z = 1–3 for conduction electrons in metals). Example: copper (ρ = 8960 kg/m³, M = 63.546 g/mol, Z = 1) gives n = 8.491e+28 free electrons/m³.

Number density counts particles (n, in m⁻³); mass density weighs them (ρ, in kg/m³). They are linked by the particle mass: ρ = n·M/NA, with M the molar mass. For air at STP, n = 2.687e+25 m⁻³ corresponds to ρ ≈ 1.293 kg/m³ with M = 28.97 g/mol. Two gases can share the same number density but have very different mass densities when their molecules differ in mass.

Kinetic theory gives λ = 1 / (√2·π·d²·n): the mean free path is inversely proportional to number density — pack molecules closer and each one collides sooner. This calculator computes n, the input; our mean free path calculator continues from temperature, molecular diameter, and pressure to find how far molecules travel between collisions. They are companion tools, not duplicates.

Molar density (concentration) is c = n / NA. Air at STP has n = 2.687e+25 m⁻³, which is 44.62 mol/m³ or 0.04462 mol/L. Conversely, a 1 mol/L solution holds 6.02214076×10²³ × 1000 = 6.022e+26 particles/m³. Chemists usually work in mol/L; physicists and vacuum engineers usually work in m⁻³ — this calculator shows both.

Air at STP (101325 Pa, 0 °C): 2.687e+25 m⁻³. Air at room temperature (101325 Pa, 20 °C): 2.503e+25 m⁻³. A rough laboratory vacuum (100 Pa, 20 °C): 2.471e+22 m⁻³. A high vacuum (10⁻⁴ Pa, 20 °C): 2.471e+16 m⁻³. Liquid water (ρ = 1000 kg/m³, M = 18.015 g/mol): 3.343e+28 molecules/m³. Copper's free electrons: 8.491e+28 /m³. See the reference table above for the full comparison in m⁻³ and cm⁻³.

Vacuum technology (pump-down targets), plasma physics and fusion (electron density), semiconductor doping profiles, atmospheric science, and astrophysics — the interstellar medium sits near 10⁶ particles/m³, about 19 orders of magnitude thinner than air at STP. Combined with the molecular diameter d, n also determines the mean free path via λ = 1/(√2·π·d²·n), and hence the Knudsen number.

Pressure: pascals (Pa), kilopascals (kPa), bar, standard atmospheres (atm), psi, or torr. Temperature: kelvin (K), degrees Celsius (°C), or degrees Fahrenheit (°F). Volume: cubic meters (m³), cubic centimeters (cm³), or liters (L). Mass density: kg/m³, g/L (identical to kg/m³), or g/cm³; molar mass in g/mol. Results are shown in m⁻³, cm⁻³, and L⁻¹ side by side.