# Gases and the ideal gas equation

> Chemistry · CIE A-Level
> Source: https://www.owlsprep.com/study/cie-9701-u4-gases-and-ideal-gas-equation/

This sub-topic covers the ideal gas equation, underlying assumptions of ideal gases, calculations for unknown gas properties, and origins of deviations between real and ideal gas behaviour, a core foundation for physical chemistry.

**Prerequisites:** [Basic gas laws (Boyle's, Charles', Avogadro's)](https://www.owlsprep.com/study/cie-9701-as-physical-chemistry-gas-laws/); [Mole concept and molar mass calculations](https://www.owlsprep.com/study/cie-9701-u1-moles-and-calculations/)

## Learning objectives

- State the four core assumptions of an ideal gas
- Manipulate the ideal gas equation to solve for unknown gas properties
- Use molar volume at r.t.p. for gas mass/volume calculations
- Explain why and when real gases deviate from ideal behaviour

## Key Assumptions of an Ideal Gas

An ideal gas is a hypothetical model that simplifies gas behaviour for calculations. All gas laws and the ideal gas equation are built on four core assumptions about gas molecules.

**Ideal Gas** — A hypothetical gas that obeys all ideal gas assumptions and the ideal gas equation under all temperature and pressure conditions.

*Example:* No perfectly ideal gas exists, but most real gases behave almost ideally at low pressure and high temperature.

- Gas molecules have negligible volume compared to the total volume of the gas container
- There are no attractive intermolecular forces between gas molecules
- All collisions between molecules and container walls are perfectly elastic (no net energy loss)
- Gas molecules move randomly in straight lines at a range of speeds

**Worked example:** State the four core assumptions of an ideal gas, required for full marks in a CIE exam question

1. 1. Gas molecules have negligible volume compared to the total volume of the container
2. 2. There are no attractive intermolecular forces between gas molecules
3. 3. All collisions between molecules are perfectly elastic, with no net energy loss
4. 4. Gas molecules move randomly at a range of different speeds

> **Exam tip:** CIE mark schemes award one mark per assumption; always list all four when asked, do not just give two or three.

## The Ideal Gas Equation and Calculations

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, which relates all four measurable properties of a gas.

$$PV = nRT$$

Where: $P$ = pressure, $V$ = volume, $n$ = number of moles, $R$ = the universal gas constant ($8.31$ J K$^{-1}$ mol$^{-1}$ for SI units), $T$ = absolute temperature in Kelvin.

> **warning**
>
> Always check your units before substituting! CIE examiners regularly trap students who use cm$^3$, °C or atm without converting to match the units of $R$.

**Worked example:** A 0.250 g sample of CO$_2$ occupies 142 cm$^3$ at 100 °C. Calculate the pressure of the sample, using $R = 8.31$ J K$^{-1}$ mol$^{-1}$.

1. Step 1: Calculate moles of CO$_2$ and convert all units to SI units

   $$n = \frac{mass}{M_r} = \frac{0.250}{44.0} = 0.00568 \text{ mol} \\ V = 142 \text{ cm}^3 = 142 \times 10^{-6} \text{ m}^3 \\ T = 100 + 273 = 373 \text{ K}$$
2. Step 2: Rearrange the ideal gas equation to solve for $P$

   $$P = \frac{nRT}{V}$$
3. Step 3: Substitute values and calculate the final pressure

   $$P = \frac{0.00568 \times 8.31 \times 373}{142 \times 10^{-6}} = 124000 \text{ Pa} = 124 \text{ kPa}$$

## Molar Volume at Room Temperature and Pressure

At r.t.p. (defined by CIE as 25 °C / 298 K and 1 atm / 101325 Pa), one mole of any ideal gas occupies a fixed volume. This value is given in the CIE data booklet, but memorising it speeds up calculations significantly.

**Molar Volume at r.t.p.** — The volume occupied by one mole of any gas at r.t.p.

*Notation:* V_m

*Example:* V_m = 24.0 dm$^3$ mol$^{-1}$ (24000 cm$^3$ mol$^{-1}$) at r.t.p.

**Worked example:** Calculate the mass of 3.0 dm$^3$ of oxygen gas ($O_2$) at r.t.p.

1. Step 1: Calculate moles of $O_2$ using the molar volume at r.t.p.

   $$n = \frac{V}{V_m} = \frac{3.0}{24.0} = 0.125 \text{ mol}$$
2. Step 2: Calculate mass from moles and molar mass of $O_2$

   $$mass = n \times M_r = 0.125 \times 32.0 = 4.0 \text{ g}$$

## Deviations of Real Gases from Ideal Behaviour

The ideal gas assumptions do not hold for real gases. Two core assumptions break down under extreme conditions, leading to measurable deviations from the ideal gas equation $PV = nRT$.

- Molecular volume is not negligible: At high pressure, molecules are pushed close together, so the volume of the molecules themselves becomes a significant fraction of total volume
- Intermolecular forces are not zero: At low temperature, molecules move slower, so attractive intermolecular forces become significant, pulling molecules away from container walls

The largest deviations occur at high pressure, low temperature, and for gases with larger molecular size or stronger intermolecular forces.

**Worked example:** Explain why propane deviates more from ideal behaviour than helium at the same temperature and pressure.

1. Step 1: Identify the key difference between the two gases
2. Propane has a larger molecular size and much stronger intermolecular forces than helium.
3. Step 2: Link to broken ideal gas assumptions
4. Both the assumption of negligible molecular volume and the assumption of no intermolecular forces are less valid for propane, leading to greater deviation from ideal behaviour.

> **Exam tip:** When comparing deviations between two gases, always link the degree of deviation to intermolecular force strength and molecular size, not just mass.

## Common pitfalls

- **Wrong:** Forgetting to convert temperature from °C to Kelvin
  - Why it fails: The ideal gas equation requires absolute temperature; using °C gives a value for $T$ that is off by 273, leading to a completely incorrect result.
  - Correct: Always add 273 to any temperature given in °C before substituting into the ideal gas equation.
- **Wrong:** Using mismatched units for pressure and volume with $R = 8.31$
  - Why it fails: $R = 8.31$ J K$^{-1}$ mol$^{-1}$ requires pressure in Pa and volume in m$^3$; using incorrect units gives a result with the wrong order of magnitude.
  - Correct: Convert cm$^3$ to m$^3$ by multiplying by $10^{-6}$, dm$^3$ to m$^3$ by $10^{-3}$, and kPa to Pa by multiplying by $10^3$.
- **Wrong:** Only listing two assumptions of an ideal gas when asked for all
  - Why it fails: CIE 3-4 mark questions for this topic award one mark per assumption, so you will lose marks for incomplete answers.
  - Correct: Memorise all four assumptions and always list every one when asked to describe an ideal gas.
- **Wrong:** Claiming high temperature increases deviation from ideal behaviour
  - Why it fails: At high temperature, intermolecular forces are negligible because molecules move too fast to be attracted, so behaviour is closer to ideal.
  - Correct: Remember that low temperature and high pressure increase deviation from ideal behaviour for real gases.

## Cheatsheet

| Quantity | Required SI Unit | Common Conversions |
| --- | --- | --- |
| Pressure ($P$) | Pascal (Pa) | 1 atm = 101325 Pa; 1 kPa = 1000 Pa |
| Volume ($V$) | Cubic meter (m$^3$) | 1 cm$^3$ = 10$^{-6}$ m$^3$; 1 dm$^3$ = 10$^{-3}$ m$^3$ |
| Temperature ($T$) | Kelvin (K) | $T \text{ (K)} = T \text{ (°C)} + 273$ |
| Gas constant ($R$) | J K$^{-1}$ mol$^{-1}$ | $R = 8.31$ for SI unit calculations |
| Molar volume at r.t.p. | dm$^3$ mol$^{-1}$ | $V_m = 24.0$ dm$^3$ mol$^{-1}$ |

## What's next

Mastering the ideal gas equation is a core foundation for all further physical chemistry topics in CIE A-Level, including calculations for gaseous equilibria, enthalpy changes, and reaction kinetics. The concepts of intermolecular forces and deviations from ideal behaviour you learned here connect directly to the study of other states of matter, and help explain bulk properties like boiling point and volatility. Getting comfortable with unit conversions for ideal gas calculations also prevents simple, costly errors in all future physical chemistry questions involving gases, which appear frequently in both multiple choice and structured questions.

- [Liquids and Solids](https://www.owlsprep.com/study/cie-9701-u4-liquids-and-solids/)
- [Crystal structures](https://www.owlsprep.com/study/cie-9701-u4-crystal-structures/)
- [Chemical energetics](https://www.owlsprep.com/study/cie-9701-u5-overview/)

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