# Ideal gas behaviour

> IB Chemistry Higher Level · S1: Models of the particulate nature of matter
> Source: https://www.owlsprep.com/study/ib-chemistry-hl-u1-ideal-gas-behaviour/

This module covers the kinetic molecular theory postulates for ideal gases, the ideal gas equation, and how to use it to calculate unknown gas properties. You will also learn when real gases deviate from ideal predicted behaviour.

**Prerequisites:** [Moles and molar mass calculations](https://www.owlsprep.com/study/ib-chemistry-hl-u1-molar-calculations/); SI unit conversion and absolute temperature scale

## Learning objectives

- State the postulates of kinetic molecular theory for ideal gases
- Apply the ideal gas equation to calculate unknown gas properties
- Derive rearranged forms of the ideal gas equation for molar mass and density
- Predict when real gases deviate from ideal behaviour and explain why

## Postulates of Kinetic Molecular Theory

An ideal gas is a hypothetical model that simplifies the behaviour of real gases to allow predictable calculations based on particle motion. The kinetic molecular theory (KMT) outlines 5 core assumptions that define an ideal gas.

**Ideal Gas** — A hypothetical gas that follows all postulates of kinetic molecular theory exactly, with no intermolecular interactions and negligible particle volume

*Example:* Most real gases approximate ideal behaviour at low pressure and high temperature

- Gases consist of large numbers of tiny particles far apart relative to their size, so individual particle volume is negligible
- All collisions between particles and container walls are elastic (no net loss of kinetic energy)
- Gas particles are in constant, random, rapid motion
- There are no attractive or repulsive intermolecular forces between gas particles
- Average kinetic energy of particles is proportional to absolute temperature

> **Exam tip**
>
> Short answer questions often ask for 2-4 postulates, so memorize the key assumptions about particle volume and intermolecular forces specifically.

## The Ideal Gas Equation

Combining Boyle's law ($P \propto 1/V$), Charles' law ($V \propto T$), and Avogadro's law ($V \propto n$) gives the combined ideal gas equation that relates all measurable properties of an ideal gas.

$$PV = nRT$$

> **warning**
>
> Always convert temperature to Kelvin before using the ideal gas equation: $T(\text{K}) = T(^\circ\text{C}) + 273.15$. Units of $P$ and $V$ must match the units of the gas constant $R$ you use.

**Worked example:** A 0.250 mol sample of nitrogen gas occupies 12.0 L at 30.0 °C. Calculate the pressure of the gas, given $R = 8.314 \text{ kPa L mol}^{-1} \text{K}^{-1}$.

1. Convert temperature from Celsius to Kelvin:
2. $$T = 30.0 + 273.15 = 303.15 \text{ K}$$
3. Rearrange the ideal gas equation to solve for pressure $P$:
4. $$P = \frac{nRT}{V}$$
5. Substitute values and calculate:
6. $$P = \frac{(0.250 \text{ mol})(8.314 \text{ kPa L mol}^{-1} \text{K}^{-1})(303.15 \text{ K})}{12.0 \text{ L}} = 52.6 \text{ kPa}$$

**Check your understanding**

Check your unit conversion understanding

1. What absolute temperature should you use for a gas at 25 °C?

   - 25 K
   - 298 K
   - 248 K
   - -248 K

   *Why:* Correct: 25 + 273 = 298 K. The ideal gas equation only works with absolute temperature.

## Calculating Molar Mass and Gas Density

The ideal gas equation can be rearranged to calculate the molar mass or density of an unknown gas, using the relationship $n = \frac{m}{M}$, where $m$ is mass of gas and $M$ is molar mass.

**Derivation:** Derive the formula for molar mass of an unknown gas

*Starting from:* $PV = nRT$ and $n = \frac{m}{M}$

1. Substitute $n = \frac{m}{M}$ into the ideal gas equation:
2. $$PV = \frac{mRT}{M}$$
3. Rearrange to isolate $M$:
4. $$M = \frac{mRT}{PV}$$
5. Substitute density $\rho = \frac{m}{V}$ to get the density relationship:
6. $$M = \frac{\rho RT}{P} \implies \rho = \frac{PM}{RT}$$

*Conclusion:* These rearranged formulas are commonly used to identify unknown gases from experimental data in IB exams.

**Worked example:** An unknown gas has a mass of 1.25 g, occupies 0.550 L at 100.0 kPa and 25 °C. Calculate its molar mass, using $R = 8.314 \text{ kPa L mol}^{-1} \text{K}^{-1}$.

1. Convert temperature to Kelvin:
2. $$T = 25 + 273.15 = 298.15 \text{ K}$$
3. Substitute values into the derived formula for $M$:
4. $$M = \frac{(1.25 \text{ g})(8.314 \text{ kPa L mol}^{-1} \text{K}^{-1})(298.15 \text{ K})}{(100.0 \text{ kPa})(0.550 \text{ L})} = 56.3 \text{ g mol}^{-1}$$

## Deviations from Ideal Gas Behaviour

The ideal gas model is a simplification: all real gases deviate from ideal behaviour because the two core KMT assumptions are never perfectly true for real particles.

- Real gas particles have non-negligible volume: this becomes significant at high pressure when particles are forced close together
- Real gas particles have attractive intermolecular forces: this becomes significant at low temperature when particles move slowly enough for forces to act

Deviations are largest at high pressure and low temperature. Gases with larger molecular size and stronger intermolecular forces deviate more than small, nonpolar gases.

**Worked example:** Which gas shows the greatest deviation from ideal behaviour at 10 atm and 100 K: He, H₂, or NH₃? Explain your answer.

1. Compare intermolecular forces and molecular size for each gas:
2. He and H₂ are small, nonpolar molecules with very weak London dispersion forces. NH₃ is polar with strong hydrogen bonding between molecules.
3. At low temperature and high pressure, intermolecular forces are highly significant.
4. Conclusion: NH₃ deviates most from ideal behaviour because of its strong intermolecular forces.

> **Exam tip**
>
> Always link deviations to the specific broken assumption, not just list conditions, to get full marks in explanations.

## Common pitfalls

- **Wrong:** Using temperature in Celsius instead of Kelvin in the ideal gas equation
  - Why it fails: The ideal gas equation relies on absolute temperature where 0 K = 0 kinetic energy. Celsius values give incorrect proportionality.
  - Correct: Always add 273.15 to Celsius temperature to get Kelvin before substituting into the equation.
- **Wrong:** Mismatching units of pressure/volume with the units of R
  - Why it fails: R has different numerical values for different unit sets, so mismatches give wrong orders of magnitude for results.
  - Correct: Check that units of P and V match the units of R you use, convert units if needed before calculating.
- **Wrong:** Claiming all deviations are caused only by intermolecular forces
  - Why it fails: At very high pressure, non-negligible particle volume is the dominant cause of deviation, not intermolecular forces.
  - Correct: Distinguish between causes: low temperature deviations come from intermolecular forces, high pressure deviations come from particle volume.
- **Wrong:** Stating deviations are largest at low pressure and high temperature
  - Why it fails: These are the conditions where ideal assumptions are closest to true, so deviations are smallest here.
  - Correct: Remember deviations are largest at high pressure and low temperature, when particles are close and moving slowly.

## Cheatsheet

| Concept | Key Formula / Fact |
| --- | --- |
| Ideal Gas Equation | $PV = nRT$ |
| Molar Mass of Unknown Gas | $M = \frac{mRT}{PV}$ |
| Gas Density | $\rho = \frac{PM}{RT}$ |
| Core Assumption 1 | Negligible individual particle volume |
| Core Assumption 2 | No intermolecular forces between particles |
| Core Assumption 3 | Average KE $\propto$ absolute temperature |
| Minimum Deviation Conditions | Low pressure, high temperature |
| Maximum Deviation Conditions | High pressure, low temperature |
| Most Deviant Gas Type | Strong intermolecular forces, large molecules |

## What's next

Understanding ideal gas behaviour is the foundation for all gas-related topics in IB Chemistry, including reaction stoichiometry involving gaseous products, entropy calculations, and acid-base reactions with gaseous reactants. The distinction between ideal and real gas behaviour also underpins your later study of intermolecular forces, where you will explore how particle interactions affect bulk properties of matter. The calculation skills you developed here are regularly tested in both Paper 1 multiple choice and Paper 2 short answer questions, so regular practice rearranging the ideal gas equation for different unknowns is key to exam success.

- [Intermolecular Forces](https://www.owlsprep.com/study/ib-chemistry-hl-u2-intermolecular-forces/)
- [Kinetic molecular theory](https://www.owlsprep.com/study/ib-chemistry-hl-u1-kinetic-molecular-theory/)
- [AHL: Advanced mass spectrometry interpretation](https://www.owlsprep.com/study/ib-chemistry-hl-u1-ahl-advanced-mass-spectrometry-interpretation/)

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