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Mean Free Path Calculator

Enter temperature, pressure, and molecular diameter to find the mean free path.

How it is calculated

The tool applies kinetic theory for a single pure gas: λ = k·T ÷ (√2 · π · d² · P). Here k = 1.380649×10−23 J/K is the Boltzmann constant, T is the absolute temperature in kelvin, d is the molecular diameter and P is the pressure in pascals.

Enter the diameter in picometres; it is converted to metres (×10−12) before use. The answer is given in metres in scientific notation with four decimals, and again in nanometres. Temperature, pressure and diameter must all be above zero, otherwise no result appears. The conditions line repeats your three inputs at their own precision (a diameter of 364.5 pm stays 364.5 pm); the calculation itself is never rounded before the final result.

Worked example

Take nitrogen-like gas at 300 K and 101,325 Pa with a diameter of 364 pm.

Numerator: k·T = 1.380649×10−23 × 300 ≈ 4.142×10−21 J. Diameter in metres: 3.64×10−10, so d² ≈ 1.325×10−19 m². Denominator: √2·π·d²·P ≈ 4.4429 × 1.325×10−19 × 101,325 ≈ 5.965×10−14. Dividing gives 6.9442 × 10⁻⁸ m, which is 69.4417 nm.

Try this example →

Common mistakes

  • Entering the diameter in nanometres or metres. The field expects picometres, so 0.364 nm must be typed as 364.
  • Entering temperature in Celsius. The formula needs kelvin, so 27 °C is 300 K, not 27. A zero or negative value gives no result.
  • Giving pressure in bar, atm or torr instead of pascals. One atmosphere is 101,325 Pa; typing 1 gives a result about 100,000 times too long.
  • Using the atomic radius instead of the molecular diameter. Because d is squared, halving it makes the answer four times too long.
  • Reading the result for a gas mixture such as air as exact. It uses a single diameter, so it is an estimate from a pure-gas model.

About this calculator

The mean free path is the average distance a gas molecule travels between successive collisions with other molecules — a core quantity in the kinetic theory of gases that connects microscopic molecular motion to macroscopic properties like viscosity, thermal conductivity, and diffusion. It’s calculated as λ = kT / (√2 · π · d² · P), where k is the Boltzmann constant, T is absolute temperature, d is the molecular diameter, and P is pressure — the √2 factor accounts for the relative motion between colliding molecules rather than treating other molecules as stationary targets.

Mean free path shrinks as pressure or molecular size increases and grows as temperature rises, which is why it becomes a critical design number in vacuum engineering (deciding whether a system behaves as a continuum flow or as free molecular flow, judged by the Knudsen number), plasma physics, and semiconductor fabrication processes like sputtering and thin-film deposition that rely on molecules traveling long, uninterrupted distances in a vacuum chamber. This calculator computes it from temperature, pressure, and molecular diameter.

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Comments (1)

  • Editorial team

    Compare the result with the size of your container. For the nitrogen-like gas at 300 K and a 364 pm diameter, 1 Pa gives 7.0362 × 10⁻³ m and 0.1 Pa gives 7.0362 × 10⁻² m, a tenfold step because the path grows as pressure falls. Once the path exceeds the chamber dimension, molecules strike walls more than each other. That is the point where continuum assumptions stop working. What pressure range and chamber size are you working with?

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