The combined gas law, P1V1/T1 = P2V2/T2, describes how a fixed amount of gas changes when pressure, volume, and temperature all shift at once. It merges Boyle’s, Charles’s, and Gay-Lussac’s laws. Convert both temperatures to kelvin, plug in five known values, and solve for the sixth.
The Formula and What Each Symbol Means
P1V1/T1 = P2V2/T2 tracks one sample of gas across two states. State 1 is the starting condition; state 2 is the final condition. The subscripts keep them straight:
- P1, V1, T1: pressure, volume, and temperature before the change
- P2, V2, T2: pressure, volume, and temperature after the change
The amount of gas never appears because it is assumed constant: no gas enters or leaves between the two states. Temperatures must be in kelvin, as always. Pressure and volume can use any units, but P1 and P2 must share a unit, and V1 and V2 must share a unit, because the equation only compares ratios.
Three Laws in One
The combined gas law is really three famous laws wearing one coat. Hold any one variable constant and the other two reproduce the classic relationship:
| Held constant | What remains | Classic name |
|---|---|---|
| Temperature (T1 = T2) | P1V1 = P2V2 | Boyle’s law |
| Pressure (P1 = P2) | V1/T1 = V2/T2 | Charles’s law |
| Volume (V1 = V2) | P1/T1 = P2/T2 | Gay-Lussac’s law |
This is why the combined form is so handy. Instead of memorizing which law applies, write P1V1/T1 = P2V2/T2, cross out the variable that does not change, and solve what is left. One equation covers every “before and after” gas problem with a fixed amount of gas.
Worked Example: Tire Pressure on a Hot Day
Problem: A car tire holds 32 psi at 20 C in the morning. After highway driving, the air inside reaches 60 C. The tire volume barely changes. What is the new pressure?
Step 1, convert temperatures: T1 = 20 + 273 = 293 K; T2 = 60 + 273 = 333 K.
Step 2, simplify: volume is constant, so V1 and V2 cancel, leaving Gay-Lussac’s form P1/T1 = P2/T2.
Step 3, isolate and compute: P2 = P1 x T2 / T1 = 32 x 333 / 293 = 32 x 1.1365 = 36.4 psi.
The pressure climbs more than 4 psi from heat alone, which is why tire gauges read high after driving and why mechanics check pressure on cold tires. Any time two of the three variables move together, this is the equation to reach for.
Worked Example: Balloon in the Cold
Problem: A balloon holds 2.0 L of helium at 25 C. It is moved into a 0 C cold room. Pressure stays at 1 atm. What is the new volume?
Step 1, convert temperatures: T1 = 25 + 273 = 298 K; T2 = 0 + 273 = 273 K.
Step 2, simplify: pressure is constant, so P1 and P2 cancel, leaving Charles’s form V1/T1 = V2/T2.
Step 3, isolate and compute: V2 = V1 x T2 / T1 = 2.0 x 273 / 298 = 2.0 x 0.9154 = 1.83 L.
The balloon shrinks by about 8 percent. Nothing leaked; the molecules simply move slower and need less room. For single-state questions, asking what P, V, n, or T is right now rather than how it changed, an ideal gas law calculator using PV = nRT is the faster tool.
The 4-Step Method for Every Problem
- List the six values. Write P1, V1, T1, P2, V2, T2 and fill in the five you know. Mark the unknown with a question mark.
- Convert both temperatures to kelvin. Do this before anything else. A Celsius temperature in a ratio silently destroys the answer.
- Cross out the constant variable. If the container is rigid, volume cancels. If it is open to the air, pressure cancels. If nothing about temperature changes, both T terms cancel.
- Isolate the unknown and check the direction. Cross-multiply, solve, then ask whether the answer makes sense: heating at constant volume should raise pressure, cooling at constant pressure should shrink volume.
Combined Gas Law vs Ideal Gas Law: When to Use Each
- Use the combined gas law when a fixed sample of gas moves from one set of conditions to another and you know five of the six values. Typical phrasing: “a gas at 2.0 atm and 300 K is compressed to…”
- Use the ideal gas law (PV = nRT) when the amount of gas can change or when you need moles. Typical phrasing: “how many grams of gas are in…” or “what is the pressure of 0.5 mol…”
- They agree with each other. The combined law is just PV = nRT with nR canceled on both sides. If you know the moles, either route reaches the same answer.
Frequently Asked Questions
What is the combined gas law formula?
P1V1/T1 = P2V2/T2. It relates the pressure, volume, and temperature of a fixed amount of gas before (state 1) and after (state 2) a change. Temperatures must be in kelvin.
When do I use the combined gas law instead of PV = nRT?
Use the combined law for before-and-after problems with a fixed amount of gas. Use PV = nRT when the amount of gas is unknown, changes, or needs to be found in moles or grams.
Do temperatures have to be in kelvin in the combined gas law?
Yes. The equation compares ratios of temperature, and only the kelvin scale starts at true zero. Convert with K = C + 273.15 before substituting.
What is the difference between the combined gas law and the ideal gas law?
The combined law tracks changes between two states of a fixed gas sample; the ideal gas law describes a single state and includes the amount of gas (n). The combined law is derived from the ideal law by canceling the constant nR.
Can pressure be in psi in the combined gas law?
Yes, any pressure unit works as long as P1 and P2 use the same one, because only their ratio matters. The same goes for volume: liters, milliliters, or cubic feet are all fine if V1 and V2 match.
Why does tire pressure rise on hot days?
The tire volume is nearly fixed, so heating the air follows P1/T1 = P2/T2. A rise from 293 K to 333 K multiplies pressure by 333/293 = 1.14, turning 32 psi into about 36 psi.
