The Maxwell Boltzmann Distribution (College Board AP® Chemistry): Revision Note

The Maxwell Boltzmann Distribution

Average Velocity (v) and Average Kinetic Energy

• According to the kinetic molecular theory, the average kinetic energy of gas molecules is directly proportional to their absolute temperature

  • However, some molecules will have more than the average kinetic energy and others will have below the average

  • Similar effects are observed for the velocities of the gas molecules

Distribution of Molecular Velocity: Maxwell-Boltzmann Distribution

• While the temperature of a gas indicates the average velocity or kinetic energy of the gas molecules, at any instant, gas molecules are moving at a variety of velocities

  • For example, at room temperature, the average speed of N₂ molecules is about 475 m/s but not all molecules have this speed at that temperature

  • Given the chaotic motion of gas molecules, the velocity of a particle is constantly changing

  • Over a period of time, the velocity will vary from almost zero to some very high value, considerably above the average

• In 1860 James Clerk Maxwell, a Scottish physicist, showed that different possible velocities are distributed among particles in a definite way

  • He developed a mathematical expression for this distribution, now referred to as the Maxwell-Boltzmann distribution

  • A graphical representation of his results is shown below:

Maxwell-Boltzmann Curve

A graph showing the general distribution of the speed of gas molecules

Distribution of Molecular Velocity and Temperature

• A key part of Maxwell’s observation was the dependence of the gas molecules’ speed on temperature

  • At any given temperature, the distribution curve shows the number of molecules moving at a certain speed

  • Consider the graph showing the distribution of O₂ molecules at two different temperatures:

Distribution curve for O₂

A graph showing the distribution of molecular velocities of oxygen molecules at 25℃ and 1000℃. At a higher temperature, more molecules of oxygen move at higher velocities.

• From this graph the following deduction can be made:

  • The peak of the curve represents the most probable velocity which describes the speed of the largest number of molecules

    • Note that the most probable velocity increases as temperature increases (the peak shifts toward the right at 1000℃)

  • Furthermore, the curve also begins to flatten out with increasing temperature, indicating that larger numbers of molecules are moving at greater speed

  • The area under each curve represents the total number of molecules, so it stays the same when the temperature changes: the curve gets lower as it gets broader