# B.2 Gas laws

> IB Physics HL · Theme B: The particulate nature of matter
> Source: https://www.owlsprep.com/study/ib-physics-hl-u2-b-2-gas-laws/

This module covers the fundamental gas laws describing ideal gas behaviour, relating pressure, volume, temperature and amount of gas. We will practice exam-style problem solving and address common misconceptions tested in IB exams.

**Prerequisites:** [Kinetic molecular theory of gases](https://www.owlsprep.com/study/ib-physics-hl-u2-b-1-kinetic-theory/); SI unit conversion for pressure, volume and temperature

## Learning objectives

- State and apply the individual gas laws and ideal gas law
- Solve problems involving changing gas conditions
- Interpret gas law graphs common in IB exams
- Compare ideal and real gas behaviour and explain deviations

## The individual gas laws

**Ideal gas** — A hypothetical gas that obeys all gas laws at all conditions of pressure and temperature, with negligible molecular volume and no intermolecular forces between particles.

*Example:* Most real gases behave like ideal gases at low pressure and high temperature.

Each individual gas law describes the relationship between two gas properties when the other two properties are held constant:

- Boyle's Law: Pressure and volume are inversely proportional at constant temperature and amount of gas: $P \propto 1/V$ → $P_1V_1 = P_2V_2$ (isothermal process)
- Charles' Law: Volume and absolute temperature are directly proportional at constant pressure and amount of gas: $V \propto T$ → $V_1/T_1 = V_2/T_2$ (isobaric process)
- Gay-Lussac's Law: Pressure and absolute temperature are directly proportional at constant volume and amount of gas: $P \propto T$ → $P_1/T_1 = P_2/T_2$ (isochoric process)

**Worked example:** A 2.0 L sample of gas at 1.0 atm pressure is compressed to 0.50 L at constant temperature. What is the new pressure?

1. Since temperature is constant, use Boyle's Law $P_1V_1 = P_2V_2$. Identify known values:
2. $$P_1 = 1.0 \text{ atm}, \quad V_1 = 2.0 \text{ L}, \quad V_2 = 0.50 \text{ L}$$
3. Rearrange to solve for $P_2$:
4. $$P_2 = \frac{P_1V_1}{V_2} = \frac{(1.0)(2.0)}{0.50} = 4.0 \text{ atm}$$

> **Exam tip:** Always convert all temperatures to Kelvin for all gas law calculations. Celsius will always give incorrect results.

## Ideal gas and combined gas law

Combining the three individual gas laws gives the general ideal gas equation of state that relates all four variables for an ideal gas.

**Ideal gas law** — The equation of state for an ideal gas, relating all four gas properties with the universal gas constant $R$.

*Notation:* PV = nRT

An alternative form using number of molecules $N$ and Boltzmann constant $k_B$ is: $PV = Nk_B T$. When the amount of gas $n$ is constant, the combined gas law relates two different states of the same gas:

$$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$

> **info**
>
> Match the units of $R$ to your pressure and volume: use 8.31 J mol⁻¹K⁻¹ for pascals and m³, 0.0821 L atm mol⁻¹K⁻¹ for atm and liters.

**Worked example:** A sealed balloon with 0.10 mol of helium gas has a volume of 2.5 L at 27°C. What is the pressure inside the balloon?

1. First convert temperature from Celsius to Kelvin:
2. $$T = 27 + 273 = 300 \text{ K}$$
3. Rearrange the ideal gas law to solve for pressure, using $R = 0.0821 \text{ L atm mol}^{-1}\text{K}^{-1}$ to match our units:
4. $$P = \frac{nRT}{V} = \frac{(0.10)(0.0821)(300)}{2.5} = 0.99 \text{ atm} \approx 1.0 \text{ atm}$$

## Gas law graphs

IB exams regularly ask you to interpret or sketch gas law graphs. The table below summarises the most common relationships:

| Relationship | Constant quantities | Graph shape |
| --- | --- | --- |
| P vs V | n, T | Downward-sloping hyperbola |
| P vs 1/V | n, T | Straight line through origin |
| V vs T (K) | n, P | Straight line through origin at 0 K |
| P vs T (K) | n, V | Straight line through origin at 0 K |

**Worked example:** Sketch two P-V curves for a fixed mass of gas at 200 K and 400 K, and label the higher temperature curve.

1. From the ideal gas law, $P = \frac{nRT}{V}$. For any fixed volume, higher temperature gives higher pressure.
2. Both curves are inverse hyperbolas, and the 400 K curve lies entirely above the 200 K curve at all volumes.

> **Exam tip:** When extrapolating V-T or P-T graphs, the intercept at zero volume/pressure is absolute zero (-273.15°C / 0 K), not 0°C.

## Ideal vs real gases

Real gases deviate from ideal gas behaviour because the two core assumptions of kinetic molecular theory for ideal gases do not hold at all conditions:

- Real gas molecules have non-zero volume, so the available volume for movement is less than the total container volume
- Intermolecular attractive forces exist between real gas molecules, which reduces the pressure on the container walls

> **warning**
>
> Deviations from ideal behaviour are largest at high pressure (molecules are close together) and low temperature (molecules move slowly, intermolecular forces are significant).

**Worked example:** Explain why the pressure of a real gas is lower than the pressure predicted by the ideal gas law at high pressure.

1. At high pressure, molecules are packed very close together, so intermolecular attractive forces become much stronger than at low pressure.
2. These attractive forces pull molecules away from the container walls, reducing the force of molecular collisions with the wall. This lowers the measured pressure compared to the ideal prediction.

## Common pitfalls

- **Wrong:** Using Celsius instead of Kelvin for temperature in gas law calculations
  - Why it fails: All gas law proportionalities are based on absolute temperature, so Celsius does not work even for ratio problems
  - Correct: Always add 273.15 to any Celsius temperature to convert to Kelvin before starting calculations
- **Wrong:** Using the wrong value of $R$ that does not match pressure/volume units
  - Why it fails: $R$ has different numerical values for different unit systems, so mismatched units give incorrect final results
  - Correct: Check that units of $R$ cancel out to give the desired unit for your final answer
- **Wrong:** Applying the combined gas law when the amount of gas changes
  - Why it fails: The combined gas law assumes $nR$ is constant, which is only true if no gas is added or removed
  - Correct: Use the full ideal gas law $PV = nRT$ to solve for unknowns when $n$ changes
- **Wrong:** Assuming ideal gas law applies to real gases at high pressure and low temperature
  - Why it fails: Ideal assumptions break down when molecules are close and moving slowly, so large deviations occur
  - Correct: Recognise when deviations occur and be prepared to explain why real gases differ from ideal behaviour
- **Wrong:** Drawing a Charles' law V vs T graph that intercepts V=0 at 0°C
  - Why it fails: Zero volume for an ideal gas occurs at absolute zero (0 K = -273°C), not 0°C
  - Correct: Extrapolate the straight line back to V=0 at T = -273°C (0 K)

## Cheatsheet

| Gas Law | Conditions | Equation |
| --- | --- | --- |
| Boyle's | Constant $n, T$ | $P_1V_1 = P_2V_2$ |
| Charles' | Constant $n, P$ | $V_1/T_1 = V_2/T_2$ |
| Gay-Lussac's | Constant $n, V$ | $P_1/T_1 = P_2/T_2$ |
| Combined | Constant $n$ | $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ |
| Ideal Gas | Any ideal gas | $PV = nRT = Nk_B T$ |

## What's next

Gas laws form the foundation of all thermal physics in IB Physics HL, and are applied in almost all subsequent thermal topics. Understanding gas behaviour and the ideal gas law is essential for studying thermodynamics, heat engines, and phase changes, which are all heavily tested in both Paper 1 and Paper 2 exams. Mastery of gas law calculations and the difference between ideal and real gases will also help you tackle extension questions that require you to explain deviations from expected behaviour. The concepts you learned here build directly on kinetic molecular theory and lead into the study of thermal properties of matter.

- [B.3 Kinetic theory of gases](https://www.owlsprep.com/study/ib-physics-hl-u2-b-3-kinetic-theory-of/)
- [B.4 Mass, energy and matter structure](https://www.owlsprep.com/study/ib-physics-hl-u2-b-4-mass-energy-and/)
- [B.5 Current and electric circuits (AHL)](https://www.owlsprep.com/study/ib-physics-hl-u2-b-5-current-and-electric/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-physics-hl-u2-b-2-gas-laws/