# Gases

> IB Physics SL · IB DP Physics SL 2025
> Source: https://www.owlsprep.com/study/ib-physics-sl-u2-gases/

This sub-topic covers the ideal gas model, the ideal gas equation, and the kinetic molecular explanation of gas behaviour. You will learn to relate measurable macroscopic properties to the microscopic motion of individual gas molecules.

**Prerequisites:** [Particulate nature of matter](https://www.owlsprep.com/study/ib-physics-sl-u2-particulate-nature-of-matter/); [Basic thermal concepts](https://www.owlsprep.com/study/ib-physics-sl-u2-thermal-concepts/)

## Learning objectives

- State the core assumptions of the ideal gas kinetic model
- Apply the ideal gas law to solve problems for gas properties
- Relate absolute temperature to average molecular kinetic energy
- Describe deviations of real gases from ideal gas behaviour

## Assumptions of the Ideal Gas Model

**Ideal Gas** — A hypothetical gas that follows all the assumptions of kinetic molecular theory exactly, with no intermolecular forces and negligible molecular volume.

- All molecules are identical point particles with negligible total volume compared to the container volume.
- There are no attractive or repulsive intermolecular forces between molecules.
- Molecules move randomly in straight lines in all directions at a range of speeds.
- Collisions between molecules and container walls are perfectly elastic (no net kinetic energy loss).
- Collision duration is negligible compared to the time between collisions.

**Worked example:** A student claims: "All molecules in an ideal gas move at the same speed." Is this correct? Explain.

1. This statement is incorrect. One core assumption of ideal gas theory is that molecules move at a range of different speeds.
2. Random collisions between molecules constantly change individual molecular speeds, so only the bulk average speed is well-defined for the gas.

> **Exam tip:** Always list all five assumptions when asked, you will lose a mark for any missed assumption.

## The Ideal Gas Law

The ideal gas law relates the four measurable macroscopic properties of an ideal gas: pressure ($P$), volume ($V$), absolute temperature ($T$) and amount of substance ($n$).

$$PV = nRT$$

Where $R = 8.31\ \text{J mol}^{-1}\text{K}^{-1}$ is the molar gas constant. In terms of number of molecules $N$, the equation can also be written as:

$$PV = Nk_B T$$

Where $k_B = \frac{R}{N_A}$ is the Boltzmann constant, and $N_A$ is Avogadro's constant.

**Worked example:** A fixed mass of gas at 1.0 × 10⁵ Pa pressure occupies 2.0 m³ at 300 K. How many moles of gas are present?

1. List known values:
2. $$P = 1.0 \times 10^5\ \text{Pa}, V = 2.0\ \text{m}^3, T = 300\ \text{K}, R = 8.31\ \text{J mol}^{-1}\text{K}^{-1}$$
3. Rearrange the ideal gas law to solve for $n$:
4. $$n = \frac{PV}{RT}$$
5. Substitute values:
6. $$n = \frac{(1.0 \times 10^5)(2.0)}{(8.31)(300)} \approx 80\ \text{mol}$$

> **warning**
>
> Always convert temperature to Kelvin before using the ideal gas law. Celsius values will always give incorrect results.

## Temperature and Average Molecular Kinetic Energy

Kinetic theory connects the absolute temperature of an ideal gas to the average kinetic energy of its molecules. The average translational kinetic energy per molecule is directly proportional to absolute temperature.

**Average Translational Kinetic Energy** — The mean kinetic energy of one gas molecule, given by $\langle E_k \rangle = \frac{3}{2}k_B T$. This is independent of the type of gas.

*Notation:* \langle E_k \rangle

**Worked example:** Calculate the average kinetic energy of a gas molecule at room temperature (20°C).

1. First convert temperature from Celsius to Kelvin:
2. $$T = 20 + 273 = 293\ \text{K}$$
3. Substitute into the kinetic energy relationship:
4. $$\langle E_k \rangle = \frac{3}{2}k_B T = 1.5 \times (1.38 \times 10^{-23}) \times 293 \approx 6.1 \times 10^{-21}\ \text{J}$$
5. Note: This result is the same for all ideal gas molecules at 20°C, regardless of their mass.

## Real Gases vs Ideal Gases

Real gases deviate from ideal gas behaviour because the ideal gas assumptions do not hold under all conditions:

- At high pressure: Molecules are pushed close together, so molecular volume is no longer negligible compared to container volume.
- At low temperature: Intermolecular attractive forces become significant, slowing molecules and reducing measured pressure.

**Worked example:** Under which conditions does a real gas behave most like an ideal gas?

1. A real gas behaves most ideally when the ideal gas assumptions are closest to reality.
2. Low pressure means molecules are far apart, so their individual volume is negligible and intermolecular forces are weak.
3. High temperature means molecules have high kinetic energy, so intermolecular forces are negligible compared to molecular kinetic energy.
4. Answer: Low pressure and high temperature.

## Common pitfalls

- **Wrong:** Using Celsius temperature instead of Kelvin in gas law calculations
  - Why it fails: The ideal gas law relies on absolute temperature, and Celsius is not an absolute temperature scale. This is one of the most common marking errors.
  - Correct: Always add 273 to any Celsius temperature to convert to Kelvin before substituting into gas equations.
- **Wrong:** Claiming heavier molecules have higher average kinetic energy at the same temperature
  - Why it fails: Many students associate higher mass with higher kinetic energy, but this ignores the proportional relationship between average kinetic energy and temperature only.
  - Correct: Remember that $\langle E_k \rangle = \frac{3}{2}k_B T$: only temperature affects average kinetic energy. Heavier molecules just move slower on average.
- **Wrong:** Mixing up $R$ and $k_B$ in equations
  - Why it fails: Students often use the wrong constant when switching between moles and number of molecules, leading to incorrect orders of magnitude.
  - Correct: Memorise the rule: $n$ (moles) uses $R$, $N$ (number of molecules) uses $k_B$.
- **Wrong:** Listing only 3-4 kinetic theory assumptions when asked for all
  - Why it fails: Examiners allocate one mark per assumption, so missing any assumption will cost you a mark even if the others are correct.
  - Correct: Memorise all five core assumptions and write them all out when prompted in an exam question.

## Cheatsheet

| Concept | Equation | Key Notes |
| --- | --- | --- |
| Ideal gas law (moles) | $PV = nRT$ | $R = 8.31$ J mol⁻¹ K⁻¹ |
| Ideal gas law (molecules) | $PV = Nk_B T$ | $k_B = 1.38 × 10⁻²³$ J K⁻¹ |
| Average KE per molecule | $\langle E_k \rangle = \frac{3}{2}k_B T$ | Only depends on absolute $T$ |
| Temperature conversion | $T$(K) = $T$(°C) + 273 | Always convert to Kelvin first |
| Ideal behaviour deviation | N/A | Deviates at high $P$, low $T$ |

## What's next

Understanding the behaviour of gases is a foundational concept for thermal physics, which underpins many topics in IB Physics SL, from thermodynamic cycles to astrophysics. The ideal gas model developed here connects microscopic particle behaviour to measurable macroscopic properties, a core pattern you will see repeated across the entire study of the particulate nature of matter. Next, you will build on this knowledge to explore thermal processes including heat transfer and the first law of thermodynamics, which describe how gases exchange energy with their surroundings and do work. Many IB exam questions combine gas behaviour with first law calculations, so mastering this sub-topic is critical for full marks in those later topics.

- [Thermal properties of matter](https://www.owlsprep.com/study/ib-physics-sl-u2-thermal-properties-of-matter/)
- [Wave behaviour](https://www.owlsprep.com/study/ib-physics-sl-u3-overview/)

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