AP Chem 3.5: Kinetic Molecular Theory

Gases through the motion of their molecules

Lesson 2.8.7 introduced kinetic molecular theory and Graham’s law. AP topic 3.5 asks you to use the model to explain all sorts of observations: why light molecules move faster at the same temperature, why the pressure changes, how the speed distribution changes with temperature and with the gas, and what is left after a gas mixture leaks.

The main points of kinetic molecular theory

Gas molecules are in constant random motion, colliding with one another and with the walls.
The volume of the molecules themselves is negligible compared with the volume of the gas.
There are no attractions or repulsions between molecules (except during collisions), and collisions don’t lose energy.
The average kinetic energy of the molecules depends only on the absolute temperature, not on the identity of the gas: the higher the temperature, the greater the average kinetic energy.

Same temperature, same kinetic energy, different speeds

At the same temperature, the molecules of every gas have the same average kinetic energy, $\tfrac{1}{2}mv^2$ . The smaller the mass, the greater the speed must be: at 300 K, the root-mean-square speed of He atoms is about 1370 m/s and that of Ar atoms only about 430 m/s, yet their average kinetic energies are equal.

Pressure comes from molecules colliding with the walls, and depends on how often they collide and how hard each collision is:

Smaller volume at constant temperature: more molecules per unit volume, so collisions are more frequent; each collision is just as hard, and the pressure rises.
Higher temperature at constant volume: the molecules move faster, so collisions are both more frequent and harder, and the pressure rises.

When two gases at different temperatures are mixed, they exchange energy through collisions until they reach the same temperature: the molecules of the gas that was hotter end up with lower average kinetic energy and lower average speed, and those of the colder gas with higher.

The Maxwell–Boltzmann distribution of speeds

At any moment, gas molecules have a range of speeds, which form a single-peaked curve; the area under the curve represents all the molecules (the total doesn’t change).

Higher temperature: the peak shifts right (faster speeds), and the curve becomes lower and broader (speeds more spread out).
Same temperature, larger molar mass: the peak is farther left, higher, and narrower.

Diagram A: three gases at the same temperature (300 K)

Diagram B: one gas (O₂) at three temperatures

Note: the curve with the tallest peak is at the lowest temperature (or is the heaviest gas), not the highest.

Choose one to three gases and temperatures and compare their speed distributions:

Try it: the distribution of molecular speeds

Gases (up to three)

Temperature

Diffusion, effusion, and Graham’s law

The gradual mixing of gas molecules as they move through another gas is diffusion扩散; the escape of a gas through a tiny hole into a vacuum is effusion泻流. At the same temperature, lighter molecules move faster, so they diffuse and effuse faster:

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

With ammonia at one end of a long glass tube and concentrated hydrochloric acid at the other, $\ce{NH3}$ (17) diffuses faster than HCl (36.5), so the white ring of $\ce{NH4Cl}$ forms nearer the HCl end.
When a gas mixture leaks, the lightest gas escapes fastest, so in what remains it has the lowest partial pressure and the smallest mole fraction.

Try it: Graham's law race

Left end

Right end

Try it: a leaking container

Example 1: He and Ar

Compare the average kinetic energy and the root-mean-square speed of 1 mol of He and 1 mol of Ar at 300 K.

Step-by-step solution

Average kinetic energy depends only on temperature. At the same temperature the two gases have the same average kinetic energy, per mole
$\begin{aligned} \tfrac{3}{2}RT &= 1.5 \times 8.314 \times 300 \\ &= \boxed{3.74 \times 10^{3}\ \text{J}} \end{aligned}$
Root-mean-square speed. $u = \sqrt{3RT/M}$ . With R = 8.314 J/(mol·K), M must be in kg/mol (He 0.00400, Ar 0.03995):
$\begin{aligned} u_\text{He} &= \sqrt{\frac{3(8.314)(300)}{0.00400}} \\ &= \boxed{1.37 \times 10^{3}\ \text{m/s}} \\ u_\text{Ar} &= \sqrt{\frac{3(8.314)(300)}{0.03995}} \\ &= \boxed{433\ \text{m/s}} \end{aligned}$
Compare. The speed ratio depends only on the molar masses:
$\frac{u_\text{He}}{u_\text{Ar}} = \sqrt{\frac{39.95}{4.00}} = 3.16$
He has 1/10 the mass of Ar and moves about 3.16 times as fast, so ½mv² comes out equal.
A common mistake. Using M in g/mol makes the speeds √1000 ≈ 31.6 times too small.

Example 2: A mixture after a leak

A container holds He, $\ce{N2}$ , and $\ce{CO2}$ , each at a partial pressure of 1.00 atm. Gas effuses into a vacuum through a pinhole in the wall. After a while, which gas has the lowest partial pressure? What is the ratio of the effusion rates of He and $\ce{CO2}$ ?

Step-by-step solution

Compare the molar masses. He 4.00, $\ce{N2}$ 28.02, $\ce{CO2}$ 44.01 g/mol.
Compare how fast they escape. At the same temperature all the molecules have the same average kinetic energy, so the lighter the gas, the faster its molecules move and the more of them reach the hole each second. He escapes fastest, so the least He remains and it has the lowest partial pressure; the most $\ce{CO2}$ remains.
Find the rate ratio (Graham’s law).
$\begin{aligned} \frac{r_\text{He}}{r_{\ce{CO2}}} &= \sqrt{\frac{M_{\ce{CO2}}}{M_\text{He}}} = \sqrt{\frac{44.01}{4.00}} \\ &= \boxed{3.32} \end{aligned}$
Result. In the gas left behind, $\ce{CO2}$ has the largest mole fraction and He the smallest; all three partial pressures fall, but He’s falls fastest.

Practice: which is larger?

He at 300 K or Ar at 300 K: average kinetic energy
He at 300 K or Ar at 300 K: average speed
$\ce{N2}$ at 300 K or $\ce{N2}$ at 600 K: average kinetic energy
$\ce{H2}$ or $\ce{O2}$ at the same temperature: rate of diffusion
One gas at 200 K or at 500 K: height of the speed-distribution peak
He or $\ce{CO2}$ at the same temperature: height of the speed-distribution peak
$\ce{CH4}$ or $\ce{O2}$ at the same temperature: average kinetic energy per mole
At constant volume, 300 K or 400 K: how often molecules hit the walls
At constant temperature, 2.0 L or 1.0 L: average force of each collision
Effusion at the same temperature: $\ce{NH3}$ or HCl

Common mistakes

Mistake 1: “At the same temperature, heavier molecules have more kinetic energy.” At the same temperature, the molecules of every gas have the same average kinetic energy. Heavy molecules move slowly and light ones quickly, so $\tfrac{1}{2}mv^2$ comes out the same.

Mistake 2: “The taller the peak of the speed distribution, the higher the temperature.” It’s the reverse. The area under the curve represents the total number of molecules, so at a low temperature, with speeds concentrated in a narrow range, the peak is higher, narrower, and farther left.

Mistake 3: “At a given temperature, all molecules have the same speed.” Molecular speeds vary, and the average kinetic energy is only an average. Even at low temperature a few molecules move very fast.

Quick check

Diagram A above shows the speed distributions of He, $\ce{N2}$ , and $\ce{CO2}$ at the same temperature. Which gas is curve 3?

Diagram B above shows the speed distributions of $\ce{O2}$ at three temperatures. Which curve is at the highest temperature, and why?

Compare 1 mol of He and 1 mol of Ar at 300 K. Which statement is correct?

At constant temperature, a sample of gas is compressed to half its volume and its pressure doubles. Which is the best explanation in terms of molecular motion?

He at 100 °C and Ar at 20 °C are mixed in an insulated container and reach a common temperature. Compared with before mixing, how has the average kinetic energy of the He atoms changed?

Cotton balls soaked in concentrated ammonia and concentrated hydrochloric acid are placed at the two ends of a long glass tube at the same time, and a white ring of $\ce{NH4Cl}$ appears. Where does it form, and why?

A container holds $\ce{H2}$ , $\ce{N2}$ , and $\ce{CO2}$ at equal partial pressures, and the wall has a tiny hole. After a while, which gas has the lowest partial pressure?

AP-style free response

(a) A rigid container of $\ce{O2}$ is heated from 300 K to 600 K. Describe how its Maxwell–Boltzmann distribution of speeds changes.

(b) Compare the average kinetic energies of $\ce{O2}$ and He at 300 K, and justify your answer.

(c) Use kinetic molecular theory to explain why the pressure in the container in (a) increases on heating.

(d) He and $\ce{O2}$ effuse through the same tiny hole at the same temperature. Calculate the ratio of their effusion rates.

Model answer and scoring

(a) The peak shifts right (faster speeds), and the curve becomes lower and broader; the area under the curve is unchanged, because the number of molecules is unchanged.

(b) They are equal. The average kinetic energy of gas molecules depends only on absolute temperature, not on the identity of the gas; the lighter He atoms move faster.

(c) At the higher temperature, the molecules’ average speed is greater, so they hit the walls more often and harder with each collision. The volume is fixed, so the pressure increases.

(d) rate(He) ÷ rate( $\ce{O2}$ ) = √(32.00 ÷ 4.00) ≈ 2.83.

Scoring: (a) shifts right, lower and broader, same area; (b) “equal, depends only on temperature”; (c) states both “more often” and “harder”; (d) about 2.83.

Summary

Average kinetic energy depends only on absolute temperature; at the same temperature, lighter molecules move faster.
Pressure comes from collisions with the walls: a smaller volume means more frequent collisions; a higher temperature means more frequent and harder collisions.
Speed distributions: a higher temperature or a lighter gas shifts the peak right and makes it lower and broader; the tallest peak belongs to the lowest temperature (or the heaviest gas).
Diffusion and effusion: the ratio of rates = $\sqrt{M_2/M_1}$ ; when a mixture leaks, the lightest gas remains in the smallest amount.

AP tip: When explaining a pressure change, separate “how often” molecules collide from “how hard”. When comparing gases, separate average kinetic energy (temperature only) from average speed (also molar mass). To read a speed distribution, look at where the peak is before how high it is.

Kinetic Molecular Theory分子运动论

Learning goals

Before you start

Gases through the motion of their molecules

The main points of kinetic molecular theory

Same temperature, same kinetic energy, different speeds

The Maxwell–Boltzmann distribution of speeds

Diffusion, effusion, and Graham’s law

Example 1: He and Ar

Example 2: A mixture after a leak

Common mistakes

Quick check

AP-style free response

Summary

Feedback / Report a problem