Root Mean Square Speed Calculator for Ideal Gas

Background.

Temperature in a gas is molecular motion. In the air of a room at about 27 °C, nitrogen molecules are flying in random directions at around 500 metres per second, faster than sound travels through that same air. No single speed describes them all: the Maxwell–Boltzmann distribution spreads them over a wide range, from almost at rest to more than double the typical value, with most somewhere in between.

The root-mean-square speed is the summary that connects directly to temperature. Square every molecule's speed, average the squares and take the square root. A molecule moving at that speed carries exactly the average translational kinetic energy of the gas, (3/2)kT, so kinetic theory can turn it into a clean formula: v_rms = √(3RT/M).

Two patterns follow. Speed grows only with the square root of absolute temperature, so doubling the kelvin temperature makes the molecules about 41% faster, not twice as fast. And at a given temperature lighter molecules move faster, in proportion to 1/√M. At 300 K hydrogen's rms speed is about 1,927 m/s, about 3.7 times nitrogen's 517 m/s. The same dependence lies behind Graham's law of effusion and is one reason hydrogen and helium leak from the top of Earth's atmosphere into space.

The rms speed is not the most likely speed of a molecule. For any ideal gas the most probable speed, the mean speed and the rms speed stand in the ratio √2 : √(8/π) : √3, about 1.414 : 1.596 : 1.732, so rms is the largest of the three.

The model is the classical ideal gas, a single species of point-like molecules that interact only by colliding. Dense or strongly interacting gases, mixtures and very cold quantum gases need more than this.

What is root mean square speed calculator for ideal gas?

The root-mean-square speed of a gas is the square root of the average of the squared speeds of its molecules. For an ideal gas in thermal equilibrium it depends only on the absolute temperature and the molar mass. It is the speed at which a molecule would carry the gas's average translational kinetic energy of (3/2)kT, which is why it, rather than the simple mean speed, appears in kinetic theory. It is reported in metres per second.

How to use this calculator.

1. 01Enter the temperature in kelvin. Add 273.15 to a Celsius reading: 25 °C is 298.15 K.
2. 02Enter the molar mass in kilograms per mole, which is the g/mol figure divided by 1,000. Nitrogen's 28.0134 g/mol becomes 0.0280134 kg/mol.
3. 03Use one gas at a time. In a mixture every component has its own rms speed at the same temperature, so run each separately.
4. 04Read the rms speed in m/s. Multiply it by 0.921 for the mean speed or by 0.816 for the most probable speed of the same gas.

The formula.

v_rms = √(3RT ÷ M), where T is the absolute temperature in kelvin, M the molar mass in kg/mol and R the molar gas constant, 8.31446261815324 J/(mol·K). R is exact in the SI, the product of the fixed Boltzmann and Avogadro constants. Per molecule the same result reads √(3kT ÷ m). It comes from kinetic theory: the pressure of molecules bouncing off a wall is P = ⅓ × (N/V) × m × ⟨v²⟩, and setting that equal to the ideal-gas law PV = NkT gives ⟨v²⟩ = 3kT ÷ m. Since a joule is a kg·m²/s², RT ÷ M comes out in m²/s² only when M is in kilograms per mole. Temperature and molar mass both sit under the square root, so quadrupling T or quartering M doubles the speed.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, University Physics Volume 2, section 2.1, Molecular Model of an Ideal Gas.
2. [2]BIPM, SI Brochure, exact Boltzmann and Avogadro constants defining R.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.