Gas atoms or molecules act almost independently of each other in comparison to liquids or solids, particles of which have greater correlation. This is because a gas may occupy thousands of times more volume than the corresponding liquid. The root-mean-square velocity of gas particles varies directly with temperature, according to the “Maxwell Speed Distribution.” That equation enables the calculation of velocity from temperature.

## Derivation of Maxwell Speed Distribution Equation

Learn the derivation and application of the Maxwell Speed Distribution equation. That equation is based on and derived from the Ideal Gas Law equation:

PV = nRT

where P is pressure, V is volume (not velocity), n is the number of moles of gas particles, R is the ideal gas constant and T is the temperature.

Study how this gas law is combined with the formula for kinetic energy:

KE = 1/2 m v^2 = 3/2 k T.

Appreciate the fact that the velocity for a single gas particle cannot be derived from the temperature of the composite gas. In essence, each particle has a different velocity and so has a different temperature. This fact has been taken advantage of to derive the technique of laser cooling. As a whole or unified system, however, the gas has a temperature that can be measured.

Calculate the root-mean-square velocity of gas molecules from the temperature of gas using the following equation:

Vrms = (3RT/M)^(1/2)

Make sure to use units consistently. For example, if the molecular weight is taken to be in grams per mole and the value of the ideal gas constant is in joules per mole per degree Kelvin, and the temperature is in degrees Kelvin, then the ideal gas constant is in joules per mole-degree Kelvin, and the velocity is in meters per second.

Practice with this example: if the gas is helium, the atomic weight is 4.002 grams/mole. At a temperature of 293 degrees Kelvin (about 68 degrees Fahrenheit) and with the ideal gas constant being 8.314 joules per mole-degree Kelvin, the root-mean-square velocity of the helium atoms is:

(3 x 8.314 x 293/4.002)^(1/2) = 42.7 meters per second.

Use this example to calculate velocity from temperature.