1. What Is RMS Speed?
2. Formula for RMS Speed
The RMS speed for molecules of an ideal gas is:
v_{rms} = \sqrt{\dfrac{3RT}{M}}
where:
- R = universal gas constant
- T = absolute temperature (in Kelvin)
- M = molar mass of the gas (in kg/mol)
2.1. Alternate Form (Using Molecular Mass)
v_{rms} = \sqrt{\dfrac{3k_B T}{m}}
Here, m is the mass of one molecule, and kB is the Boltzmann constant.
3. How Temperature Affects RMS Speed
The RMS speed increases when temperature increases because molecules gain kinetic energy.
v_{rms} \propto \sqrt{T}
3.1. Example
If temperature becomes four times larger, the RMS speed becomes twice as large.
4. How Molecular Mass Affects RMS Speed
Heavier gases move slower, and lighter gases move faster. RMS speed decreases with increase in molar mass:
v_{rms} \propto \dfrac{1}{\sqrt{M}}
4.1. Examples of Trend
- Helium (very light) → very high RMS speed
- Oxygen (heavier) → moderate RMS speed
- Xenon (very heavy) → low RMS speed
5. Connection with Kinetic Energy
The RMS speed directly links to the average kinetic energy of molecules:
\dfrac{1}{2} m v_{rms}^2 = \dfrac{3}{2} k_B T
This equation shows that temperature is a measure of the average kinetic energy of gas molecules.
5.1. Interpretation
Higher temperature → faster molecules → higher kinetic energy.
6. Typical Values of RMS Speed
- Helium at room temperature: ~1250 m/s
- Air (mostly N2 and O2): ~500 m/s
- Carbon dioxide: ~400 m/s
These speeds are surprisingly high, even though we do not feel individual molecules!
7. Why RMS Speed Is Useful
RMS speed helps explain real observations such as:
- why lighter gases escape Earth's atmosphere more easily
- why smell spreads faster for lighter molecules
- why diffusion and effusion depend on molecular mass
7.1. Simple Understanding
Faster molecules mix, spread, and react more quickly.