Core Concept

Maxwell-Boltzmann Distribution describes the range of speeds that gas particles can have at a given temperature. It provides insight into how molecular speed and energy vary among particles in a gas.

Key Concepts

1. Distribution of Molecular Speeds:

   • Not all particles in a gas move at the same speed. Some move faster, and some slower, depending on factors like temperature and the mass of the gas particles.

   • The Maxwell-Boltzmann Distribution curve shows the range of speeds, with most particles clustering around a most probable speed.

2. Temperature and Kinetic Energy:

   • Temperature directly affects the distribution of molecular speeds.

   • As temperature increases, the average speed and kinetic energy of particles increase, and the distribution curve flattens and shifts to the right (indicating higher speeds).

3. Effect of Particle Mass:

   • Heavier gas particles move more slowly on average than lighter gas particles at the same temperature.

   • Lighter gases, like helium, have broader distributions and higher average speeds compared to heavier gases like xenon at the same temperature.

Maxwell-Boltzmann Distribution Curve

1. Shape of the Curve:

   • The curve is asymmetrical, starting at the origin (0 speed, where no particles exist) and rising to a peak that represents the most probable speed.

   • The curve then tails off to the right, indicating a small number of particles with very high speeds.

2. Key Points on the Curve:

   • Most Probable Speed ($u_{\text{mp}}$​): The speed at which the largest number of particles are moving. This is the peak of the distribution.

   • Average Speed ($u_{\text{avg}}$​): The mean speed of all particles, slightly to the right of the most probable speed.

   • Root Mean Square Speed ($u_{\text{rms}}$​): A statistical measure of the speed of particles, even farther to the right. It takes into account the square of particle speeds.

   Typically, these speeds relate as follows:

   $u_{\text{mp}} < u_{\text{avg}} < u_{\text{rms}}$

3. Effect of Temperature on the Curve:

   • At higher temperatures, the curve becomes broader and flatter, shifting to the right. This shows that more particles have higher speeds.

   • At lower temperatures, the curve is narrower and steeper, with more particles near the most probable speed.

Maxwell-Boltzmann Equation (Advanced Concept)

The Maxwell-Boltzmann distribution of speeds in a gas can be described mathematically (optional for AP Chemistry):

$f(u) = 4 \pi \left( \frac{m}{2 \pi k T} \right)^{3/2} u^2 e^{-\frac{mu^2}{2kT}}$f

Where:

• f(u) is the fraction of particles with speed u.

• m is the mass of a gas particle.

• k is the Boltzmann constant (1.38×10−23J/K).

• T is the temperature in Kelvin.

This equation is not necessary to memorize but helps to understand that speed distribution depends on particle mass and temperature.

Visualizing the Maxwell-Boltzmann Distribution Curve

1. Comparing Temperatures:

   • For the same gas, a distribution curve at a higher temperature is broader and flatter, indicating a greater range of particle speeds.

2. Comparing Gases at the Same Temperature:

   • Lighter gases have a wider distribution (more particles with high speeds) than heavier gases, which have a narrower distribution.