Kinetic Molecular Theory and Graham's Law

Faster than a bullet, yet slow to spread

At room temperature, nitrogen molecules in the air move at about 500 m/s on average, faster than many handgun bullets. Yet when someone opens a bottle of perfume in one corner of a classroom, it takes a while before anyone across the room smells it. Lesson 1.3.1 explained diffusion with moving particles. This lesson makes the picture quantitative: how fast do gas molecules move, how does that depend on temperature and mass, and why does something so fast spread so slowly?

This lesson has two parts:

Kinetic molecular theory and molecular speeds: the five postulates, how they explain the gas laws, and how temperature sets the average kinetic energy and the spread of speeds.
Diffusion, effusion, and Graham’s law: why lighter gases spread and escape faster, and how rates can identify a gas.

A quick check follows Part 1; get it right before you start Part 2.

Part 1: Kinetic molecular theory and molecular speeds

Kinetic molecular theory

The theory that explains the behavior of gases by the motion of their particles is kinetic molecular theory分子运动论 (KMT). For an ideal gas it has five postulates:

A gas is made of a huge number of particles that are far apart compared with their size, so the particles’ own volume is negligible.
The particles move constantly and randomly, in straight lines between collisions.
Collisions between particles and with the walls are elastic: no kinetic energy is lost overall.
Except during collisions, the particles neither attract nor repel each other.
The average kinetic energy of the particles is directly proportional to the kelvin temperature.

Two physics words first: the energy a moving object has is its kinetic energy, KE = ½mv², where m is the mass and v is the speed: the heavier and faster something is, the more kinetic energy it has. Motion of a whole particle from one place to another is called translational motion. Molecules also spin and vibrate, but a gas’s pressure and temperature depend only on translation, so the kinetic energy below means translational kinetic energy.

Chinese physics textbooks state a related, shorter theory in three points: matter is made of huge numbers of molecules; the molecules move ceaselessly and randomly; and forces act between molecules. They add that temperature is the measure of average molecular kinetic energy. Postulate 4 seems to contradict the third point, but it is the ideal-gas approximation: gas molecules are so far apart that the forces between them are negligible.

Explaining the gas laws

Change	What the particles do	Result
Constant T, smaller volume	More particles per liter, so more hits on each patch of wall	Higher pressure (Boyle’s law)
Constant V, higher temperature	Faster particles hit more often and harder	Higher pressure (Gay-Lussac’s law)
Constant P, higher temperature	Particles must spread out to keep the hits from getting stronger	Larger volume (Charles’s law)
Constant T and P, more particles	More room is needed to keep the hit rate the same	Larger volume (Avogadro’s law)

Temperature and molecular speed

The average kinetic energy of one mole of gas particles is

\overline{KE} = \frac{3}{2}RT

It depends only on temperature, not on which gas it is. Since KE = ½mv², at the same kinetic energy lighter particles must move faster. One kind of average speed is the root-mean-square speed: square each particle’s speed, take the mean of those squares, then take the square root. Its formula is

u_\text{rms} = \sqrt{\frac{3RT}{M}}

with R = 8.314 J/(mol·K) and M in kg/mol, so the speed comes out in m/s.

The particles in a gas don’t all move at the same speed; their speeds spread out along a curve called the Maxwell–Boltzmann distribution麦克斯韦–玻尔兹曼分布. At a higher temperature the curve is lower and broader and its peak shifts right; at the same temperature, a lighter gas has a broader curve shifted farther right.

Besides the rms speed, two other speeds are often marked on the curve. The speed at the peak is the most probable speed最概然速率: more particles have this speed than any other. Adding up all the speeds and dividing by the number of particles gives the average speed平均速率. On any one curve the order is always most probable speed < average speed < rms speed. Squaring before averaging gives the fast particles extra weight, so the rms speed comes out largest.

Choose one to three gases and drag the temperature to see how the speed distributions change. The first gas shows three kinds of average speed:

Try it: the distribution of molecular speeds

Gases (up to three)

Temperature

Part 1 check

When a gas is heated from 300 K to 1200 K, its root-mean-square speed becomes how many times as large?

Part 2: Diffusion, effusion, and Graham’s law

Gas molecules are fast, but in air each one collides with others billions of times per second, so it zigzags instead of traveling straight; that is why perfume spreads slowly. The spreading of one gas through another is diffusion扩散. The escape of a gas through a tiny hole into a vacuum is effusion泻流. In the 1830s and 1840s, Thomas Graham found that at the same temperature and pressure, the rate of effusion (or diffusion) is inversely proportional to the square root of the molar mass. This is Graham’s law格雷厄姆定律:

\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}

Hydrogen’s molar mass is 1/16 of oxygen’s, so hydrogen effuses 4 times as fast. Small differences like this are used to separate isotopes: uranium for nuclear reactors was first enriched using the tiny difference in effusion rates of ²³⁵UF₆ and ²³⁸UF₆.

Postulates 1 and 4 break down at high pressure and low temperature. When particles are crowded together, their own volume and the attractions between them matter, and real gases deviate from PV = nRT.

This lesson uses the atomic masses on the AP periodic table (H 1.008, He 4.00, C 12.01, N 14.01, O 16.00, S 32.06, Cl 35.45).

Run a Graham’s law race. Two gases start from opposite ends of a glass tube at the same moment. Where will they meet?

Try it: Graham's law race

Left end

Right end

Example 1: how fast is a nitrogen molecule?

Find the root-mean-square speed of N₂ molecules at 25 °C. How many times as fast are H₂ molecules at the same temperature?

Step-by-step solution

Formula and units. u_rms = √(3RT/M). Here R = 8.314 J/(mol·K), and 1 J = 1 kg·m²/s², so M must be in kg/mol for the answer to come out in m/s.
Convert. T = 298 K and M(N₂) = 28.02 g/mol = 0.02802 kg/mol.
Substitute.
$\begin{aligned} u_\text{rms} &= \sqrt{\frac{3RT}{M}} \\ &= \sqrt{\frac{3 \times 8.314 \times 298}{0.02802}} \\ &= \boxed{515\ \text{m/s}} \end{aligned}$
That’s about 1150 mph.
Compare. At the same temperature, u_rms is inversely proportional to √M:
$\begin{aligned} \frac{u(\ce{H2})}{u(\ce{N2})} &= \sqrt{\frac{M(\ce{N2})}{M(\ce{H2})}} \\ &= \sqrt{\frac{28.02}{2.016}} \\ &= \boxed{3.73} \end{aligned}$
Hydrogen molecules are about 3.7 times as fast.
A common mistake. Substituting M = 28.02 (in g/mol) gives 16.3 m/s, about 32 times too slow (√1000 ≈ 31.6).

Example 2: identifying a gas by its rate

At the same temperature and pressure, an unknown gas effuses 1/4 as fast as H₂. What is its molar mass, and what might it be?

Step-by-step solution

The rule. Graham’s law: at the same T and P, the rate of effusion is inversely proportional to the square root of the molar mass.
Set up.
$\begin{aligned} \frac{\text{rate(X)}}{\text{rate}(\ce{H2})} &= \sqrt{\frac{M(\ce{H2})}{M(\text{X})}} \\ &= \frac{1}{4} \end{aligned}$
Square both sides and solve.
$\begin{aligned} \frac{M(\ce{H2})}{M(\text{X})} &= \frac{1}{16} \\ M(\text{X}) &= 16 \times 2.016 \\ &= \boxed{32.3\ \text{g/mol}} \end{aligned}$
Conclusion. A gas with a molar mass near 32 g/mol could be O₂ (32.00 g/mol).
Check. √(2.016 ÷ 32.3) = 0.250 = 1/4, as given.
A common mistake. Forgetting to square, giving M(X) = 4 × 2.016 = 8.06 g/mol. The rate ratio is the square root of the molar-mass ratio, so the molar-mass ratio is the rate ratio squared.

Practice: compare the two quantities.

Same T: average kinetic energy of H₂ vs. O₂ molecules
Same T: average speed of H₂ vs. O₂ molecules
Average kinetic energy of N₂ at 300 K vs. 600 K
Same T and P: effusion rate of CO₂ vs. CH₄
Root-mean-square speed of He vs. Ar, both at 25 °C
Same T and P: effusion rate of N₂ vs. CO
Average speed of air molecules at 0 °C vs. 100 °C
Same T: total kinetic energy of 1 mol He vs. 1 mol CO₂

Common mistakes

Mistake 1: “At a higher temperature, every molecule moves faster.” Temperature reflects the average kinetic energy. At any temperature some molecules are fast and some are slow; heating shifts the whole distribution to higher speeds, but slow molecules still exist.

Mistake 2: “Heavier molecules have more kinetic energy.” At the same temperature, all gas particles have the same average kinetic energy. Heavier particles move more slowly, so ½mv² comes out the same.

Mistake 3: “Effusion rate is inversely proportional to molar mass.” It is inversely proportional to the square root of molar mass. O₂ is 16 times as heavy as H₂, but H₂ effuses only 4 times as fast, not 16.

Quick check

At the same T and P, what is the ratio of the effusion rates of CH₄ and O₂, approximately?

Summary

KMT: gas particles have negligible volume, move randomly, collide elastically, don’t attract each other, and have an average kinetic energy proportional to T. Pressure comes from their collisions with the walls.
The average kinetic energy per mole is (3/2)RT, the same for every gas at a given T; u_rms = √(3RT/M), so lighter molecules are faster. Molecular speeds follow the Maxwell–Boltzmann distribution.
Graham’s law: rate₁/rate₂ = √(M₂/M₁). Real gases deviate from ideal behavior at high pressure and low temperature.

On to AP (for later)

AP Chemistry Unit 3 asks you to read Maxwell–Boltzmann distribution graphs: compare curves for two temperatures or two gases, say which peak is higher or farther right, and explain it with “average kinetic energy is proportional to T.” The AP equation sheet gives the kinetic energy of a molecule, KE = ½mv². Unit 3 also covers when and why real gases deviate from ideal behavior.

Kinetic Molecular Theory and Graham's Law分子运动论与格雷厄姆扩散定律

Learning goals

Before you start

Faster than a bullet, yet slow to spread

Part 1: Kinetic molecular theory and molecular speeds

Kinetic molecular theory

Explaining the gas laws

Temperature and molecular speed

Part 1 check

Part 2: Diffusion, effusion, and Graham’s law

Example 1: how fast is a nitrogen molecule?

Example 2: identifying a gas by its rate

Common mistakes

Quick check

Summary

Feedback / Report a problem