RMS Gas Speed Calculator: v = sqrt(3RT / M)

The typical speed of gas molecules, from the temperature and the molar mass.

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

  • v_{rms} = \sqrt{\frac{3RT}{M}}
  • T = \frac{Mv_{rms}^2}{3R}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the rms speed: rms speed = (sqrt(3R) x temperature^0.5) / (molar mass^0.5)
  2. Put the values in (every value in its SI unit): rms speed = (sqrt(3R) x 293^0.5) / (0.02897^0.5)
  3. The answer: rms speed: 502.27 m/s
  4. The same answer in other units: 1808.17 km/h

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

A second worked example

A different case, with these numbers: What do you want to find? Temperature; RMS speed (leave it as it is if this is what you are finding) 502 m/s; Molar mass (leave it as it is if this is what you are finding) 0.02897 kg/mol.

  1. The formula, arranged to find the temperature: temperature = ((rms speed x molar mass^0.5) / (sqrt(3R)))^2
  2. Put the values in (every value in its SI unit): temperature = ((502 x 0.02897^0.5) / (sqrt(3R)))^2
  3. The answer: temperature: 292.685 K

Common questions

What is the RMS speed?

The square root of the average of the squares of the molecular speeds. It is a little more than the mean speed.

How does temperature change the speed?

The speed rises with the square root of the absolute temperature.

Why must the molar mass be in kg per mole?

Because the gas constant R is in joules per mole kelvin, so the answer comes out in metres per second only with SI units.

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

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