Ideal Gas Law

Imagine pumping air into a bicycle tyre: the more air you push in, the higher the pressure — yet if you heat the tyre on a hot summer day, the pressure rises even without adding more air. These everyday observations are captured precisely by the Ideal Gas Law, one of the most powerful and widely used equations in all of physics. It relates three macroscopic properties of a gas — pressure (P), volume (V), and absolute temperature (T) — to the amount of gas present, expressed as the number of moles (n) or the number of molecules (N). The word 'ideal' signals that the law rests on two simplifying assumptions: gas molecules occupy negligible volume compared with the container, and they exert no attractive or repulsive forces on each other between collisions. Real gases — oxygen, nitrogen, carbon dioxide — obey these assumptions very closely at everyday temperatures and pressures, making the Ideal Gas Law astonishingly accurate for a vast range of practical calculations, from meteorology and engineering to chemistry and astrophysics. The Ideal Gas Law is an example of an equation of state: a single mathematical relationship that completely specifies the equilibrium state of a system. Understanding it means understanding how the microscopic chaos of billions of rapidly moving molecules gives rise to orderly, predictable macroscopic behaviour — a triumph of nineteenth-century science that still underpins modern thermal physics.

Key takeaways

Boyle's Law — Pressure and Volume at Constant Temperature
Charles's Law — Volume and Temperature at Constant Pressure
Gay-Lussac's Law — Pressure and Temperature at Constant Volume
Combining the Three Laws
The Microscopic Picture — Kinetic Theory

Boyle's Law — Pressure and Volume at Constant Temperature

Robert Boyle (1662) found that, for a fixed amount of gas held at constant temperature, pressure and volume are inversely proportional: P ∝ 1/V, or equivalently PV = constant. Think of squeezing a sealed syringe: halving the volume doubles the pressure because the same number of molecules must collide with a smaller surface area.

Charles's Law — Volume and Temperature at Constant Pressure

Jacques Charles (1787) discovered that, at constant pressure, the volume of a gas is directly proportional to its absolute temperature: V ∝ T. The key word is absolute — temperature must be measured in Kelvin (K = °C + 273.15). A gas at 300 K occupies exactly twice the volume it would at 150 K. Physically, hotter molecules move faster, pushing walls outward until pressure rebalances.

Gay-Lussac's Law — Pressure and Temperature at Constant Volume

Joseph Gay-Lussac (1808) showed that, at constant volume, pressure is proportional to absolute temperature: P ∝ T. A sealed metal canister heated on a stove builds pressure because faster molecules strike the walls more often and with greater force.

Combining the Three Laws

Combining all three proportionalities — P ∝ 1/V, V ∝ T, and P ∝ T — gives PV ∝ T. To turn this into an equation we introduce the amount of gas. For n moles, experiment shows that PV = nRT, where R is the universal gas constant (8.314 J mol⁻¹ K⁻¹). Alternatively, using the total number of molecules N and Boltzmann's constant k_B = R/N_A ≈ 1.381 × 10⁻²³ J K⁻¹, we write PV = Nk_BT. Both forms are equally valid and exactly equivalent.

The Microscopic Picture — Kinetic Theory

Kinetic theory derives PV = nRT from first principles: gas molecules travel in straight lines, collide elastically with walls, and exert pressure through the impulse of those collisions. The derivation yields P = (1/3)(N/V)m⟨v²⟩, and identifying (1/2)m⟨v²⟩ = (3/2)k_BT reproduces the Ideal Gas Law exactly. This links temperature directly to the average kinetic energy of molecules, giving a beautifully concrete meaning to an otherwise abstract quantity.

The Key Equation

PV = nRT

P — Pressure (Pascal (Pa)): Force per unit area exerted by gas molecules colliding with the container walls. 1 atm ≈ 101 325 Pa.
V — Volume (Cubic metre (m³)): The space occupied by the gas. Be careful with unit conversions: 1 L = 10⁻³ m³.
n — Amount of substance (Mole (mol)): Number of moles of gas. One mole contains exactly 6.022 × 10²³ molecules (Avogadro's number).
R — Universal gas constant (J mol⁻¹ K⁻¹): R = 8.314 J mol⁻¹ K⁻¹. The same value applies to every ideal gas — it is truly universal.
T — Absolute temperature (Kelvin (K)): Temperature measured from absolute zero. Always convert Celsius to Kelvin before substituting: T(K) = T(°C) + 273.15.

Worked Example

Problem. A weather balloon contains 40.0 mol of helium gas at a pressure of 1.01 × 10⁵ Pa and a temperature of 20 °C. What volume does the gas occupy? If the balloon rises to an altitude where the temperature falls to −40 °C and the pressure drops to 2.50 × 10⁴ Pa, what is the new volume?

V_1 \approx 0.965\ \text{m}^3,\quad V_2 \approx 3.10\ \text{m}^3

Answer. Initial volume ≈ 0.965 m³; volume at altitude ≈ 3.10 m³

Figure

Diagram

Origins · 1 minÉmile Clapeyron and how this was discovered

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Émile Clapeyron

1799–1864 · French

Clapeyron was the first to combine Boyle's Law and Charles's Law into a single equation of state for an ideal gas in 1834, writing it in the form PV/T = constant for a fixed mass of gas — the direct ancestor of PV = nRT. He also expressed Carnot's ideas about heat engines in mathematical form, making them accessible and launching the formal development of thermodynamics. Clapeyron was a railroad engineer by profession and applied thermodynamic reasoning to improve steam locomotive design — one of the earliest examples of fundamental physics driving industrial technology.

Kinetic Theory of Gases — Kinetic theory provides the molecular-level derivation of PV = nRT, linking temperature to average molecular kinetic energy.
First Law of Thermodynamics — The Ideal Gas Law is used alongside the first law to analyse work done by gases in thermodynamic processes such as isothermal and adiabatic expansions.
Boyle's Law — Boyle's Law (PV = constant at fixed T and n) is a special case of the Ideal Gas Law and historically one of its key building blocks.
Absolute Temperature and the Kelvin Scale — The Ideal Gas Law requires temperature in Kelvin; understanding absolute zero and the Kelvin scale is essential for correct application.