1. Why Molecules Have Different Speeds
In a gas, molecules do not move with the same speed. Collisions continually change their speeds, creating a spread of slow, moderate, and fast molecules.
The Maxwell distribution helps define three important speeds that describe this spread.
2. Most Probable Speed (v<sub>mp</sub>)
The most probable speed is the speed possessed by the largest number of molecules. It corresponds to the peak of the Maxwell speed distribution curve.
v_{mp} = \sqrt{\dfrac{2RT}{M}}
2.1. Meaning
If you pick a random molecule in the gas, it is most likely to have a speed close to vmp.
2.2. Example
For air at room temperature, vmp is roughly 400–450 m/s.
3. Average Speed (v<sub>avg</sub>)
The average speed is the arithmetic mean of all molecular speeds. It is given by:
v_{avg} = \sqrt{\dfrac{8RT}{\pi M}}
3.1. Why It Is Higher Than v<sub>mp</sub>
The presence of some very fast molecules pulls the average slightly higher than the most probable speed.
4. RMS Speed (v<sub>rms</sub>)
The root-mean-square speed is calculated from the square root of the average of squared speeds. It is directly linked to kinetic energy.
v_{rms} = \sqrt{\dfrac{3RT}{M}}
4.1. Why It Is the Largest
Squaring gives greater weight to high speeds, so vrms becomes the highest among the three.
5. Relation Between the Three Speeds
The three characteristic speeds always follow the relation:
v_{mp} < v_{avg} < v_{rms}
This ordering arises because of the long tail of high-speed molecules in the Maxwell distribution.
5.1. Physical Interpretation
Most molecules are around vmp, but some fast ones push the average and RMS values higher.
6. Effect of Temperature on Speeds
All three characteristic speeds increase with temperature:
v \propto \sqrt{T}
As temperature rises, molecules move faster on average.
6.1. Example
Heating a gas from 300 K to 600 K increases vmp, vavg, and vrms by a factor of \( \sqrt{2} \).
7. Effect of Molecular Mass
All three speeds decrease when molecular mass increases:
v \propto \dfrac{1}{\sqrt{M}}
Lighter gases move faster; heavier gases move slower.
7.1. Examples
- Helium: very high speeds
- Nitrogen: moderate speeds
- Xenon: slow speeds
8. Why These Speeds Matter
The three characteristic speeds help explain several gas properties:
- Rate of effusion and diffusion
- Escape of gases from planetary atmospheres
- Reaction rates in gases
- Thermal conductivity
8.1. Simple Understanding
The faster the molecules move, the more active the gas becomes in spreading, mixing, and transferring energy.