Ideal Gas Law — meaning, derivation, applications and worked examples
The Ideal Gas Law, usually written as PV = nRT, connects the macroscopic properties of a gas: pressure (P), volume (V), temperature (T) and amount in moles (n), with the universal gas constant R. It unifies three classical gas laws — Boyle’s, Charles’s and Avogadro’s — and provides a simple but powerful model of gas behaviour under many ordinary conditions.
In practice, the law is both an empirical compression of observed behaviour and a theoretical consequence of the kinetic molecular theory: an ideal gas is imagined as a large number of point-like particles that move randomly, collide elastically with container walls, and do not interact otherwise. Real gases approximate this behaviour when pressure is moderate and temperature is not too low.
What each term means
Pressure (P) is force per unit area due to molecular collisions with the container walls — measured in pascals (Pa) in SI. Volume (V) is the space occupied by the gas in cubic metres (m³). Temperature (T) must be the absolute temperature in kelvin (K), because kinetic energy scales with absolute temperature. Moles (n) measure the amount of gas (1 mole = Avogadro’s number of particles). Finally, R is the universal gas constant that provides the right proportionality between these variables.
How it is derived
Empirically, three classical laws were observed: Boyle’s law (P ∝ 1/V at constant T and n), Charles’s law (V ∝ T at constant P and n), and Avogadro’s law (V ∝ n at constant P and T). Combining these proportionalities gives V ∝ nT/P; introducing the universal constant R yields PV = nRT.
Microscopic interpretation
Kinetic molecular theory links the macroscopic law to microscopic motion: pressure is proportional to the average translational kinetic energy of molecules. The average kinetic energy of a molecule is (3/2)kT where k is Boltzmann’s constant; converting per-molecule energy into per-mole energy introduces R = N_A k (Avogadro’s number times Boltzmann’s constant), yielding PV = nRT at the macroscopic scale.
Worked examples
Example 1 — Pressure: A 0.02 m³ container holds 0.5 mol of a gas at 300 K. Using R = 8.314 J·mol⁻¹·K⁻¹, P = nRT / V = (0.5 × 8.314 × 300) / 0.02 = 62,355 Pa (≈ 0.615 atm).
Example 2 — Temperature: A 1.0 L (0.001 m³) flask contains 0.05 mol of gas at 2.0×10⁵ Pa. T = PV/(nR) = (2.0e5 × 0.001) / (0.05 × 8.314) ≈ 481 K.
Example 3 — Volume with different R units: If pressure is given in atm and R = 0.082057 L·atm·mol⁻¹·K⁻¹, ensure V is in litres. For P = 2 atm, n = 0.25 mol and T = 350 K, V = nRT / P = (0.25 × 0.082057 × 350) / 2 ≈ 3.58 L.
When the ideal assumption breaks down
At high pressures the finite volume of gas molecules becomes important; at low temperatures intermolecular attractions cause condensation tendencies — both create measurable deviations from the ideal law. The Van der Waals equation introduces two constants a and b to correct for these effects: (P + a(n/V)²)(V - nb) = nRT. Empirical or semi-empirical equations of state (Redlich–Kwong, Peng–Robinson) are used in engineering where accuracy matters over wide conditions.
Practical tips for calculations
- Always use absolute temperature in kelvin.
- Keep units consistent with the chosen R value (J/mol·K with Pa·m³ units; L·atm units with R = 0.082057).
- Convert volumes (L ↔ m³) and pressures (atm ↔ Pa) carefully: 1 m³ = 1000 L, 1 atm ≈ 101325 Pa.
- Watch significant figures — report answers with appropriate precision based on input data.
Applications across disciplines
Chemists use PV = nRT for stoichiometric problems with gases; meteorologists apply it to atmospheric calculations; mechanical and chemical engineers use it for compressors, engines and reactors; and planetary scientists use it to estimate atmospheric behaviour on other worlds.
Connections to thermodynamics
The ideal gas law is foundational to classical thermodynamics: internal energy and enthalpy of an ideal gas depend only on temperature (for ideal monoatomic and polyatomic gases under ideal behaviour), simplifying analyses of heat engines and refrigeration cycles. Many textbook problems rely on the ideal assumption to build intuition before introducing real-gas corrections.
Measuring and experimental checks
Experimentally, you can verify PV = nRT by measuring P, V and T for a known n and checking proportionality. Precise measurements reveal systematic deviations where corrections are needed; these deviations are themselves informative about intermolecular forces and molecular size.
Summary
The Ideal Gas Law is an elegant, compact relation that captures the gross behaviour of gases under ordinary conditions. While not universally precise, it forms the backbone of gas calculations in science and engineering and provides a clear bridge between microscopic molecular motion and macroscopic observables.