# Kinetic molecular theory

> IB Chemistry HL · S1: Models of the particulate nature of matter
> Source: https://www.owlsprep.com/study/ib-chemistry-hl-u1-kinetic-molecular-theory/

This sub-topic develops the microscopic kinetic molecular theory (KMT) model of gas behavior, linking molecular motion to observable macroscopic properties. We also use KMT to explain why real gases deviate from ideal gas behavior.

**Prerequisites:** [Ideal gas laws and basic gas properties](https://www.owlsprep.com/study/ib-chemistry-hl-u1-ideal-gas-laws/)

## Learning objectives

- State the core postulates of the kinetic molecular theory (KMT) for ideal gases
- Use KMT to explain macroscopic gas properties such as pressure and temperature
- Calculate root mean square speed of gas molecules from KMT relationships
- Explain deviations of real gases from ideal behavior using KMT

## Core Postulates of KMT

Kinetic molecular theory is a microscopic model that describes the behavior of ideal gases, built on 5 core testable postulates about gas molecules and their motion.

**Ideal gas** — A hypothetical gas that follows all KMT postulates exactly. Most real gases behave nearly ideally at low pressure and high temperature.

*Example:* Nitrogen gas at 1 atm and 25°C approximates ideal behavior very closely.

- Gases consist of large numbers of tiny particles, far apart relative to their own size: most volume of a gas is empty space, so molecular volume is negligible.
- Gas molecules move constantly in random, straight-line motion, colliding frequently with each other and container walls.
- All collisions are elastic: total kinetic energy is conserved, no net energy loss over time.
- There are no attractive or repulsive intermolecular forces between ideal gas molecules.
- The average kinetic energy of gas molecules is directly proportional to the absolute temperature of the sample.

**Worked example:** Which of the following is NOT a postulate of KMT for ideal gases?
A) Collisions between gas molecules are elastic
B) The volume of gas molecules is negligible compared to total volume
C) Intermolecular forces between molecules are significant
D) Average kinetic energy is proportional to absolute temperature

1. Recall the 5 core postulates of KMT. One key postulate explicitly states that there are no attractive or repulsive forces between ideal gas molecules.
2. Evaluate the options: Option C directly contradicts this postulate, so it is not a postulate of KMT.
3. Answer: C

> **Exam tip:** You can be asked to state KMT postulates for 2-3 marks in Paper 2, so memorize all five clearly.

## Linking KMT to Macroscopic Gas Properties

KMT explains why empirical gas laws work by connecting microscopic molecular behavior to the macroscopic properties we measure experimentally.

**Gas pressure** — The force per unit area exerted by gas molecules colliding with the walls of their container.

For example, Boyle's law ($P \propto 1/V$ at constant $T$) is explained by KMT: decreasing volume increases the number of collisions per unit area of container wall, increasing pressure. Similarly, increasing temperature increases average molecular speed, leading to more forceful collisions and higher pressure at fixed volume (Gay-Lussac's law).

**Worked example:** Use kinetic molecular theory to explain why the pressure of a gas in a sealed fixed-volume container increases when temperature increases.

1. From KMT postulates, absolute temperature of a gas is directly proportional to the average kinetic energy of its molecules.
2. When temperature increases, average kinetic energy increases, so molecules move faster on average.
3. Faster molecules collide with the container walls more frequently and with greater force per collision. Since volume is fixed, the total force per unit area (pressure) increases.

## Kinetic Energy and Molecular Speed

A key mathematical result derived from KMT gives the relationship between temperature, molar mass, and average molecular speed. The core derivation leads to:

$$\frac{1}{2}Nm\overline{c^2} = \frac{3}{2}nRT$$

Rearranging gives the root mean square speed, the most common measure of average molecular speed:

$$v_{rms} = \sqrt{\frac{3RT}{M}}$$

Where $R$ is the gas constant ($8.31 \text{ J K}^{-1} \text{mol}^{-1}$), $T$ is absolute temperature in Kelvin, and $M$ is molar mass in $\text{kg mol}^{-1}$. This result shows that at the same temperature, lighter gases have higher average molecular speed than heavier gases.

**Worked example:** Calculate the root mean square speed of oxygen ($\text{O}_2$) molecules at 27°C, given $R = 8.31 \text{ J K}^{-1} \text{mol}^{-1}$, molar mass of $\text{O}_2 = 32.00 \text{ g mol}^{-1}$.

1. Convert temperature to Kelvin and molar mass to $\text{kg mol}^{-1}$ to match the units of $R$:
2. $$T = 27 + 273 = 300 \text{ K}, \quad M = 32.00 \text{ g mol}^{-1} = 0.03200 \text{ kg mol}^{-1}$$
3. Substitute into the $v_{rms}$ formula:
4. $$v_{rms} = \sqrt{\frac{3 \times 8.31 \times 300}{0.03200}} = \sqrt{233718.75}$$
5. Calculate the final result:
6. $$v_{rms} \approx 480 \text{ m s}^{-1}$$

> **Exam tip:** Always check units: forgetting to convert molar mass from g to kg is a common exam mistake.

## Deviation of Real Gases from Ideal Behavior

No real gas follows KMT postulates exactly, because two core assumptions are only approximately true at low pressure and high temperature:

- Molecular volume is not always negligible: at high pressure, molecules are crowded close together, so their own volume makes up a significant fraction of total volume.
- Intermolecular forces are not zero: at low temperature, molecules move slowly enough that intermolecular attractions affect their motion.

**Compressibility factor** — A measure of deviation from ideal behavior, defined as $Z = \frac{PV}{nRT}$. $Z=1$ for ideal gases.

*Notation:* Z

**Worked example:** Use KMT to explain why carbon dioxide deviates more from ideal behavior at 0°C than at 100°C, at the same pressure.

1. At lower temperature, gas molecules have lower average kinetic energy, so they move slower.
2. Slower molecular motion allows intermolecular attractive forces, which KMT assumes to be zero for ideal gases, to have a significant effect on molecular motion.
3. At 100°C, higher average kinetic energy overcomes intermolecular attractions, so behavior is much closer to ideal. At 0°C, intermolecular forces are significant, so deviation is larger.

## Common pitfalls

- **Wrong:** Forgetting to convert molar mass from g mol⁻¹ to kg mol⁻¹ when calculating $v_{rms}$
  - Why it fails: The gas constant $R = 8.31$ J K⁻¹ mol⁻¹ has units of kg m² s⁻², so incorrect units give a result ~30x smaller than the correct value
  - Correct: Always convert molar mass to kg mol⁻¹ before substituting into the $v_{rms}$ formula
- **Wrong:** Claiming all individual gas molecules have the same speed at a given temperature
  - Why it fails: KMT states that average kinetic energy (not individual molecular speed) is proportional to temperature. There is a wide distribution of molecular speeds in any sample.
  - Correct: KMT describes average behavior of a large collection of molecules, not the speed of any single molecule
- **Wrong:** Stating that only heavy gases deviate from ideal behavior
  - Why it fails: All real gases deviate from ideal behavior under appropriate conditions, regardless of molar mass
  - Correct: Deviation increases with stronger intermolecular forces and higher molar mass, but all real gases deviate from ideal behavior at high pressure/low temperature
- **Wrong:** Mix up the conditions for maximum deviation: claiming high temperature/low pressure causes maximum deviation
  - Why it fails: These are the conditions where real gases are closest to ideal, because molecular volume is negligible and intermolecular forces are weak
  - Correct: Maximum deviation from ideal behavior occurs at high pressure and low temperature

## Cheatsheet

| Concept | Key Result | Notes for Exam |
| --- | --- | --- |
| Ideal gas KMT postulates | Elastic collisions, no IMFs, negligible molecular volume, random motion, $\overline{KE} \propto T$ | Memorize all 5 for explanation questions |
| Root mean square speed | $v_{rms} = \sqrt{\frac{3RT}{M}}$ | Convert M to kg mol⁻¹, T to Kelvin, use R = 8.31 J K⁻¹ mol⁻¹ |

## What's next

Kinetic molecular theory is the foundational microscopic model for gas behavior in IB Chemistry HL, connecting molecular motion to measurable macroscopic properties of gases. It also sets the stage for understanding why real gases deviate from ideal behavior, which depends directly on the strength of intermolecular interactions between molecules. KMT's core relationship between absolute temperature and average molecular kinetic energy also underpins key concepts in later topics, including thermochemistry (energy transfers between systems) and reaction kinetics (the effect of temperature on reaction rate). Mastering KMT is critical for answering both calculation and explanation questions across multiple units of the IB Chemistry HL syllabus.

- [AHL: Advanced mass spectrometry interpretation](https://www.owlsprep.com/study/ib-chemistry-hl-u1-ahl-advanced-mass-spectrometry-interpretation/)
- [S2: Models of bonding and structure](https://www.owlsprep.com/study/ib-chemistry-hl-u2-overview/)

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