# Maxwell Boltzmann Formula

Created by: Team Physics - Examples.com, Last Updated: May 10, 2024

## What is Maxwell Boltzmann Formula?

The Maxwell-Boltzmann formula, developed by James Clerk Maxwell and Ludwig Boltzmann in the 19th century, describes the speed distribution of particles in a gas at thermal equilibrium. This formula is pivotal in physics, especially in the study of statistical mechanics. It helps in understanding how particle speeds in a gas vary at a given temperature.

The formula is expressed as

π(π£) = ( (π / 2πππ)^ 3/ 2 ) x (π ^βππ£Β² / 2ππ)
• π is the mass of the particles.
• π is the temperature.
• π is Boltzmann’s constant.
• π£ is the speed of the particles.

This equation shows that most particles in a gas move at moderate speeds, with very few moving very quickly or very slowly, providing a statistical view of molecular motion in gases.

## Usages of Maxwell Boltzmann Formula

• Predicting Gas Properties: It calculates properties like pressure and temperature of gases, crucial for understanding gas behavior in various conditions.
• Engineering Applications: Engineers use this formula to design and optimize systems involving gas flow, such as jet engines and exhaust systems.
• Chemical Reaction Rates: The formula helps in estimating the speeds of molecules which is vital for determining how fast chemical reactions occur.
• Astrophysics: It aids in analyzing the distribution of speeds of particles in stars and planetary atmospheres, offering insights into their thermal properties.
• Material Science: Understanding the diffusion of gases through materials, important for creating efficient filters and membranes.

## Example Problems on Maxwell Boltzmann Formula

### Problem 1: Calculating Particle Speed

Question: In a container of helium gas at 300 K, estimate the most probable speed of the helium atoms. Assume the mass of a helium atom is 4Γ10β»Β²β· kg.

Solution:

The most probable speed π£πvpβ can be calculated using the formula: π£π = β 2ππ / π Where:

π=1.38Γ10β»Β²Β³βJ/K (Boltzmann constant),

π=300βK,

π=4Γ10β»Β²β·βkg.

Plugging in the values:

π£π = β( ( 2 Γ 1.38 Γ 10β»Β²Β³ Γ 300 ) / ( 4 Γ 10β»Β²β· ) ) β 1370βm/s

### Problem 2: Comparing Particle Speeds

Question: Compare the most probable speeds of hydrogen and oxygen molecules in a gas mixture at 400 K. Assume the mass of a hydrogen molecule (Hβ) is 2 Γ 10β»Β²β· kg and the mass of an oxygen molecule (Oβ) is 32Γ10β»Β²β· kg.

Solution: Calculate the most probable speed for each:

Hydrogen:

π£π, Hβ = β 2ππ / πHβ = β (( 2 Γ 1.38 Γ 10β»Β²Β³ Γ 400) / (2Γ10β»Β²β·)) β 1838βm/s

Oxygen:

π£π, Oβ = β 2ππ / πOβ = β ( (2Γ1.38Γ10β»Β²Β³ Γ 400 ) / ( 32 Γ 10β»Β²β· ) ) β 459βm/s

### Problem 3: Determining Kinetic Energy Distribution

Question: Calculate the fraction of nitrogen molecules (molecular mass 28Γ10β»Β²β· kg) moving faster than 500 m/s at a temperature of 298 K.

Solution: First, calculate the fraction using the Maxwell-Boltzmann speed distribution function for speeds greater than 500 m/s. We use the cumulative distribution function (CDF):

Fraction=β«ββββπ(π£)βππ£

Where π(π£) is the Maxwell-Boltzmann distribution function. We simplify by using an integral table or computational tools because the integral calculation involves exponential functions:

Fraction β 0.12

## What Does Maxwell-Boltzmann Show?

Maxwell-Boltzmann distribution illustrates how gas particlesβ speeds vary at a specific temperature, predicting molecular motion in thermal equilibrium.

## What is K in Boltzmann’s Formula?

In Boltzmann’s formula, πk represents the Boltzmann constant, 1.38Γ10β»Β²Β³βJ/K, linking temperature with energy.

## What is the Formula for Maxwell Boltzmann Statistics?

The Maxwell-Boltzmann formula is π(π£) = ( (π / 2πππ)^ 3/ 2 ) x (π ^βππ£Β² / 2ππ)β, defining particle speed distribution in gases.

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