Combined Gas Law Calculator
Solve for final pressure, volume, or temperature across changing gas conditions with automatic Kelvin conversions and step-by-step mathematical substitution.
Select your unknown variable and configure the initial and final states of the gas.
P₂ = (P₁ × V₁ × T₂) / (V₂ × T₁) Substitute your values to see the full mathematical expansion. This combined gas law calculator solves for the final pressure, final volume, or final temperature of a gas when its conditions change. Enter the initial state (P1, V1, T1) and any two values of the final state, and the tool finds the missing one using the formula P1V1/T1 = P2V2/T2. It shows the substituted equation and every step, so you can check your homework, verify a lab result, or learn the method.
The calculator accepts common pressure units (atm, kPa, mmHg, torr, bar, psi), volume units (L, mL, m³), and temperature units (Kelvin, Celsius, Fahrenheit). Temperatures are converted to Kelvin automatically, which removes the most common source of error in gas law problems.
Quick Reference
| Item | Value |
|---|---|
| Formula | P1V1/T1 = P2V2/T2 |
| Temperature scale required | Kelvin (K = °C + 273.15) |
| Amount of gas | Must stay constant |
| Works for | Ideal gases and real gases at moderate pressure |
| Special cases | Boyle's law, Charles's law, Gay-Lussac's law |
What Is the Combined Gas Law?
Definition
The combined gas law is a gas law that relates the pressure, volume, and absolute temperature of a fixed amount of gas in one equation. It states that the quantity (pressure × volume) ÷ temperature stays constant for a given gas sample. When any of the three properties changes, the other two adjust so that this ratio remains the same.
The law brings together three earlier discoveries: Boyle's law (1662), Charles's law (published by Joseph Louis Gay-Lussac in 1802 from work by Jacques Charles), and Gay-Lussac's pressure-temperature law. In 1834, Émile Clapeyron unified these relationships into a single expression, which later became the basis of the ideal gas law.
What It Calculates
The combined gas law predicts the new state of a gas after a change in conditions. You can use it to find:
- The final pressure of a gas after it is compressed, expanded, heated, or cooled.
- The final volume of a gas sample after pressure and temperature both change.
- The final temperature needed to reach a target pressure and volume.
- An initial value, if you know the final state and want to work backward.
This is useful when more than one variable changes at once, which is the situation where Boyle's, Charles's, and Gay-Lussac's laws alone are not enough.
Constant Amount of Gas
The combined gas law only applies when the number of moles (n) does not change. The gas sample must stay in a closed system. No gas can leak out, be added, or react chemically. Under this condition, the term nR in the ideal gas law (PV = nRT) is constant, so PV/T is constant too.
Why the Law Works
According to the kinetic molecular theory, gas pressure comes from particles colliding with the walls of a container, and temperature reflects the average kinetic energy of those particles. Compressing a gas packs particles into a smaller space, so collisions happen more often and pressure rises. Heating a gas makes particles move faster, so they hit the walls harder and more often. The combined gas law captures both effects in one relationship.
Combined Gas Law Formula
The formula is:
Where the Formula Comes From
Start with the ideal gas law:
Divide both sides by T:
For a fixed amount of gas, n is constant, and the gas constant R is always constant. So nR is a constant, which means PV/T has the same value in the initial and final states:
Variable Table
| Variable | Meaning | Common Units | Notes |
|---|---|---|---|
| P1 | Initial pressure | atm, kPa, mmHg, torr, bar, psi | Use absolute pressure, not gauge |
| V1 | Initial volume | L, mL, m³ | Same unit as V2 |
| T1 | Initial temperature | K | Must be absolute temperature |
| P2 | Final pressure | atm, kPa, mmHg, torr, bar, psi | Same unit as P1 |
| V2 | Final volume | L, mL, m³ | Same unit as V1 |
| T2 | Final temperature | K | Must be absolute temperature |
How to Use This Calculator
Follow these steps to solve any combined gas law problem:
- 1 Select the unknown. Choose whether to solve for P2, V2, T2, or an initial value.
- 2 Enter the initial state. Type in P1, V1, and T1 exactly as given in your problem.
- 3 Enter the known final values. Fill in the two final values you know. Leave the unknown field blank.
- 4 Choose units for each field. Pick atm, kPa, mmHg, or another pressure unit. Pick L, mL, or m³ for volume. Pick K, °C, or °F for temperature.
- 5 Click Calculate. The tool converts temperatures to Kelvin, rearranges the formula, and computes the answer.
- 6 Review the steps. Read the substituted equation to see how the result was found.
- 7 Run a sense check. Ask whether the answer matches the physical situation (see the tips below).
- If pressure goes up and temperature stays the same, volume must go down.
- If temperature goes up and pressure stays the same, volume must go up.
- If temperature goes up and volume stays the same, pressure must go up.
How this calculator works: The tool converts every temperature to Kelvin, converts pressure and volume inputs to a consistent internal unit, applies the rearranged formula, and converts the result back to your selected unit. No rounding happens until the final display.
What Can This Calculator Calculate?
Pressure
Find the final pressure (P2) after a gas changes in volume or temperature. Typical problems involve a sealed container that is heated, or a gas that is compressed by a piston. For a simpler interface focused on pressure only, use the Gas Pressure Calculator.
Volume
Find the final volume (V2) of a gas after pressure and temperature change. Typical problems include balloons rising into cooler, low-pressure air, or gas samples collected in a lab. For a tool focused on volume only, use the Gas Volume Calculator.
Temperature
Find the final temperature (T2) needed to reach a target pressure and volume. Typical problems ask what temperature a gas must reach after it is compressed or expanded. For a tool focused on temperature only, use the Gas Temperature Calculator.
Each dedicated page includes extra examples and practice problems for that variable.
Why Temperature Must Be in Kelvin
Short Explanation
Gas laws require absolute temperature because pressure and volume are proportional to the actual thermal energy of the gas particles. The Kelvin scale, named after Lord Kelvin (William Thomson), begins at absolute zero (0 K), the lowest possible temperature, where particle motion is at its minimum. Celsius and Fahrenheit have arbitrary zero points. Using them in the formula would produce wrong answers, division by zero, or negative results.
At 0 °C, a gas does not have zero thermal energy, so dividing by 0 or comparing ratios in Celsius makes no physical sense. Kelvin fixes this because ratios like T2/T1 reflect real changes in energy.
Celsius and Fahrenheit to Kelvin
| Celsius | Fahrenheit | Kelvin |
|---|---|---|
| 0 °C | 32 °F | 273.15 K |
| 20 °C | 68 °F | 293.15 K |
| 25 °C | 77 °F | 298.15 K |
| 37 °C | 98.6 °F | 310.15 K |
| 100 °C | 212 °F | 373.15 K |
Many textbooks round 273.15 to 273. Both are acceptable, but use the value your instructor or textbook requires. For quick conversions, use the Temperature Calculator.
Combined Gas Law Example
Here are four complete worked examples, one for each unknown, plus one with mixed units.
Example 1: Solve for Final Volume (V2)
Solve V₂Problem: A gas has a volume of 10.0 L at 2.0 atm and 27 °C. The pressure rises to 4.0 atm and the temperature rises to 127 °C. Find the new volume.
The volume falls because the pressure doubled, while the temperature increased by only one third.
Example 2: Solve for Final Pressure (P2)
Solve P₂Problem: A 2.50 L sealed container holds gas at 101.3 kPa and 20 °C. The gas is heated to 80 °C and expands to 4.00 L. Find the final pressure.
Pressure drops because the volume increased by 60%, which outweighs the temperature rise.
Example 3: Solve for Final Temperature (T2)
Solve T₂Problem: A 500 mL gas sample at 1.00 atm and 25 °C is compressed to 250 mL at 2.50 atm. Find the final temperature.
Note that mL cancels out because both volumes use the same unit.
Example 4: Mixed Units (mmHg, mL, °F)
Mixed UnitsProblem: A gas occupies 750 mL at 760 mmHg and 68 °F. Find the volume at 600 mmHg and 86 °F.
Pressure units (mmHg) and volume units (mL) cancel in the ratio, so no conversion was needed for them. Only temperature needed conversion.
Combined Gas Law Rearranged Formulas
Each variable can be isolated with basic algebra. Keep all temperatures in Kelvin.
Solve for Pressure
Solve for Volume
Solve for Temperature
Combined Gas Law Units
Short Explanation
The combined gas law is a ratio equation, so any unit works as long as it stays consistent between the initial and final states. The only mandatory unit is Kelvin for temperature. If P1 is in kPa, P2 must also be in kPa. If V1 is in mL, V2 comes out in mL.
Pressure
| Unit | Equivalent |
|---|---|
| 1 atm | 101.325 kPa |
| 1 atm | 760 mmHg |
| 1 atm | 760 torr |
| 1 atm | 1.01325 bar |
| 1 atm | 14.696 psi |
| 1 bar | 100 kPa |
Use absolute pressure. Gauge pressure (such as a tire gauge reading) excludes atmospheric pressure, so add about 14.7 psi (or 1 atm) to convert gauge to absolute.
Volume
| Unit | Equivalent |
|---|---|
| 1 L | 1000 mL |
| 1 L | 1 dm³ |
| 1 mL | 1 cm³ |
| 1 m³ | 1000 L |
Temperature
Temperature must be in Kelvin (K). Convert from Celsius or Fahrenheit before calculating, or let the calculator do it for you.
Combined Gas Law vs Other Gas Laws
The combined gas law contains three simpler laws. Each one keeps one variable constant.
| Gas Law | Formula | Held Constant | Relationship |
|---|---|---|---|
| Boyle's law | P1V1 = P2V2 | Temperature and moles | Pressure and volume are inversely related |
| Charles's law | V1/T1 = V2/T2 | Pressure and moles | Volume and temperature are directly related |
| Gay-Lussac's law | P1/T1 = P2/T2 | Volume and moles | Pressure and temperature are directly related |
| Avogadro's law | V1/n1 = V2/n2 | Pressure and temperature | Volume and moles are directly related |
Boyle's Law
Robert Boyle showed that when temperature is constant, squeezing a gas into a smaller volume raises its pressure. A syringe with the tip sealed is a classic example. Use the Boyle's Law Calculator for pressure-volume problems.
Charles's Law
Jacques Charles observed that at constant pressure, a gas expands as it is heated. A hot air balloon rises because the heated air expands and becomes less dense. Use the Charles's Law Calculator for volume-temperature problems.
Gay-Lussac's Law
Joseph Louis Gay-Lussac's pressure law states that at constant volume, pressure rises with temperature. This is why sealed aerosol cans carry warnings against heat. Use the Gay-Lussac's Law Calculator for pressure-temperature problems.
How to choose: If only two of the three variables change, use the matching single law. If pressure, volume, and temperature all change, use the combined gas law. If the amount of gas also changes, use the Ideal Gas Law Calculator.
When Can You Use the Combined Gas Law?
Use the combined gas law when all of the following are true.
The number of moles stays the same from start to finish. Nothing leaks, escapes, dissolves, or reacts.
You are following a single gas sample through a change. The law does not compare two different gases or two separate containers.
The problem describes a before state (P1, V1, T1) and an after state (P2, V2, T2). If the problem describes only one state, use the ideal gas law.
Both temperatures must be expressed in Kelvin.
Real-World Applications
- • Weather balloons: Volume increases as the balloon rises into lower pressure, partly offset by colder air.
- • Scuba diving: Gas volume in a cylinder or lung changes with depth (pressure) and water temperature.
- • Engines: Gas in a cylinder is compressed and heated during each stroke.
- • Aerosol cans and sealed containers: Pressure changes with temperature and remaining volume.
- • Chemistry labs: Gas volumes collected at room conditions are corrected to standard conditions.
Limits of the Law
The law is most accurate for ideal gases. Real gases deviate at very high pressure or very low temperature, where particle size and intermolecular attractions matter. In those cases, the van der Waals equation or a compressibility factor (Z) gives better results.
Common Combined Gas Law Mistakes
| Mistake | Why It Fails | Fix |
|---|---|---|
| Using °C or °F | Gas laws need absolute temperature | Convert with K = °C + 273.15 |
| Unit mismatch | P1 in atm and P2 in kPa breaks the ratio | Convert to one unit first |
| Swapping P1 and P2 | Puts the initial state in the final slot | Label all six variables before solving |
| Swapping V1 and V2 | Inverts the volume ratio | Pair each P, V, and T with its state |
| Using gauge pressure | Ignores atmospheric pressure | Add atmospheric pressure to get absolute |
| Changing the amount of gas | Moles must be constant | Use the ideal gas law instead |
| Rounding too early | Small errors grow through the calculation | Keep extra digits and round at the end |
| Wrong rearrangement | Algebra slip when isolating a variable | Cross-multiply first, then divide |
| Skipping the sense check | Wrong answers go unnoticed | Compare direction of change with physics |
Combined Gas Law vs Ideal Gas Law
| Feature | Combined Gas Law | Ideal Gas Law |
|---|---|---|
| Formula | P1V1/T1 = P2V2/T2 | PV = nRT |
| Number of states | Two (initial and final) | One |
| Moles (n) | Must stay constant | Can be solved for |
| Gas constant R needed | No | Yes |
| Best for | A fixed sample that changes conditions | Finding moles, mass, or one set of conditions |
Gas constant values for the ideal gas law:
Use the combined gas law when the same gas moves from one condition to another. Use the Ideal Gas Law Calculator when you need the amount of gas or only have one state.
FAQs
Frequently asked questions about the Combined Gas Law, calculations, units, and thermodynamic principles.
What is the combined gas law?
It is a gas law that relates the pressure, volume, and temperature of a fixed amount of gas through the equation P1V1/T1 = P2V2/T2. It combines Boyle's law, Charles's law, and Gay-Lussac's law.
What is the combined gas law formula?
The formula is P1V1/T1 = P2V2/T2. P is pressure, V is volume, T is absolute temperature in Kelvin, and the subscripts 1 and 2 mark the initial and final states.
How do you use the combined gas law calculator?
Enter P1, V1, and T1, then enter two of the three final values (P2, V2, T2). Select your units, leave the unknown blank, and click Calculate. The tool converts temperature to Kelvin and returns the missing value with steps.
Why must temperature be in Kelvin?
Kelvin is an absolute scale that starts at absolute zero. Gas pressure and volume are proportional to absolute temperature, so Celsius and Fahrenheit give incorrect results.
Can I use any units for pressure and volume?
Yes. Any pressure unit (atm, kPa, mmHg, torr, bar, psi) and any volume unit (L, mL, m³) works if you use the same unit for the initial and final values. Temperature must be in Kelvin.
How do I find the final volume with the combined gas law?
Use V2 = (P1 × V1 × T2) / (P2 × T1). Convert temperatures to Kelvin, substitute the values, and solve.
What is the difference between the combined gas law and the ideal gas law?
The combined gas law compares two states of one gas sample and does not need R. The ideal gas law (PV = nRT) describes one state and lets you solve for moles.
Does the combined gas law work if the amount of gas changes?
No. The number of moles must remain constant. If gas is added or removed, use the ideal gas law.
What happens if one variable stays constant?
The combined gas law reduces to a simpler law. Constant temperature gives Boyle's law, constant pressure gives Charles's law, and constant volume gives Gay-Lussac's law.
Is the combined gas law accurate for real gases?
It is accurate for real gases at low to moderate pressure and moderate to high temperature. Near condensation or at very high pressure, real gases deviate from ideal behavior.
Can the combined gas law be used with STP conditions?
Yes. STP is often used as the final or initial state. IUPAC defines STP as 273.15 K and 100 kPa. Many textbooks still use 273.15 K and 1 atm (101.325 kPa), so check which definition your course uses.
Do I need the gas constant for the combined gas law?
No. The constant nR cancels out because it has the same value in both states.