Mean Free Path Calculator: lambda = kT / (sqrt(2) pi d^2 p)

How far a gas molecule goes before it hits another.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose which quantity you want to find.

Enter the other values in the units shown.

The calculator gives the answer with the formula, the substituted numbers, and the answer in other units.

Formulas used

  • \lambda = \frac{k_BT}{\sqrt{2}\pi d^2p}
  • p = \frac{k_BT}{\sqrt{2}\pi d^2\lambda}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the mean free path: mean free path = (k / (sqrt(2) x pi) x temperature) / (diameter of a molecule^2 x pressure)
  2. Put the values in (every value in its SI unit): mean free path = (k / (sqrt(2) x pi) x 293) / (3.7e-10^2 x 101325)
  3. The answer: mean free path: 6.56396e-8 m
  4. The same answer in other units: 65.6396 nm; 0.0656396 micrometres

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What is the mean free path of air?

About 66 nm at room temperature and atmospheric pressure.

How does it change in a vacuum?

It gets much longer. At 1e-3 Pa it is about 6.6 m.

Why does it matter?

It decides when a gas stops behaving as a smooth fluid and the individual molecules matter, as in vacuum systems and small channels.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any physics or chemistry text (kinetic theory of gases).

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