This independent behavior allows us to calculate the root-mean-square velocity of gas particles directly from temperature.

## Tweaking the Ideal Gas Law

We derive this equation from another well-known equation, the Ideal Gas Law equation. Algebraically, that equation is written:PV = nRT

where P is pressure, V is volume (not velocity), n is the number of moles of gas, R is the ideal gas constant and T is the temperature.By combining the above equation with derivatives of Boltzmann’s equation, we can determine the velocity of a particle from its temperature. However, it is important to realize that the temperature and the velocity of any one particle are statistically unlikely to equal those of the composite gas. Each particle moves independently. So we speak of the root-mean-square velocity of the bulk gas using the following equation:

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

## A Sample Example

Let’s choose helium. Then the molar mass M is 0.004002 kilograms. At a temperature of 293 degrees Kelvin, with the ideal gas constant at 8.314 joules per mole-degree Kelvin, the root-mean-square velocity of the helium atom is:Vrms = (3 x 8.314 x 293/0.004002)^{1/2} = 1351 meters per second.

**References:**

Georgia State University Hyper Physics: Ideal Gas Law

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