# Ideal gas equation of state

> A-Level Physics · CIE 9702
> Source: https://www.owlsprep.com/study/cie-9702-u19-ideal-gas-equation-of-state/

This module explains the ideal gas equation of state, combining Boyle's, Charles's and Pressure laws to relate pressure, volume, temperature and amount of gas. You will learn to solve common exam problems for changing gas conditions.

**Prerequisites:** [Individual gas laws (Boyle's, Charles's, Pressure Law)](https://www.owlsprep.com/study/cie-9702-u19-gas-laws-for-ideal-gases/); Absolute temperature and the Kelvin scale

## Learning objectives

- State the ideal gas equation in all common forms
- Convert between temperature scales for gas calculations
- Solve problems involving changing gas conditions
- Relate the ideal gas constant to the Boltzmann constant
- Identify conditions for ideal gas behaviour

## Definition and forms of the ideal gas equation

The ideal gas equation of state relates the four state variables that describe a fixed amount of gas: pressure, volume, amount of substance, and absolute temperature. It is derived by combining the three individual gas laws into a single relationship.

**Ideal gas equation of state** — An equation relating the measurable properties of an ideal gas, where all intermolecular interactions are negligible and molecular volume is ignored compared to total gas volume

*Notation:* pV = nRT

*Example:* Calculate the volume of 2 moles of oxygen at room temperature and pressure

The equation can also be written in terms of the number of molecules $N$, rather than moles $n$. Since $n = \frac{N}{N_A}$ where $N_A$ is Avogadro's constant, substituting gives:

$$pV = N\frac{R}{N_A}T = NkT$$

where $k = \frac{R}{N_A}$ is the Boltzmann constant.

**Worked example:** Calculate the pressure exerted by 0.5 moles of an ideal gas with volume 0.01 m³ at a temperature of 27°C. $R = 8.31$ J mol⁻¹ K⁻¹.

1. First convert the given Celsius temperature to absolute Kelvin temperature:

   $$T = 27 + 273.15 = 300 \text{ K}$$
2. Rearrange $pV = nRT$ to isolate $p$:

   $$p = \frac{nRT}{V}$$
3. Substitute the known values into the equation:

   $$p = \frac{0.5 \times 8.31 \times 300}{0.01} = 124650 \text{ Pa} \approx 1.25 \times 10^5 \text{ Pa}$$

## Solving problems for changing gas conditions

When a fixed mass of gas undergoes changes to its pressure, volume or temperature, we can use the ideal gas equation to find the unknown quantity without needing to know the amount of gas. For fixed $n$, $nR$ is constant, so:

$$\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}$$

where subscript 1 refers to initial conditions and subscript 2 refers to final conditions.

> **Critical requirement**
>
> You must always use absolute Kelvin temperature in this calculation. Using Celsius will always give an incorrect result.

**Worked example:** A fixed mass of gas has an initial pressure of $1 \times 10^5$ Pa, volume 0.02 m³, and temperature 20°C. The gas is compressed to 0.01 m³ and heated to 100°C. Find the new pressure.

1. Convert initial $T_1$ and final $T_2$ to Kelvin:

   $$T_1 = 20 + 273 = 293 \text{ K}, \quad T_2 = 100 + 273 = 373 \text{ K}$$
2. Rearrange the combined gas law for $p_2$:

   $$p_2 = \frac{p_1 V_1 T_2}{V_2 T_1}$$
3. Insert all known values and calculate:

   $$p_2 = \frac{(1.0 \times 10^5)(0.02)(373)}{(0.01)(293)} \approx 2.55 \times 10^5 \text{ Pa}$$

**Check your understanding**

Test your understanding:

1. A fixed amount of gas at constant temperature has its pressure doubled. What happens to its volume?

   - Halved
   - Doubled
   - Unchanged
   - Quartered

   *Why:* Correct. For constant $n$ and $T$, $pV = constant$ so $p \propto 1/V$, doubling $p$ halves $V$.

## Molar volume at standard conditions

Examiners often cite standard reference conditions for gas calculations, so you need to remember the standard values for CIE exams.

**Standard Temperature and Pressure (STP)** — Defined by CIE as 273 K (0°C) and $1.00 \times 10^5$ Pa (1 atmosphere)

*Example:* Molar volume of any ideal gas at STP is approximately 22.4 dm³ (0.0224 m³)

Room temperature and pressure (r.t.p.) is also commonly used, defined as 293 K (20°C) and $1.00 \times 10^5$ Pa, with molar volume ~24 dm³ (0.024 m³).

**Worked example:** Calculate the volume of 0.25 moles of an ideal gas at STP. $R = 8.31$ J mol⁻¹ K⁻¹.

1. State standard STP values:

   $$p = 1.00 \times 10^5 \text{ Pa}, \quad T = 273 \text{ K}$$
2. Rearrange $pV = nRT$ to isolate volume $V$:

   $$V = \frac{nRT}{p}$$
3. Substitute values to get the final volume:

   $$V = \frac{0.25 \times 8.31 \times 273}{1.00 \times 10^5} \approx 0.00566 \text{ m}^3 = 5.66 \text{ dm}^3$$

## Ideal vs real gas behaviour

The ideal gas equation makes two key assumptions: 1) the volume of gas molecules themselves is negligible compared to the total volume of the gas, and 2) there are no intermolecular forces between molecules. This is never perfectly true for real gases, but is a good approximation under most conditions.

> **info**
>
> Real gases deviate most from ideal behaviour at high pressure and low temperature. At high pressure molecules are close together so their own volume is significant, and at low temperature intermolecular forces significantly affect molecular motion.

**Worked example:** State and explain whether you expect carbon dioxide gas at 500 K and $1 \times 10^5$ Pa to behave like an ideal gas.

1. Assess the given temperature and pressure conditions:
2. The temperature is high (well above the boiling point of CO₂) and pressure is low. Molecules are far apart on average, so their individual volume is negligible compared to total volume and intermolecular forces are very weak. Therefore the gas will behave approximately like an ideal gas.

## Common pitfalls

- **Wrong:** Using Celsius temperature instead of Kelvin in the ideal gas equation
  - Why it fails: The ideal gas equation is derived for absolute temperature, so any value relative to 0°C will give a wrong result
  - Correct: Always add 273 (or 273.15 if requested) to Celsius temperatures before substituting into the equation
- **Wrong:** Using inconsistent volume units (e.g. cm³ or dm³ with pressure in Pa)
  - Why it fails: Pressure in Pa has units of N m⁻², so volume must be in m³ to get consistent SI units for energy
  - Correct: Convert volumes from dm³ to m³ by dividing by 1000, and from cm³ to m³ by dividing by 1,000,000 before calculation
- **Wrong:** Using the combined gas law when the mass of gas changes
  - Why it fails: The combined gas law $\frac{p_1V_1}{T_1} = \frac{p_2V_2}{T_2}$ only applies for a fixed mass of gas
  - Correct: Use the full $pV = nRT$ form if the amount of gas changes, to account for the change in $n$
- **Wrong:** Confusing Boltzmann constant $k$ with molar gas constant $R$
  - Why it fails: $pV = NkT$ applies for number of molecules $N$, while $pV =nRT$ applies for moles $n$. Using the wrong constant gives an incorrect result
  - Correct: Check if the question asks for number of molecules (use $k$) or number of moles (use $R$), and remember $R = N_A k$

## Cheatsheet

| Relationship | Variables | Key Notes |
| --- | --- | --- |
| $pV = nRT$ | $p$ (Pa), $V$ (m³), $n$ (mol), $T$ (K) | For calculations involving moles of gas |
| $pV = NkT$ | $p$ (Pa), $V$ (m³), $N$ (molecules), $T$ (K) | For calculations involving number of molecules |
| $\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}$ | Fixed mass of gas | Only use when amount of gas is constant |
| STP: $p=1 \times 10^5$ Pa, $T=273$ K | Molar volume = 22.4 dm³ | Standard reference condition for CIE exams |
| r.t.p: $p=1 \times 10^5$ Pa, $T=293$ K | Molar volume = 24 dm³ | Common room condition reference |

## What's next

The ideal gas equation of state is the foundation for the kinetic theory of gases, which links the macroscopic properties of gas we measure to the microscopic motion of individual molecules. Understanding this equation is critical for solving all thermodynamics problems involving gases, from engine cycles to diffusion. After mastering this sub-topic, you can move on to deriving the kinetic theory equation and relating gas pressure to average molecular kinetic energy, the next core topic in the ideal gases unit.

- [Kinetic theory of gases](https://www.owlsprep.com/study/cie-9702-u19-kinetic-theory-of-gases/)
- [First law of thermodynamics](https://www.owlsprep.com/study/cie-9702-u20-first-law-of-thermodynamics/)
- [Root mean square speed](https://www.owlsprep.com/study/cie-9702-u19-root-mean-square-speed/)

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