RMS Speed — Nitrogen at Room Temperature (Worked Example)

Given

• T = 300 K

• M = 0.028 kg/mol

• R = 8.314 J/(mol·K)

Find

The rms speed v_rms.

Principle

Kinetic theory: temperature is molecular motion, v_rms = √(3RT/M).

Formula

Solution

1. Substitute: $v_{rms} = \sqrt{3 \times 8.314 \times 300/0.028}$

2. Inside: $7482.6/0.028 = 267{,}236$

3. Square root: $v_{rms} = 517\ \mathrm{m/s}$

Unit check

√(J/(mol·K) × K ÷ kg/mol) = √(m²/s²) = m/s — correct.

What this means physically

517 m/s — faster than a passenger jet, yet air feels still because molecules collide billions of times per second and go nowhere on average. Temperature is this motion.