B.2 Gas laws

IB Physics HL· Theme B, B.2· 45 min read

1. The individual gas laws★★☆☆☆⏱ 15 min

📘 Definition

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: $P \propto 1/ V$ → $P_{1} V_{1} = P_{2} V_{2}$ (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 $P_{1} V_{1} = P_{2} V_{2}$ . 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.

2. Ideal gas and combined gas law★★☆☆☆⏱ 20 min

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

📘 Definition

Ideal gas law

PV = nRT

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

An alternative form using number of molecules $N$ and Boltzmann constant $k_{B}$ is: $P V = N k_{B} T$ . 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}

📐 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 $R = 0.0821 L atm mol^{- 1} K^{- 1}$ to match our units:
4
$P = \frac{nRT}{V} = \frac{(0.10)(0.0821)(300)}{2.5} = 0.99 \text{ atm} \approx 1.0 \text{ atm}$

3. Gas law graphs★★★☆☆⏱ 15 min

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, $P = \frac{n R T}{V}$ . 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.

4. Ideal vs real gases★★★★☆⏱ 15 min

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

📐 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.

5. Common Pitfalls

Wrong move:

Using Celsius instead of Kelvin for temperature in gas law calculations

Why:

All gas law proportionalities are based on absolute temperature, so Celsius does not work even for ratio problems

Correct move:

Always add 273.15 to any Celsius temperature to convert to Kelvin before starting calculations

Wrong move:

Using the wrong value of $R$ that does not match pressure/volume units

Why:

$R$ has different numerical values for different unit systems, so mismatched units give incorrect final results

Correct move:

Check that units of $R$ cancel out to give the desired unit for your final answer

Wrong move:

Applying the combined gas law when the amount of gas changes

Why:

The combined gas law assumes $n R$ is constant, which is only true if no gas is added or removed

Correct move:

Use the full ideal gas law $P V = n R T$ to solve for unknowns when $n$ changes

Wrong move:

Assuming ideal gas law applies to real gases at high pressure and low temperature

Why:

Ideal assumptions break down when molecules are close and moving slowly, so large deviations occur

Correct move:

Recognise when deviations occur and be prepared to explain why real gases differ from ideal behaviour

Wrong move:

Drawing a Charles' law V vs T graph that intercepts V=0 at 0°C

Why:

Zero volume for an ideal gas occurs at absolute zero (0 K = -273°C), not 0°C

Correct move:

Extrapolate the straight line back to V=0 at T = -273°C (0 K)

6. Quick Reference Cheatsheet

Gas Law	Conditions	Equation
Boyle's	Constant $n, T$	$P_{1} V_{1} = P_{2} V_{2}$
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 $n$	$\frac{P _{1} V _{1}}{T _{1}} = \frac{P _{2} V _{2}}{T _{2}}$
Ideal Gas	Any ideal gas	$P V = n R T = N k_{B} T$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Ideal gas constant calculation
2024 · 2
Pressure-volume graph interpretation
2023 · 1
Boyle's law application problem

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.