1. What Is Maxwell Speed Distribution?
The Maxwell speed distribution describes how the speeds of gas molecules are spread out. In a gas, not all molecules move with the same speed—some are slow, some are very fast, and many are in between.
The distribution tells what fraction of molecules have particular speeds at a given temperature.
2. Why Do Speeds Vary?
Molecules constantly collide with each other. These collisions randomly change their speeds, so at any moment the molecules have a variety of speeds instead of a single common value.
2.1. Key Idea
Even at a fixed temperature, molecular speeds are not fixed—they are distributed over a range.
3. The Maxwell Distribution Curve
When we plot the number of molecules versus their speeds, we get a curve that rises, peaks, and then falls. This is the Maxwell distribution curve.
3.1. Shape of the Curve
- Low speeds: few molecules
- Medium speeds: most molecules (peak)
- Very high speeds: fewer molecules
The curve is asymmetric—it rises slowly and falls sharply.
4. Most Probable, Average, and RMS Speeds
The Maxwell distribution gives us three important speeds:
4.1. 1. Most Probable Speed (v<sub>mp</sub>)
The speed corresponding to the peak of the distribution curve. Most molecules have this speed.
v_{mp} = \sqrt{\dfrac{2RT}{M}}
4.2. 2. Average Speed (v<sub>avg</sub>)
The average of all molecular speeds.
v_{avg} = \sqrt{\dfrac{8RT}{\pi M}}
4.3. 3. RMS Speed (v<sub>rms</sub>)
The root-mean-square speed related to kinetic energy.
v_{rms} = \sqrt{\dfrac{3RT}{M}}
4.4. Relationship
v_{mp} < v_{avg} < v_{rms}
5. Effect of Temperature on the Distribution
Temperature has a major impact on the distribution curve.
5.1. At Higher Temperature
- The peak shifts to the right (higher speeds).
- The curve becomes broader (greater spread of speeds).
- More molecules have higher speeds.
5.2. At Lower Temperature
- The peak shifts to the left (lower speeds).
- The curve becomes narrower.
- Fewer high-speed molecules exist.
6. Importance of Maxwell Speed Distribution
The distribution helps explain many gas behaviors, such as:
6.1. 1. Effusion and Diffusion
Lighter molecules move faster and spread more quickly because a larger fraction lies in the high-speed tail.
6.2. 2. Evaporation
Only the fastest molecules can escape from a liquid’s surface.
6.3. 3. Reaction Rates
Chemical reactions occur faster when more molecules have enough kinetic energy to overcome activation barriers.
7. Real-Life Analogy
Think of people running in a marathon: some run slowly, some run very fast, and most run at a moderate pace. Molecular speeds behave similarly—the distribution describes the entire range.