Chemistry / Unit 7 Chem 1321 worked example, 60 to 90 seconds
Calculating Root-Mean-Square Speed
The idea
Calculating a gas's root-mean-square speed means substituting its molar mass in kilograms and the kelvin temperature into urms = the square root of (3RT divided by molar mass).
In practice
For nitrogen gas (N2, molar mass 28.02 g/mol) at 30 °C, first list the known quantities in the units the equation needs: R = 8.314 J/mol·K, T = 303 K, and M converted from 28.02 g/mol to 0.02802 kg/mol. Substituting gives urms = √(3 × 8.314 J/mol·K × 303 K ÷ 0.02802 kg/mol). Since a joule equals one kilogram-meter-squared per second-squared, the units under the square root reduce to meters squared per second squared, leaving units of meters per second after taking the root. Carrying the arithmetic through yields urms ≈ 517 m/s, a result in the few-hundred-meters-per-second range expected for a light gas near room temperature, with a positive value and speed units, confirming the answer is physically sensible.
Where people slip
Leaving molar mass in grams per mole instead of converting to kilograms per mole, to match R's joule-based units, produces a speed far larger than the correct value.
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