When should I use the combined gas law instead of PV = nRT?

Use the combined gas law when the amount of gas (n) is constant and you are comparing two sets of conditions. It cancels nR, so you do not need R or moles. Use PV = nRT when n changes or you need to find moles, density, or molar mass.

What happens if I use Celsius instead of Kelvin in the combined gas law?

The answer will be physically wrong.Always add 273.15 to the Celsius temperature before substituting.

Does the combined gas law work for real gases like steam or propane?

It works as an approximation at low pressures and moderate temperatures. Near the condensation point or above 10 atm, the error exceeds 5%. For steam at high pressure or propane in a tank, use the van der Waals equation or a compressibility chart.

The combined gas law gave me a negative volume, what went wrong?

You likely used a negative temperature value or forgot to convert to Kelvin. Temperature in the combined gas law must be absolute (Kelvin). Also check that n is constant; if gas was added or removed, the law does not apply.

Combined Gas Law: P₁V₁/T₁ = P₂V₂/T₂

The Combined Gas Law (and Boyle's, Charles's, Gay-Lussac's)

The combined gas law formula P₁V₁/T₁ = P₂V₂/T₂ is the most practical tool for before/after gas problems where the amount of gas stays constant. You skip the moles entirely. The law combines Boyle's, Charles's, and Gay-Lussac's laws into one equation. The single most useful fact: if you know any two of pressure, volume, and temperature before a change, and the value of one after, the law gives you the missing one. No R value required.

To use it, convert all temperatures to Kelvin. A Celsius reading of 25 °C becomes 298 K. A common failure: using Celsius in the denominator, which produces a physically meaningless answer. The law works for any gas at low pressures, under about 5 atm for most diatomic gases, and fails when the gas approaches its condensation point.

From PV = nRT to the Combined Law

The ideal gas law (PV = nRT) relates pressure (P), volume (V), amount of gas (n), and temperature (T) through the molar gas constant R. When the amount of gas does not change, no gas added, none removed, the nR term on the right is the same before and after the change. Set the two states equal: P₁V₁/T₁ = nR = P₂V₂/T₂. Cancel nR, and you get the combined gas law. The derivation takes one step.

OpenStax Chemistry 2e, ch. 9, sections 9.2-9.3 and 9.6, traces this derivation from the individual gas laws. The combined gas law is exact for an ideal gas and a good approximation for real gases at moderate conditions. The failure case: if gas is added or removed between measurements, the n changes and the combined law gives a wrong answer. Use PV = nRT instead when the amount changes.

Boyle's Law (Constant T)

Boyle's law states that pressure and volume are inversely proportional when temperature and amount of gas are held constant. Mathematically, P ∝ 1/V, or P₁V₁ = P₂V₂. Published in 1662, it was the first of the gas laws. If you double the pressure on a gas at constant temperature, the volume halves.

When to use it: the problem says temperature does not change. A common classroom example: a syringe with the outlet sealed. Push the plunger; volume decreases, pressure increases. The product of pressure and volume stays the same. Boyle's law is a special case of the combined gas law where T₁ = T₂, which cancels the temperature terms.

Charles's Law (Constant P)

Charles's law says volume is directly proportional to absolute temperature when pressure and amount are constant. V ∝ T, or V₁/T₁ = V₂/T₂. Gay-Lussac published the relationship in 1802, crediting Charles's unpublished work. If you heat a gas at constant pressure, its volume expands proportionally.

Use Charles's law when the problem holds pressure constant. A balloon in a freezer shrinks; the air inside cools, volume drops. Convert every °C to Kelvin before substituting. Charles's law is another special case of the combined gas law, obtained when P₁ = P₂.

Gay-Lussac's Law (Constant V)

Gay-Lussac's law (also called Amontons's law) states that pressure is directly proportional to absolute temperature at constant volume and constant amount. P ∝ T, or P₁/T₁ = P₂/T₂. A rigid gas cylinder left in the sun: the temperature rises, the pressure inside builds. The container volume does not change.

Apply Gay-Lussac's law when the problem specifies a rigid container. It is the third special case of the combined gas law, for V₁ = V₂. Safety note: a sealed container rated for a maximum pressure will fail if the temperature exceeds the design limit. Always check the pressure rating.

Which Law When

A single comparison table in prose covers when to use each law. For a fixed amount of gas, check what stays constant.

If temperature is constant, use Boyle's law: P₁V₁ = P₂V₂. If pressure is constant, use Charles's law: V₁/T₁ = V₂/T₂. If volume is constant, use Gay-Lussac's law: P₁/T₁ = P₂/T₂. If nothing is constant except the amount of gas, use the combined gas law: P₁V₁/T₁ = P₂V₂/T₂. The combined law works in all four cases; the individual laws are shortcuts when one variable does not change.

A student who memorises only the combined law can solve any before/after problem with constant n. The shortcut is speed, not correctness.

Worked Examples

Constant Volume: Pressure Drops With Temperature

Example 1: A gas at 2.0 atm and 5.0 L is cooled from 300 K to 200 K at constant volume. Find the final pressure. Since volume is constant, P₁/T₁ = P₂/T₂. Substitute: 2.0 atm / 300 K = P₂ / 200 K. P₂ = (2.0 × 200) / 300 = 1.33 atm. The pressure drops to 1.33 atm. Failure check: temperature must be in Kelvin.

Nothing Constant: Use the Combined Law

Example 2: A balloon at 1.0 atm and 298 K has a volume of 2.0 L. It is taken outside where the temperature is 280 K and the pressure is 0.95 atm. Find the new volume. Nothing is constant except n. Use P₁V₁/T₁ = P₂V₂/T₂. (1.0 × 2.0) / 298 = (0.95 × V₂) / 280. V₂ = (1.0 × 2.0 × 280) / (298 × 0.95) = 560 / 283.1 = 1.98 L. The volume contracts slightly because the temperature drop outweighs the pressure drop.

Leaking Container: The Law Fails

Example 3: A student uses the combined gas law but the container is leaking. If gas escapes, n changes. The combined law assumes constant n. The student gets a physically impossible result: a volume smaller than the container. The fix: use PV = nRT for each state and account for the change in moles.

When to Use the Ideal Gas Law Instead

Use PV = nRT when you need a value for n (moles), when the amount of gas changes between two states, or when you need to calculate gas density or molar mass. The ideal gas law requires R, and the choice of R value depends on your units. The most common R in general chemistry is 0.082057 L·atm/(mol·K). For pressure in mmHg, use R = 62.3637 L·mmHg/(mol·K). A common error: using 8.314 J/(mol·K) with volume in litres and pressure in atm. That mismatch produces an answer off by a factor of about 101.3.

The ideal gas law also applies when the problem asks for conditions at STP. Since 1982, IUPAC defines STP as 273.15 K and 1 bar (10⁵ Pa). The molar volume at IUPAC STP is 22.710 L/mol. The older definition (1 atm, 273.15 K) gives a molar volume of 22.414 L/mol. Check which definition your textbook or exam uses. Applying the wrong molar volume introduces a 1.3% error. For water vapor at 273.15 K and 1 atm, the ideal gas law gives a number, but water is below its condensation point; the physical state is liquid or ice, not gas. Do not apply STP shortcuts to gases above their boiling point.

For real gases near their condensation point or above 10 atm, the ideal gas law error can exceed 5%. Use the van der Waals equation or a compressibility factor Z instead. The combined gas law inherits the same limits from its parent equation.