Speeds of gas molecules

How fast do air molecules move?

Open a bottle of perfume in one corner of a room. After a while you smell it in the other corner. The molecules must move. But how fast? Let us find out from what we already know.

The rms speed

From the last lesson, $\tfrac12 m\bar{v^2} = \tfrac32 k_BT$. We want the speed, so

Here $m$ is the mass of one molecule and $M = mN_A$ is the molar mass, in kg/mol. This speed is the root mean square speed. You square every speed, take the mean, and then the square root.

For oxygen at $300$ K, with $M = 0.032$ kg/mol:

That is faster than sound in air. Hydrogen, with $M = 0.002$ kg/mol, is sixteen times lighter, so it is four times faster: about 1930 m/s.

Then why does the perfume take so long to cross the room? Each molecule hits other molecules billions of times a second. Its path is a zigzag, and it drifts forward very slowly.

Two rules from one formula

• At one temperature, $v_{rms}$ is proportional to $1/\sqrt{M}$. A molecule four times heavier moves half as fast.
• For one gas, $v_{rms}$ is proportional to $\sqrt{T}$, with $T$ in kelvin. To double the speed, multiply the temperature by four.

Not every molecule has the same speed

The molecules of a gas do not all move at $v_{rms}$. They hit each other all the time and keep changing speed. A few are almost at rest. A few are very fast. Most are somewhere in between.

James Clerk Maxwell found how the speeds spread out. The graph below is his distribution of speeds for oxygen at two temperatures. You do not need its formula in Class 11. You need to read its shape.

• The curve starts at zero. Very few molecules are nearly at rest.
• It rises to a peak. The speed at the peak is the most probable speed, $v_{mp}$.
• It has a long tail on the fast side. That tail pulls the average speed above the peak.
• At a higher temperature, the curve moves to the right, gets lower and spreads out. The area under it, the total number of molecules, stays the same.

Three speeds

For an ideal gas, the three speeds are

Compare the numbers under the roots: $2$, $8/\pi = 2.55$ and $3$. So always

Set $M$ to 2 g/mol, for hydrogen. How far to the right does the curve move? Then set $M$ to 131 g/mol, for xenon, and raise the temperature. Can you make xenon as fast as oxygen at 300 K?

Why the Moon has no air

To leave a planet for good, a molecule must move faster than the escape speed. For the Moon, the escape speed is only $2.38$ km/s. In sunlight its surface reaches about $390$ K.

At $390$ K, the rms speed of hydrogen is about $2.2$ km/s. That is close to the escape speed. A large part of the fast tail is faster than $2.38$ km/s, so hydrogen leaks away. Even heavier gases have a few very fast molecules. Over millions of years, the Moon has lost them all.

Jupiter is cold and very massive. Its escape speed is about $59.5$ km/s, far above the speed of any of its molecules. So Jupiter has kept even its hydrogen. Earth, in between, has kept nitrogen and oxygen but lost almost all its free hydrogen.

Practise with new numbers

Try it in Python

The program gives 50000 argon atoms random velocities, the way the Maxwell distribution does. It finds their average speed and their rms speed, and compares them with the formulas.

Common mistakes

• Using $M$ in g/mol. The formula needs kg/mol. With g/mol the speed comes out about 32 times too small.
• Doubling $T$ to double the speed. The speed goes as $\sqrt{T}$, so $T$ must be four times as large.
• Thinking every molecule moves at $v_{rms}$. It is a kind of average. The speeds spread over a wide range.
• Mixing up the order. $v_{mp}$ is the smallest of the three, and $v_{rms}$ the largest.

Remember these

• $v_{rms} = \sqrt{3RT/M} = \sqrt{3k_BT/m}$, with $M$ in kg/mol.
• At one temperature, lighter molecules move faster: $v_{rms}$ goes as $1/\sqrt{M}$.
• For one gas, $v_{rms}$ goes as $\sqrt{T}$.
• $v_{mp} : v_{avg} : v_{rms} = 1 : 1.128 : 1.225$.
• A higher temperature moves the speed curve to the right and makes it wider and lower.

Check yourself

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

1. Find the rms speed of helium atoms, $M = 4$ g/mol, at $300$ K.

2. Put the three speeds $v_{avg}$, $v_{rms}$ and $v_{mp}$ in order, from the smallest.

3. The kelvin temperature of a gas is made four times as large. What happens to its rms speed?

4. The top of Jupiter's clouds is near $165$ K. The rms speed of hydrogen there is about $1.4$ km/s. Will Jupiter lose its hydrogen?