Chemistry

Mean Molecular Speed

Calculate the average speed in a Maxwell–Boltzmann gas.

v ˉ = \frac{8 R T}{π M}

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Source & review

Difficulty

Intermediate

Read time

3 min

Prerequisites

Algebra, logarithms, and molar quantities

Review statusNeeds review

Source

OpenStax, Chemistry 2e, §9.5

Adapted from Teng-Jui Lin’s CHEM 145 equation sheet. The linked textbook reference was used to check the equation and its physical context. Pending editorial review.

Adapted from Teng-Jui Lin, Equation Sheets, under CC BY 4.0.

FormuLab adapted notation and chemical symbols, added variable meanings, units, applicability conditions, and an original worked example.

Definition

Understanding the core concept

The mean molecular speed averages the magnitudes of molecular velocities. At the same temperature and molar mass it is smaller than the rms speed; the average velocity vector can still be zero.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$v ˉ$	Mean molecular speed	$m/s$
$R$	Molar gas constant	$J mo l^{- 1} K^{- 1}$
$T$	Absolute temperature	$K$
$M$	Molar mass	$kg mo l^{- 1}$

Real-World Applications

Where this formula is used in practice

Collision estimates

Supply a mean speed for simple molecular-collision models.

Thermal motion

Compare average molecular speeds for different gases.

Worked Example

Step-by-step calculation with real numbers

Problem

Estimate the mean speed of helium at 300 K using molar mass 0.00400 kg/mol.

Given

T = 300 K

M = 0.00400 kg mo l^{- 1}

R = 8.314462618 J mo l^{- 1} K^{- 1}

Solution

v ˉ = \frac{8 R T}{π M}

v ˉ = \frac{8 ( 8.314462618 ) ( 300 )}{π ( 0.00400 )} m/s

v ˉ \approx 1260 m/s

Final Answer

v ˉ \approx 1260 m/s

Formula Guide

Info

DifficultyIntermediate

Read Time3 min

PrerequisitesAlgebra, logarithms, and molar quantities

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

OpenStax, Chemistry 2e, §9.5

Adapted from Teng-Jui Lin’s CHEM 145 equation sheet. The linked textbook reference was used to check the equation and its physical context. Pending editorial review.

Adapted from Teng-Jui Lin, Equation Sheets, under CC BY 4.0.

FormuLab adapted notation and chemical symbols, added variable meanings, units, applicability conditions, and an original worked example.

Assumptions

• Assume a classical ideal gas in thermal equilibrium with a Maxwell–Boltzmann speed distribution.
• Use positive kelvin temperature and molar mass in kg/mol.

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