KAHIBARO
Table of Contents
Random molecular motion
In a gas, molecules move constantly in many different directions and with many different speeds. Because the motion is random, the average velocity of all molecules in a container can be zero, even though the molecules are moving very fast. This is why we need a different kind of average to describe molecular motion.
The useful quantity is the root mean square speed, usually written as $v_{\mathrm{rms}}$. It gives a single speed value that represents the overall molecular motion in a gas.
Meaning of RMS speed
The term root mean square comes from three steps. First, square each molecular speed. Second, take the average of those squared speeds. Third, take the square root.
If the speeds of molecules are $v_1, v_2, v_3, \dots, v_N$, then
$$
v_{\mathrm{rms}} = \sqrt{\frac{v_1^2 + v_2^2 + \cdots + v_N^2}{N}}
$$
This is not the same as the ordinary average speed. Squaring gives more weight to larger speeds, so $v_{\mathrm{rms}}$ is always at least as large as the simple average speed.
The RMS speed is defined by
$$
v_{\mathrm{rms}} = \sqrt{\langle v^2 \rangle}
$$
where $\langle v^2 \rangle$ means the average of the squared speeds.
Connection to kinetic energy
In kinetic theory, the average translational kinetic energy of one molecule is related to temperature. For a molecule of mass $m$,
$$
\frac{1}{2}m \langle v^2 \rangle = \frac{3}{2}k_B T
$$
Since $\langle v^2 \rangle = v_{\mathrm{rms}}^2$, we get
$$
\frac{1}{2}m v_{\mathrm{rms}}^2 = \frac{3}{2}k_B T
$$
Solving for $v_{\mathrm{rms}}$ gives
$$
v_{\mathrm{rms}} = \sqrt{\frac{3k_B T}{m}}
$$
This formula shows two important facts. RMS speed increases when temperature increases. RMS speed decreases when molecular mass increases.
For a gas molecule of mass $m$ at absolute temperature $T$,
$$
v_{\mathrm{rms}} = \sqrt{\frac{3k_B T}{m}}
$$
This is one of the key formulas of kinetic theory.
Formula using molar mass
Sometimes it is more convenient to use molar mass $M$ instead of the mass of one molecule. Using the gas constant $R$, the formula becomes
$$
v_{\mathrm{rms}} = \sqrt{\frac{3RT}{M}}
$$
Here, $M$ must be in $\mathrm{kg/mol}$, not in $\mathrm{g/mol}$.
For example, if a gas has molar mass $28\,\mathrm{g/mol}$, then in the formula we must use
$$
M = 0.028\,\mathrm{kg/mol}
$$
When using
$$
v_{\mathrm{rms}} = \sqrt{\frac{3RT}{M}}
$$
always use temperature in kelvin and molar mass in $\mathrm{kg/mol}$.
What RMS speed tells us physically
RMS speed is a measure of how energetic the molecular motion is. Since kinetic energy depends on $v^2$, the RMS speed naturally appears in the energy relation.
A higher RMS speed means molecules are, on average, moving more vigorously. This leads to more energetic collisions with the container walls. The detailed link to pressure belongs to the molecular interpretation of pressure, but RMS speed is one of the central quantities behind that idea.
Temperature dependence
Because
$$
v_{\mathrm{rms}} \propto \sqrt{T}
$$
the RMS speed does not double when temperature doubles. Instead, it increases by the square root of the temperature factor.
If temperature changes from $T_1$ to $T_2$, then
$$
\frac{v_{\mathrm{rms},2}}{v_{\mathrm{rms},1}} = \sqrt{\frac{T_2}{T_1}}
$$
So if the temperature becomes four times larger, the RMS speed becomes twice as large.
Effect of molecular mass
Lighter molecules move faster than heavier molecules at the same temperature. This follows from
$$
v_{\mathrm{rms}} \propto \frac{1}{\sqrt{m}}
$$
or equivalently
$$
v_{\mathrm{rms}} \propto \frac{1}{\sqrt{M}}
$$
Hydrogen molecules have a much larger RMS speed than oxygen molecules at the same temperature because hydrogen molecules are much lighter.
Comparison of gases at the same temperature
For two gases at the same temperature,
$$
\frac{v_{\mathrm{rms},1}}{v_{\mathrm{rms},2}} = \sqrt{\frac{M_2}{M_1}}
$$
This relation is useful because the temperature factor cancels out.
Suppose gas 1 is helium with $M_1 = 0.004\,\mathrm{kg/mol}$ and gas 2 is nitrogen with $M_2 = 0.028\,\mathrm{kg/mol}$. Then
$$
\frac{v_{\mathrm{rms,He}}}{v_{\mathrm{rms,N_2}}}
= \sqrt{\frac{0.028}{0.004}}
= \sqrt{7}
\approx 2.65
$$
So helium molecules move about $2.65$ times faster in RMS speed than nitrogen molecules at the same temperature.
Example calculation
Let us find the RMS speed of nitrogen gas at $T = 300\,\mathrm{K}$. The molar mass of nitrogen is approximately
$$
M = 0.028\,\mathrm{kg/mol}
$$
Using
$$
v_{\mathrm{rms}} = \sqrt{\frac{3RT}{M}}
$$
with $R = 8.31\,\mathrm{J/(mol \cdot K)}$, we get
$$
v_{\mathrm{rms}} = \sqrt{\frac{3(8.31)(300)}{0.028}}
$$
$$
v_{\mathrm{rms}} = \sqrt{267107.14}
\approx 517\,\mathrm{m/s}
$$
So the RMS speed of nitrogen molecules at room temperature is about $517\,\mathrm{m/s}$.
RMS speed, average speed, and most probable speed
A gas does not have one single molecular speed. Instead, it has a spread of speeds. Because of this, several different characteristic speeds are used.
For an ideal gas, the three common ones are the most probable speed, the average speed, and the RMS speed. Their exact distribution is covered elsewhere, but it is useful to know their order.
| Speed type | Symbol | Relative size |
|---|---|---|
| Most probable speed | $v_{mp}$ | Smallest |
| Average speed | $\bar{v}$ | Middle |
| RMS speed | $v_{\mathrm{rms}}$ | Largest |
For an ideal gas,
$$
v_{mp} < \bar{v} < v_{\mathrm{rms}}
$$
This happens because RMS speed gives extra weight to larger speeds.
Visual idea
The molecules in a gas move in all directions, and their speeds are spread over a range of values. RMS speed is a single number that summarizes that motion in a way that matches kinetic energy.
Common mistakes
A common mistake is to use Celsius temperature in the RMS formula. Temperature must always be in kelvin.
Another common mistake is to use molar mass in grams per mole instead of kilograms per mole. Since the SI unit is $\mathrm{kg/mol}$, you must convert properly.
Students also sometimes confuse speed with velocity. RMS speed uses the magnitude of molecular motion, not direction. Even if the average velocity is zero, the RMS speed is not zero.
Important checks for RMS speed calculations:
$$
v_{\mathrm{rms}} = \sqrt{\frac{3k_B T}{m}} = \sqrt{\frac{3RT}{M}}
$$
Use $T$ in kelvin.
Use $m$ in kilograms for one molecule, or $M$ in $\mathrm{kg/mol}$ for one mole.
Do not confuse average velocity with RMS speed.
Summary
RMS molecular speed is a statistical measure of how fast gas molecules move. It is especially useful because it connects directly to molecular kinetic energy. For an ideal gas,
$$
v_{\mathrm{rms}} = \sqrt{\frac{3k_B T}{m}} = \sqrt{\frac{3RT}{M}}
$$
It increases with the square root of temperature and decreases with the square root of molecular mass. Lighter gases move faster than heavier gases at the same temperature, and hotter gases have larger RMS speeds than cooler gases.