Ideal gas equation of state

A-Level Physics· 19.2 Ideal gas equation of state· 15 min read

1. Definition and forms of the ideal gas equation★★☆☆☆⏱ 4 min

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.

📘 Definition

Ideal gas equation of state

pV = nRT

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

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 $p V = n R T$ to isolate $p$ :
$p = \frac{nRT}{V}$
3
Substitute the known values into the equation:
$p = \frac{0.5 \times 8.31 \times 300}{0.01} = 124650 \text{ Pa} \approx 1.25 \times 10^5 \text{ Pa}$

2. Solving problems for changing gas conditions★★★☆☆⏱ 5 min

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 $n$ , $n R$ 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.

📐 Worked Example

A fixed mass of gas has an initial pressure of $1 \times 1 0^{5}$ 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:
$p_2 = \frac{(1.0 \times 10^5)(0.02)(373)}{(0.01)(293)} \approx 2.55 \times 10^5 \text{ Pa}$

✓ Quick check

Test your understanding:

A fixed amount of gas at constant temperature has its pressure doubled. What happens to its volume?
- Halved
- Doubled
- Unchanged
- Quartered
Reveal answer
Halved —
Correct. For constant $n$ and $T$ , $p V = co n s t an t$ so $p \propto 1/ V$ , doubling $p$ halves $V$ .

3. Molar volume at standard conditions★★☆☆☆⏱ 3 min

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

📘 Definition

Standard Temperature and Pressure (STP)

Defined by CIE as 273 K (0°C) and $1.00 \times 1 0^{5}$ 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 $1.00 \times 1 0^{5}$ 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 $p V = n R T$ to isolate volume $V$ :
$V = \frac{nRT}{p}$
3
Substitute values to get the final volume:
$V = \frac{0.25 \times 8.31 \times 273}{1.00 \times 10^5} \approx 0.00566 \text{ m}^3 = 5.66 \text{ dm}^3$

4. Ideal vs real gas behaviour★★★☆☆⏱ 3 min

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.

📐 Worked Example

State and explain whether you expect carbon dioxide gas at 500 K and $1 \times 1 0^{5}$ 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.

5. Common Pitfalls

Wrong move:

Using Celsius temperature instead of Kelvin in the ideal gas equation

Why:

The ideal gas equation is derived for absolute temperature, so any value relative to 0°C will give a wrong result

Correct move:

Always add 273 (or 273.15 if requested) to Celsius temperatures before substituting into the equation

Wrong move:

Using inconsistent volume units (e.g. cm³ or dm³ with pressure in Pa)

Why:

Pressure in Pa has units of N m⁻², so volume must be in m³ to get consistent SI units for energy

Correct move:

Convert volumes from dm³ to m³ by dividing by 1000, and from cm³ to m³ by dividing by 1,000,000 before calculation

Wrong move:

Using the combined gas law when the mass of gas changes

Why:

The combined gas law $\frac{p _{1} V _{1}}{T _{1}} = \frac{p _{2} V _{2}}{T _{2}}$ only applies for a fixed mass of gas

Correct move:

Use the full $p V = n R T$ form if the amount of gas changes, to account for the change in $n$

Wrong move:

Confusing Boltzmann constant $k$ with molar gas constant $R$

Why:

$p V = N k T$ applies for number of molecules $N$ , while $p V = n R T$ applies for moles $n$ . Using the wrong constant gives an incorrect result

Correct move:

Check if the question asks for number of molecules (use $k$ ) or number of moles (use $R$ ), and remember $R = N_{A} k$

6. Quick Reference Cheatsheet

Relationship	Variables	Key Notes
$p V = n R T$	$p$ (Pa), $V$ (m³), $n$ (mol), $T$ (K)	For calculations involving moles of gas
$p V = N k T$	$p$ (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: $p = 1 \times 1 0^{5}$ Pa, $T = 273$ K	Molar volume = 22.4 dm³	Standard reference condition for CIE exams
r.t.p: $p = 1 \times 1 0^{5}$ Pa, $T = 293$ K	Molar volume = 24 dm³	Common room condition reference

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.

2022 · 12
Multiple choice gas volume change
2023 · 22
Calculate number of moles of gas
2024 · 11
Compare ideal/real gas conditions

Going deeper

syllabusCIE 9702 Official SyllabusCheck section 19 for latest requirements

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.