Calculate particle number density n = N/V from particle count, pressure and temperature, or mass density. Log-scale comparison from interstellar space to solids.
Number density n is the count of particles per unit volume (#/m³ or #/cm³). It is fundamental in gas kinetics, plasma physics, semiconductor doping, and astrophysics. This calculator offers four input modes to determine number density from whatever information you have.
Mode 1: n = N/V from a known particle count and volume. Mode 2: n = P/(k_BT) from the ideal gas law for a gas at known pressure and temperature. Mode 3: n = ρN_A/M from mass density and molar mass. Mode 4: enter number density directly for unit conversion and comparison.
Results include number density in #/m³ and #/cm³, mean free path (assuming air-like collision cross-section), inter-particle distance, a regime classification (vacuum to condensed matter), and a logarithmic-scale comparison spanning 24 orders of magnitude—from interstellar space (10⁶/m³) to solid iron (8.5×10²⁸/m³). Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
Number density bridges the macroscopic (pressure, temperature, mass density) and microscopic (particle count, mean free path) worlds. It is essential in kinetic theory, vacuum technology, plasma engineering, semiconductor physics, and astrophysics.
The log-scale comparison spanning 24 orders of magnitude provides immediate physical context for any value—whether you are working with interstellar gas or liquid water.
Number density: n = N/V (count/volume), n = P/(k_BT) (ideal gas), n = ρN_A/M (from mass density). Mean free path: λ = 1/(√2 π d² n). Inter-particle distance: ℓ ≈ n^(−1/3). k_B = 1.381×10⁻²³ J/K, N_A = 6.022×10²³ /mol.
Result: 2.687 × 10²⁵ /m³
n = 101325 / (1.381e-23 × 273.15) = 2.687 × 10²⁵ /m³. This is the Loschmidt number—the number density of an ideal gas at STP.
The range of number densities encountered in nature spans over 30 orders of magnitude:
| Environment | n (#/m³) | log₁₀(n) | |---|---|---| | Intergalactic space | ~1 | 0 | | Interstellar medium | ~10⁶ | 6 | | Solar wind at 1 AU | ~7×10⁶ | 7 | | Best laboratory vacuum | ~10¹² | 12 | | Low Earth orbit (400 km) | ~10¹⁴ | 14 | | Mars atmosphere (surface) | ~2×10²³ | 23 | | Earth atmosphere (sea level) | 2.5×10²⁵ | 25 | | Liquid water | 3.3×10²⁸ | 28 | | Solid iron | 8.5×10²⁸ | 29 | | White dwarf core | ~10³⁶ | 36 | | Neutron star | ~10⁴⁴ | 44 |
The ratio of mean free path to a characteristic length gives the Knudsen number: Kn = λ/L. When Kn < 0.01, the gas behaves as a continuum (Navier-Stokes applies). When Kn > 10, molecular flow dominates (individual particle trajectories matter). This is critical for vacuum system design, microfluidics, and hypersonic aerodynamics.
It is the number density of an ideal gas at STP: n_L ≈ 2.687 × 10²⁵ /m³. One mole of ideal gas at STP occupies 22.414 L, so n = N_A / 0.022414 = 2.687 × 10²⁵.
Molar concentration (mol/L) counts moles; number density counts individual particles. Convert: n (#/m³) = c (mol/L) × N_A × 1000.
Mean free path depends on number density and collision cross-section: λ = 1/(√2 π d² n). Higher particle density → shorter mean free path.
Plasma behavior—Debye shielding, plasma frequency, collision rates—are all determined by electron and ion number densities. Use this as a practical reminder before finalizing the result.
At 20 °C and 1 atm: n = 101325 / (1.381e-23 × 293.15) ≈ 2.50 × 10²⁵ /m³. The mean free path is about 68 nm.
Yes—silicon doping levels are expressed as number density. Intrinsic Si has n_i ≈ 1.5 × 10¹⁶ /m³ at 300 K. Typical doping: 10²¹–10²⁵ /m³.