Kinetic molecular theory — AP Chemistry
1. Core Postulates of Kinetic Molecular Theory ★★☆☆☆ ⏱ 3 min
Kinetic molecular theory (KMT) is a microscopic model that connects the behavior of individual gas molecules to measurable bulk gas properties like pressure, volume, and temperature. Unlike empirical gas laws like $PV = nRT$, which come from experimental observation, KMT builds gas behavior from first principles about molecular motion, and only applies strictly to ideal gases.
- Gases consist of large numbers of tiny particles separated by large distances relative to their size; the total volume of the molecules themselves is negligible compared to the container volume.
- Gas molecules are in constant, random, straight-line motion, colliding frequently with each other and the container walls; measured pressure is the force of these collisions per unit area of the container wall.
- All collisions between gas molecules (and between molecules and container walls) are elastic: no net kinetic energy is lost during collisions, so total kinetic energy of the gas remains constant as long as temperature is constant.
- There are no attractive or repulsive intermolecular forces between gas molecules; molecules interact only during collisions, not between collisions.
- The average kinetic energy of a collection of gas molecules is directly proportional to the absolute (Kelvin) temperature of the gas, and this proportionality holds for all gases at the same temperature.
Exam tip: When asked about deviations from ideal behavior: high pressure/low temperature breaks the negligible molecular volume postulate, strong intermolecular forces break the no intermolecular forces postulate.
2. Temperature and Average Kinetic Energy ★★☆☆☆ ⏱ 2 min
One of the most important results KMT gives us is the direct relationship between absolute temperature and average molecular kinetic energy. For 1 mole of gas, this relationship is written as:
\overline{KE} = \frac{3}{2}RT
where $R = 8.314 \ \text{J/(mol·K)}$ (the gas constant in energy units) and $T$ is absolute temperature in Kelvin. The key takeaway for the AP exam, tested more often than any other KMT concept, is: *all gases have the same average kinetic energy at the same absolute temperature*. Lighter gases move faster on average to compensate for their lower mass, while heavier gases move slower, but their average kinetic energy per mole is identical at the same temperature. Temperature is, by definition, a measure of average molecular kinetic energy.
Exam tip: If an AP question asks you to compare average kinetic energy of two gases, the only thing you need to check is their temperature. Same temperature = same average KE, no exceptions.
3. Root-Mean-Square (rms) Speed ★★★☆☆ ⏱ 4 min
Unlike average kinetic energy, the average speed of gas molecules depends on the molar mass of the gas at a given temperature. Root-mean-square speed ($v_{\text{rms}}$) is defined as the speed of a molecule that has the average kinetic energy of the sample. Derived from KMT, the formula is:
v_{\text{rms}} = \sqrt{\frac{3RT}{M}}
where $R = 8.314 \ \text{J/(mol·K)} = 8.314 \ \text{kg·m}^2/(\text{s}^2·\text{mol·K})$, $T$ is absolute temperature in Kelvin, and $M$ is molar mass in **kilograms per mole**. The unit requirement for $M$ is the most common mistake students make on these calculations. The intuition is straightforward: to maintain the same average kinetic energy ($KE = \frac{1}{2}mv^2$), a lower mass requires a higher speed, so lighter gases are faster on average at the same temperature.
Exam tip: If your calculated $v_{\text{rms}}$ is less than 100 m/s for a gas near room temperature, you almost certainly forgot to convert molar mass to kg/mol. Double-check the unit conversion immediately.
4. Graham's Law of Effusion ★★★☆☆ ⏱ 4 min
Effusion is the process of gas escaping through a tiny hole into a vacuum, while diffusion is the mixing of two gases. Graham's law relates the rate of effusion (or diffusion) of two gases to their molar masses, and it is derived directly from the $v_{\text{rms}}$ formula (since rate is proportional to average speed). The formula is:
\frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}}
where $\text{Rate}_1$ and $\text{Rate}_2$ are the effusion rates of gas 1 and 2, and $M_1, M_2$ are their molar masses. Like $v_{\text{rms}}$, the law confirms that lighter gases effuse faster than heavier gases at the same temperature. Common AP questions ask you to calculate the effusion rate ratio or find the molar mass of an unknown gas from its effusion rate.
Exam tip: Always check your result with the rule: faster gas = lower molar mass, slower gas = higher molar mass. If your result contradicts this, you flipped the ratio.
5. AP-Style Concept Check ★★★★☆ ⏱ 3 min
Common Pitfalls
Why: Students confuse average kinetic energy with average speed, incorrectly carrying over the molar mass dependence of speed to kinetic energy
Why: Students are used to working with Celsius for everyday temperatures and forget KMT relationships depend on absolute temperature
Why: Gases are almost always reported with molar mass in g/mol in problems, so students forget the unit requirement for the R constant
Why: Students memorize the ratio backwards, forgetting that faster speed correlates to lower mass
Why: The postulate says average KE is proportional to T, so students incorrectly assume all molecules have the same speed
Why: Students confuse intermolecular forces with collision forces, mixing up deviations from ideal behavior with the origin of pressure