Combined Gas Law

Combines Boyle's and Charles's laws: handles gas samples whose pressure, volume, AND temperature all change at once.

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

P1V1/T1 = P2V2/T2

What it means

The combined gas law is Boyle's law and Charles's law folded into one expression: it says the ratio PV/T is invariant for a fixed amount of gas, no matter which of pressure, volume, or temperature change. If only pressure and volume change it collapses to Boyle's law; if only volume and temperature change it collapses to Charles's law. It is really just the ideal gas law with n and R canceled out, which is why temperature must be in kelvin. When to use it: whenever a gas sample changes between two states and you know five of the six values P1, V1, T1, P2, V2, T2 — the standard 'before and after' gas problem.

Variables

Symbol	Name	Unit	Meaning
$P1$	Initial pressure	atm (or Pa, kPa)	Pressure of the gas in its initial state.
$V1$	Initial volume	L (or m^3)	Volume of the gas in its initial state.
$T1$	Initial temperature	K	Absolute temperature in the initial state — kelvin required.
$P2$	Final pressure	atm (or Pa, kPa)	Pressure of the gas in its final state.
$V2$	Final volume	L (or m^3)	Volume of the gas in its final state.
$T2$	Final temperature	K	Absolute temperature in the final state — kelvin required.

Assumptions

The amount of gas (n) is unchanged between the two states.
Both temperatures must be in kelvin.
The gas behaves ideally in both states.

Worked example

Problem

A 2.50 L gas sample at 1.00 atm and 300 K is compressed to 2.00 atm and heated to 600 K. What is the new volume?

Given:

P1 = 1.00 atm, V1 = 2.50 L, T1 = 300 K
P2 = 2.00 atm, T2 = 600 K, n constant

Write the combined gas law: P1V1/T1 = P2V2/T2.
Solve for V2: V2 = P1V1T2 / (T1P2).
Substitute: V2 = (1.00 atm)(2.50 L)(600 K) / (300 K)(2.00 atm).
Numerator: 1.00 × 2.50 × 600 = 1500. Denominator: 300 × 2.00 = 600.
Divide: 1500 / 600 = 2.50 L.

V2 = 2.50 L

Doubling the pressure and doubling the absolute temperature cancel each other's effect on volume, so the gas ends at the volume it started with.