Combined gas law

Formula and variables

In words, the combined gas law states that the ratio of the product of pressure and volume to absolute temperature remains constant for a given sample of gas: PV/T = k. In its comparative form this becomes (P1 V1)/T1 = (P2 V2)/T2, where:

• P denotes pressure,
• V denotes volume,
• T denotes temperature measured on the absolute scale (Kelvin, K),
• k is a constant for that fixed amount of gas.

Derivation from three simple gas laws

The combined gas law is obtained by combining three classical relationships that each hold when one variable is kept constant. Charles's law asserts that volume is proportional to temperature at constant pressure (isobaric conditions). Boyle's law shows that pressure and volume are inversely proportional at constant temperature. Gay-Lussac's law (often also called Gay-Lussac or Amontons) expresses that pressure is proportional to temperature when volume is fixed. Combining these three proportionalities eliminates the restriction of holding one variable strictly constant, yielding PV/T = constant.

Applications and examples

The combined gas law is widely used in laboratory practice and engineering when a sealed sample of gas is subjected to changes in conditions. Typical uses include calculating the change in pressure when a gas is compressed and warmed, predicting how volume changes when temperature drops under constant pressure, and estimating behaviour in sealed containers exposed to temperature swings. It is often the first tool for solving problems that involve two states of the same gas without invoking the gas constant or molar amount explicitly.

Relation to the ideal gas law and limitations

Adding Avogadro's law (which relates volume to amount of substance at fixed temperature and pressure) extends the combined gas law into the full ideal gas law, PV = nRT, where n is moles and R is the gas constant. The combined gas law assumes ideal behaviour: molecules have negligible volume and no intermolecular forces. Real gases deviate under high pressures and low temperatures, so corrections (for example, the van der Waals equation) are used when precision is required.

Notable points and distinctions

Several practical cautions help avoid mistakes: always use absolute temperature (Kelvin) when applying the law; ensure the amount of gas is unchanged between the two states; and remember that pressure units must be consistent on both sides of the equation. While the combined gas law is a compact and helpful relation for many problems, it is a special-case result of the more general ideal gas law and loses accuracy when gases behave nonideally.

For further reading on each component law and on how to apply the combined relation in calculations, consult introductory resources on gas behaviour and thermodynamics: combined gas law, ideal gases, pressure, volume, temperature, Charles's law, isobaric descriptions, Boyle's law, Kelvin scale, Avogadro's law and the ideal gas law.