Maxwell Speed Distribution Calculator
Enter temperature and molar mass to compute the most probable, mean and RMS molecular speeds of an ideal gas.
Input Data
Results
At a glance:The Maxwell-Boltzmann distribution describes the spread of molecular speeds in an ideal gas at temperature T. Three characteristic speeds are commonly quoted: the most probable speed v_p = √(2RT/M) (peak of the distribution), the mean (average) speed v_mean = √(8RT/πM), and the RMS speed v_rms = √(3RT/M). Here R ≈ 8.314 J/(mol·K) is the gas constant and M is the molar mass in kg/mol. Order: v_p < v_mean < v_rms. The RMS speed is used in kinetic theory for average kinetic energy: ½m v_rms² = 3/2 k_B T.
Formula
Most probable: v_p = √(2RT/M)
Mean: v_mean = √(8RT/πM)
RMS: v_rms = √(3RT/M)
$$v_p = \sqrt{\frac{2kT}{m}}, \quad v_{avg} = \sqrt{\frac{8kT}{\pi m}}, \quad v_{rms} = \sqrt{\frac{3kT}{m}}$$How to Use
- Enter the temperature T (K).
- Enter the molar mass M (kg/mol).
- The calculator returns v_p, v_mean and v_rms.
Case Studies
Nitrogen at room temperature
T = 300 K, M = 0.028 kg/mol.
v_p ≈ 419 m/s, v_mean ≈ 476 m/s, v_rms ≈ 517 m/s.
Close to the speed of sound in air.
FAQ
Why are the three speeds different?
The distribution is skewed, so the peak (most probable), the arithmetic average (mean) and the square-root-of-average-of-squares (RMS) all differ. They satisfy v_p < v_mean < v_rms.
Which speed matters for energy?
The RMS speed: average translational kinetic energy is ½m·v_rms² = 3/2 k_B T, linking temperature to molecular motion.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.