Molecular speeds and the Maxwell distribution

A crowd with many speeds

Think of a busy station. Some people run, many walk and a few stand still. A gas is like this. At any moment the molecules have different speeds. When two molecules collide, one may speed up and the other slow down. Their total energy stays the same, and the speeds change. So even if all molecules started at one speed, the collisions would spread the speeds out within a tiny fraction of a second.

James Clerk Maxwell worked out the spread in 1860. Ludwig Boltzmann later put it on a wider base. The result is the Maxwell distribution, also called the Maxwell-Boltzmann distribution, of molecular speeds.

The formula and what it means

For a gas of molar mass $M$ at temperature $T$, the fraction of molecules with speeds between $u$ and $u + du$ is $f(u)\,du$, where

$M$ is in kg/mol. The unit of $f(u)$ is seconds per metre, because $f(u)\,du$ is a pure fraction. The area under the whole curve is $1$, since every molecule is somewhere.

The shape comes from two factors. The factor $u^2$ grows with speed, because more directions of motion are available at higher speed. The factor $e^{-Mu^2/2RT}$ is the Boltzmann factor: $\tfrac{1}{2}mu^2$ is the kinetic energy, and a molecule with a high energy is rare. The first factor wins at low speed and the second at high speed. The curve rises from zero, reaches a peak and then falls in a long tail. Zero molecules have exactly zero speed.

Three speeds

One number cannot describe a spread, and three are in use:

• The most probable speed $u_{\text{mp}} = \sqrt{\dfrac{2RT}{M}}$ is the peak of the curve. We find it by setting the slope of $f(u)$ to zero.
• The average speed $\bar{u} = \sqrt{\dfrac{8RT}{\pi M}}$ is the mean of all the speeds.
• The root mean square speed $u_{\text{rms}} = \sqrt{\dfrac{3RT}{M}}$ is the square root of the mean of the squares. It is the one that gives the kinetic energy.

They are always in the same ratio, for every gas and every temperature:

So $u_{\text{rms}}$ is the largest, because squaring gives extra weight to fast molecules. For nitrogen, $M = 28.02\ \text{g/mol}$, at $300\ \text{K}$, the three speeds are $422\ \text{m/s}$, $476\ \text{m/s}$ and $517\ \text{m/s}$.

Because the ratios are fixed, we can also say how many molecules are faster than each of the three. For every gas at every temperature, $57.2\%$ are faster than $u_{\text{mp}}$, $46.7\%$ are faster than $\bar{u}$ and $39.2\%$ are faster than $u_{\text{rms}}$. Only $4.6\%$ are faster than twice $u_{\text{mp}}$, and $0.044\%$ are faster than three times $u_{\text{mp}}$.

Set $T = 300\ \text{K}$ and $M = 28\ \text{g/mol}$. The green line is the leftmost of the three lines. Now set $M = 4\ \text{g/mol}$, and watch the peak fall and move far to the right.

The effect of temperature

At a higher temperature, the whole curve moves to the right. The peak speed grows as $\sqrt{T}$, so it moves from $422\ \text{m/s}$ at $300\ \text{K}$ to $731\ \text{m/s}$ at $900\ \text{K}$, which is $\sqrt{3} = 1.73$ times. The area stays $1$. So the peak gets lower and the curve gets wider.

The tail changes much more than the peak does. For nitrogen, the fraction of molecules faster than $1000\ \text{m/s}$ is $1.05\%$ at $300\ \text{K}$ and $29\%$ at $900\ \text{K}$. Tripling the temperature makes this fraction $28$ times larger.

This is how a small rise in temperature can speed up a reaction. A reaction needs molecules with enough energy. Only the fast tail has it. The fraction of molecules with an energy above $E$ is roughly $e^{-E/RT}$ when $E$ is much larger than $RT$. A small rise in $T$ makes this number rise steeply. The kinetics chapter uses this in the Arrhenius equation.

The effect of mass

At the same temperature all gases have the same average kinetic energy. A light gas must move faster to have it. So hydrogen has a low and wide curve with a peak at $1573\ \text{m/s}$, while carbon dioxide has a tall and narrow curve with a peak at $337\ \text{m/s}$. The right half of the picture shows this. A light gas has a long tail. For hydrogen at $300\ \text{K}$, about $6.4\%$ of the molecules are faster than $3000\ \text{m/s}$, while for helium it is $0.24\%$. The tail explains why light gases leave the atmosphere of a planet.

Counting the molecules in a range

The fraction of molecules faster than a speed $u_0$ has a closed form. Let $x = \dfrac{u_0}{u_{\text{mp}}}$. Then

where $\text{erfc}$ is the complementary error function, which Python and every scientific calculator know. The fraction between two speeds is the difference of two such values. The program below does the same sum by adding up the curve in small steps.

Practise with new numbers

Try it in Python

The program adds up the Maxwell curve for nitrogen at $300\ \text{K}$ in small steps. It checks that the area is $1$ and that the three speeds agree with the formulas.

Common mistakes

• Using one speed for every molecule. The three speeds are different, and none of them is the speed of every molecule.
• Mixing up the three speeds. $u_{\text{mp}} < \bar{u} < u_{\text{rms}}$. The factors under the root are $2$, $\tfrac{8}{\pi}$ and $3$.
• Thinking that the area changes with temperature. The area is always $1$. A hotter gas has a lower and wider curve.
• Thinking that the peak moves to the right by the same factor as the temperature. The speed goes as $\sqrt{T}$.
• Forgetting that $M$ is in kg/mol. The speeds come out about $32$ times too small if the molar mass is in g/mol.

Remember these

• Fraction of molecules between $u$ and $u + du$: $f(u)\,du$, with total area $1$.
• $u_{\text{mp}} = \sqrt{2RT/M}$, $\bar{u} = \sqrt{8RT/\pi M}$, $u_{\text{rms}} = \sqrt{3RT/M}$. Their ratio is $1 : 1.128 : 1.225$.
• A higher temperature moves the curve to the right and lowers the peak. The fast tail grows a lot.
• A lighter gas has a higher peak speed, a lower peak and a wider curve.
• $57.2\%$ of molecules are faster than $u_{\text{mp}}$, and $39.2\%$ are faster than $u_{\text{rms}}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Which is the largest of the three speeds, and which is the smallest?

2. What happens to the area under the Maxwell curve when the temperature is raised?

3. Methane, $M = 16.04\ \text{g/mol}$, is at $400\ \text{K}$. What is its average speed?

4. Hydrogen and oxygen are at the same temperature. Which gas has the taller peak in its distribution?

5. Why does a rise of only $10\ \text{K}$ make some reactions go much faster?

References

• OpenStax, Chemistry 2e, Chapter 9, section 9.5 (molecular speed distribution).
• The fractions above multiples of the most probable speed are computed with the error function.