The Combined Gas Law Formula
A complete reference for P₁V₁/T₁ = P₂V₂/T₂. Where it comes from, how to rearrange it, what units it requires, and the mistakes that trip people up most often.
Need to solve a specific problem right now? Use the interactive combined gas law calculator on the homepage, enter any five of the six variables and get the sixth instantly, with full step-by-step working.
Open the Combined Gas Law CalculatorThe Formula
The combined gas law states that for a fixed amount (moles) of an ideal gas:
P₁V₁ / T₁ = P₂V₂ / T₂
Subscript 1 denotes the gas's initial state, its starting pressure, volume, and absolute temperature. Subscript 2 denotes its final state, after any combination of compression, expansion, heating, or cooling. The equation says the ratio PV/T stays constant throughout, no matter how pressure, volume, and temperature individually change, as long as the amount of gas does not.
Where the Formula Comes From
The combined gas law isn't a separately-discovered law of nature. It's an algebraic combination of three laws discovered independently, each describing what happens when exactly one variable is held fixed:
- Boyle's Law (1662), at constant temperature, P₁V₁ = P₂V₂
- Charles' Law (1780s), at constant pressure, V₁/T₁ = V₂/T₂
- Gay-Lussac's Law (1808), at constant volume, P₁/T₁ = P₂/T₂
Each of these three laws is a special case of the combined gas law with one variable frozen. Set T₁ = T₂ in the combined gas law and the temperature terms cancel, leaving Boyle's Law exactly. Set P₁ = P₂ and it reduces to Charles' Law. Set V₁ = V₂ and it reduces to Gay-Lussac's Law. The combined gas law is simply the general statement that works when none of the three variables are held fixed.
The Extended Form with Moles
The standard combined gas law assumes the amount of gas (n, in moles) never changes. If gas is added, removed, produced, or consumed, common in chemical reactions, use the extended form:
P₁V₁ / (n₁T₁) = P₂V₂ / (n₂T₂)
This form is itself just a rearrangement of the ideal gas law, PV = nRT, applied to both states and divided through. Since R is a universal constant, it cancels out of the ratio entirely, which is why neither the standard nor extended combined gas law ever needs you to know the value of R.
Solving for Each Variable
Rearranged for each of the six variables:
| Solve for | Formula |
|---|---|
| P₁ | P₁ = P₂V₂T₁ / (V₁T₂) |
| V₁ | V₁ = P₂V₂T₁ / (P₁T₂) |
| T₁ | T₁ = P₁V₁T₂ / (P₂V₂) |
| P₂, dedicated calculator | P₂ = P₁V₁T₂ / (V₂T₁) |
| V₂, dedicated calculator | V₂ = P₁V₁T₂ / (P₂T₁) |
| T₂, dedicated calculator | T₂ = P₂V₂T₁ / (P₁V₁) |
Unit Requirements
The combined gas law places exactly one hard requirement on units: temperature must be in an absolute scale, Kelvin (K) or Rankine (°R), never Celsius or Fahrenheit directly. This is because the formula relies on direct and inverse proportions with T, and those proportions are only true when zero on the scale actually means zero thermal energy. Convert with K = °C + 273.15 before substituting anything into the formula.
Pressure and volume have no such restriction. You can use any consistent pressure unit (Pa, atm, kPa, mmHg, psi, and so on) and any consistent volume unit (L, m³, mL, ft³, and so on) as long as you use the same unit for P₁ and P₂, and the same unit for V₁ and V₂. The units don't have to match SI, they just have to match each other within the equation. See our pressure unit converter and Kelvin converter if you need to switch a value from one unit system to another before solving.
Common Mistakes
- Forgetting to convert temperature to Kelvin. By far the most common error. Plugging °C directly into the formula gives a completely wrong answer, since the proportion breaks down on a scale with an arbitrary zero.
- Mixing pressure or volume units between states. If P₁ is in atm, P₂ must also be in atm (or converted before substituting). You cannot mix atm and kPa within the same side of the equation.
- Applying the standard formula when moles change. If gas is added, removed, or reacted away, the standard three-variable formula silently gives a wrong answer, you need the extended moles form or the ideal gas law instead.
- Cross-multiplying incorrectly when rearranging. P₁V₁/T₁ = P₂V₂/T₂ rearranges by cross-multiplication. A frequent algebra slip is dividing by the wrong term when isolating a variable. Use the table above, or let the calculator do the rearrangement for you.
Try It Yourself
Rather than rearranging the formula by hand every time, use the combined gas law calculator on the homepage. Enter any five of P₁, V₁, T₁, P₂, V₂, T₂ and it solves for the sixth instantly, with the full substitution shown step by step so you can check your own hand-worked answer against it. For worked examples with numbers already plugged in, see our Combined Gas Law Examples page.