Ideal Gas Equation

Gases are unique among the states of matter because their physical behaviour can be described with remarkable accuracy by a single, elegant equation. The ideal gas equation, PV = nRT, unifies three historically separate experimental gas laws — Boyle's Law, Charles's Law, and Avogadro's Law — into one compact relationship that predicts how pressure, volume, temperature, and amount of gas are interrelated. Although real gases deviate from this model at very high pressures and very low temperatures, the ideal gas equation serves as the essential foundation for all gas-phase chemistry at ordinary conditions. The concept of an 'ideal gas' rests on two simplifying assumptions: (1) the molecules themselves occupy negligible volume compared to the container, and (2) intermolecular forces between molecules are zero — molecules interact only through perfectly elastic collisions. These assumptions make the mathematics tractable and produce a law that is accurate to within a few percent for most common gases (N₂, O₂, H₂, noble gases) at standard laboratory conditions of around 25 °C and 1 atm pressure. Beyond the basic PV = nRT form, the ideal gas equation can be rearranged to connect macroscopic, measurable quantities — density and molar mass — to gas behaviour through the relationship PM = dRT. This extension is enormously practical: it allows chemists to determine the molar mass of an unknown gas simply by measuring its density at a known temperature and pressure, a technique used routinely in identifying volatile compounds and verifying molecular formulae.

Key takeaways
  • Boyle's Law: Pressure–Volume Relationship (constant T, n)
  • Charles's Law: Volume–Temperature Relationship (constant P, n)
  • Avogadro's Law: Volume–Amount Relationship (constant T, P)
  • Introducing the Universal Gas Constant R
  • Deriving the Density–Molar Mass Form: PM = dRT

Boyle's Law: Pressure–Volume Relationship (constant T, n)

Robert Boyle (1662) showed experimentally that at constant temperature and constant amount of gas, pressure and volume are inversely proportional: P ∝ 1/V, or equivalently PV = constant. Doubling the pressure halves the volume. Mathematically: V ∝ 1/P (at constant n, T).

Charles's Law: Volume–Temperature Relationship (constant P, n)

Jacques Charles (1787) demonstrated that at constant pressure, the volume of a gas is directly proportional to its absolute temperature (in Kelvin): V ∝ T (at constant n, P). Combining with Boyle's Law gives V ∝ T/P.

Avogadro's Law: Volume–Amount Relationship (constant T, P)

Amedeo Avogadro (1811) proposed that equal volumes of all gases at the same temperature and pressure contain equal numbers of molecules. Therefore V ∝ n (at constant T, P). This gives the combined proportionality: V ∝ nT/P.

Introducing the Universal Gas Constant R

Replacing the proportionality sign with an equality requires a constant of proportionality. Writing V = R·(nT/P) and rearranging gives PV = nRT. The constant R is called the universal (or molar) gas constant. Its experimentally determined value is R = 8.314 J mol⁻¹ K⁻¹ (equivalently 8.314 Pa·m³ mol⁻¹ K⁻¹, or 0.08206 L·atm mol⁻¹ K⁻¹). The same value of R applies to every ideal gas, which is why it is called 'universal'.

Deriving the Density–Molar Mass Form: PM = dRT

Start with PV = nRT. The number of moles n = m/M, where m is mass (g) and M is molar mass (g mol⁻¹). Substituting: PV = (m/M)RT. Rearrange to P = (m/V)·(RT/M). Since density d = m/V, this gives P = d·RT/M, or rearranging: PM = dRT. This shows that at fixed T and P, density is directly proportional to molar mass — heavier gases are denser. Solving for molar mass: M = dRT/P, enabling experimental determination of M from a density measurement.

The Key Equation

PV=nRT⟹PM=dRTPV = nRT \quad \Longrightarrow \quad PM = dRT
  • P — Pressure (Pa (or atm)): The force exerted per unit area by gas molecules colliding with the container walls.
  • V — Volume (m³ (or L)): The space occupied by the gas; must match units of R chosen.
  • n — Amount of substance (mol): Number of moles of gas particles present.
  • R — Universal gas constant (8.314 J mol⁻¹ K⁻¹): A universal proportionality constant that makes the equation exact for an ideal gas.
  • T — Absolute temperature (K (Kelvin)): Temperature measured from absolute zero; T(K) = T(°C) + 273.15. Must always be in Kelvin.
  • M — Molar mass (kg mol⁻¹ (or g mol⁻¹)): Mass per mole of substance; links the macroscopic density to molecular identity.
  • d — Density (kg m⁻³ (or g L⁻¹)): Mass per unit volume of the gas at the conditions specified.

Worked Example

Problem. A 2.50 L flask at 27 °C contains a gas at a pressure of 1.50 atm. (a) How many moles of gas are present? (b) If the density of the gas under these conditions is 1.96 g L⁻¹, what is its molar mass? Identify the gas if it is a pure diatomic element.

n=0.152 mol,M=dRTP=32.0 g mol−1⇒Gas=\ceO2n = 0.152 \text{ mol}, \quad M = \frac{dRT}{P} = 32.0 \text{ g mol}^{-1} \Rightarrow \text{Gas} = \ce{O2}

Answer. 0.152 mol of gas are present; molar mass = 32.0 g mol⁻¹, identifying the gas as molecular oxygen, O₂.

Figure
Diagram
Graphical Summary: Ideal Gas Laws Leading to PV = nRTBoyle's LawPV = k₁(const. T, n)P ∝ 1/VCharles's LawV/T = k₂(const. P, n)V ∝ TAvogadro's LawV/n = k₃(const. T, P)V ∝ nV ∝ nT/P→ PV = nRTDensity–Molar Mass FormPM = dRT → M = dRT/P(substitute n = m/M, d = m/V)R = 8.314 J mol⁻¹ K⁻¹ | T always in Kelvin
Origins · 1 minÉmile Clapeyron and how this was discovered
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Émile Clapeyron
1799–1864 · French

Émile Clapeyron was a French engineer and physicist who in 1834 first combined Boyle's Law and Charles's Law into a single equation of state for an ideal gas, writing it in a form equivalent to PV = nR'T. His 1834 paper, 'Mémoire sur la puissance motrice de la chaleur,' also placed the earlier work of Sadi Carnot on thermodynamic cycles onto rigorous mathematical footing. The universal gas constant R was later given its precise value by scientists including Henri Victor Regnault. Clapeyron was trained primarily as a civil engineer and worked on railway construction in Russia and France — yet his single theoretical paper on gas behaviour became one of the most cited results in physical chemistry.

  • Boyle's Law — Boyle's Law (PV = constant at fixed T and n) is one of the three empirical laws whose combination directly produces the ideal gas equation.
  • Charles's Law — Charles's Law (V ∝ T at fixed P and n) contributes the temperature dependence to PV = nRT and introduces the necessity of using the Kelvin scale.
  • Dalton's Law of Partial Pressures — For gas mixtures, each component behaves ideally and contributes a partial pressure; Dalton's Law is a direct consequence of the ideal gas equation applied to mixtures.
  • Kinetic Molecular Theory of Gases — KMT provides the microscopic, molecular-level justification for why PV = nRT holds — it derives the ideal gas equation from first principles of molecular motion and elastic collisions.
  • Real Gases and the Van der Waals Equation — The Van der Waals equation corrects PV = nRT for intermolecular attractions and finite molecular volume, making it applicable when the ideal gas assumptions break down at high P or low T.