Gay-Lussac's Law Calculator
Find the final pressure or final temperature of a gas at constant volume using P1/T1 = P2/T2 with automatic Kelvin conversions.
Isochoric Transformation (Constant Volume: V₁ = V₂)
Step-by-Step Mathematical Derivation:
1. Formula: P₁ / T₁ = P₂ / T₂ → P₂ = P₁ × (T₂ / T₁)
2. Absolute Temperature Conversion:
• T₁ = 20 °C + 273.15 = 293.15 K
• T₂ = 80 °C + 273.15 = 353.15 K
3. Substitution: P₂ = 2.0 atm × (353.15 K / 293.15 K)
4. Final Output: P₂ = 2.4093 atm
This Gay-Lussac's law calculator finds the final pressure or final temperature of a gas when volume and the amount of gas stay constant. Enter three of the four values (P1, T1, P2, T2), leave the unknown blank, and the tool solves P1/T1 = P2/T2 with full steps. It accepts atm, kPa, mmHg, torr, bar, and psi for pressure and K, °C, and °F for temperature, and it converts every temperature to Kelvin automatically, so you avoid the most common error in gas law problems.
Quick Reference
| Item | Value |
|---|---|
| Formula | P1/T1 = P2/T2 |
| Relationship | Pressure and absolute temperature are directly proportional |
| Held constant | Volume and amount of gas (moles) |
| Solves for | P1, T1, P2, or T2 |
| Temperature scale required | Kelvin (K = °C + 273.15) |
What Is Gay-Lussac's Law?
Gay-Lussac's law is a gas law stating that the pressure of a fixed amount of gas is directly proportional to its absolute temperature when volume stays constant. If you double the Kelvin temperature of a gas in a rigid container, its pressure doubles. If you cool it to half its Kelvin temperature, its pressure halves.
The law is named after the French chemist and physicist Joseph Louis Gay-Lussac, who studied the behavior of gases in the early 1800s. The pressure-temperature relationship was first observed earlier by the French physicist Guillaume Amontons around 1702, so the law is also called Amontons's law. Some textbooks use the name "Gay-Lussac's law" for the volume-temperature relationship, which most sources call Charles's law. Gay-Lussac also formulated a separate law of combining volumes for gases in chemical reactions, so check which law your course means. This page covers the pressure-temperature law.
Gay-Lussac's law is one of the three classic gas laws, along with Boyle's law and Charles's law. Together they form the combined gas law, and all of them follow from the ideal gas law.
Why the Law Works
According to the kinetic molecular theory, temperature measures the average kinetic energy of gas particles. Heating a gas makes its particles move faster, so they strike the container walls harder and more often. When the container is rigid and its volume cannot change, the only result is a rise in pressure. Cooling the gas reverses the effect and lowers the pressure.
Gay-Lussac's Law Formula
The formula is:
You can also write it as P/T = k, where k is a constant for a given gas sample in a container of fixed volume.
Where the Formula Comes From
Start with the ideal gas law:
Rearrange it to isolate P/T:
When the amount of gas (n) and volume (V) are constant, and R is the universal gas constant, nR/V is a constant. That means P/T is the same in the initial and final states:
Variable Table
| Variable | Meaning | Common Units |
|---|---|---|
| P1 | Initial pressure | atm, kPa, mmHg, torr, bar, psi |
| T1 | Initial absolute temperature | K |
| P2 | Final pressure | atm, kPa, mmHg, torr, bar, psi |
| T2 | Final absolute temperature | K |
How to Use This Gay-Lussac's Law Calculator
Select what to solve for
Choose P1, T1, P2, or T2.
Enter the three known values
Fill in the initial pressure, initial temperature, and one final value.
Pick the units
Choose atm, kPa, mmHg, torr, bar, or psi for pressure, and K, °C, or °F for temperature.
Click Calculate
The tool converts temperatures to Kelvin, rearranges P1/T1 = P2/T2, and returns the missing value.
Read the steps
Check the substituted equation to see how the answer was found.
Run a sense check
If temperature went up, pressure must have gone up. If temperature went down, pressure must have gone down.
What Can This Calculator Calculate?
Final pressure (P2)
The pressure of a gas in a rigid container after it is heated or cooled.
Final temperature (T2)
The temperature a gas must reach to produce a target pressure.
Initial pressure (P1)
The starting pressure when you know the final state.
Initial temperature (T1)
The starting temperature when you know the final state.
If pressure and volume change together at constant temperature, use the Boyle's Law Calculator. If volume and temperature change at constant pressure, use the Charles's Law Calculator. If all three change, use the Combined Gas Law Calculator.
Why Temperature Must Be in Kelvin
Gay-Lussac's law only works with absolute temperature. The Kelvin scale, named after Lord Kelvin (William Thomson), starts at absolute zero, where particle motion is at its minimum. Celsius and Fahrenheit have arbitrary zero points, so using them gives wrong ratios. For example, heating a gas from 10 °C to 20 °C looks like a doubling in Celsius, but in Kelvin the change is only from 283.15 K to 293.15 K, about 3.5%. Gas pressure follows the Kelvin change.
Convert to Kelvin with these formulas:
| 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 |
Gay-Lussac's Law Rearranged Formulas
Each variable can be isolated with one step of algebra. Keep all temperatures in Kelvin.
| Solve For | Formula |
|---|---|
| Final pressure | P2 = (P1 × T2) / T1 |
| Final temperature | T2 = (P2 × T1) / P1 |
| Initial pressure | P1 = (P2 × T1) / T2 |
| Initial temperature | T1 = (P1 × T2) / P2 |
Gay-Lussac's Law Examples
Solve for Final Pressure (P2)
Problem: A rigid container holds gas at 2.0 atm and 300 K. The gas is heated to 450 K. Find the final pressure.
The Kelvin temperature increased by 50%, so the pressure increased by 50%.
Solve for Final Temperature (T2)
Problem: A sealed steel tank holds gas at 1.0 atm and 20 °C. At what temperature will the pressure reach 1.5 atm?
Cooling a Gas
Problem: A sealed rigid container holds gas at 101.3 kPa and 25 °C. It is cooled to −20 °C. Find the final pressure.
Pressure falls by about 15% because the Kelvin temperature falls by the same fraction.
Fahrenheit Input with mmHg
Problem: A rigid vessel holds gas at 760 mmHg and 68 °F. It is heated to 212 °F. Find the final pressure.
The mmHg unit carries through because P1 and P2 use the same unit. Only temperature needed conversion.
Solve for Initial Pressure (P1)
Problem: A gas in a fixed-volume container ends at 3.0 atm and 450 K after being heated. It started at 300 K. What was the initial pressure?
Gay-Lussac's Law Units
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 |
Use the same pressure unit for P1 and P2. Use absolute pressure, not gauge pressure. A tire gauge reads pressure above the atmosphere, so add about 14.7 psi (1 atm) to convert it to absolute before applying the law.
Temperature must be in Kelvin (K). Convert from Celsius or Fahrenheit before calculating, or let the calculator do it for you.
Volume is not in the formula, but it must stay constant. This is called an isochoric process (also called isovolumetric or isometric).
Gay-Lussac's Law Graph: The Direct Relationship
Plotting pressure against absolute temperature gives a straight line that passes through the origin. This is what "directly proportional" means. The slope of the line depends on the amount of gas and the volume of the container. A smaller container or more gas gives a steeper line.
| Kelvin Temperature Change | Pressure Change |
|---|---|
| Temperature doubles | Pressure doubles |
| Temperature halves | Pressure halves |
| Temperature increases by 10% | Pressure increases by 10% |
| Temperature drops to one third | Pressure drops to one third |
Extending the line back to zero pressure points to absolute zero, 0 K or −273.15 °C. Real gases condense before reaching that point, but the extrapolation is one reason gas laws use the Kelvin scale.
Real-World Examples of Gay-Lussac's Law
Pressure cookers
The sealed lid keeps volume fixed, so heating the contents raises pressure and lets water reach temperatures above 100 °C.
Aerosol cans
Heat raises the pressure of the propellant inside, which is why cans carry warnings against heat and fire.
Car tires
Tire pressure rises as tires warm up while driving and drops in cold weather, because tire volume changes very little.
Propane tanks and gas cylinders
Pressure increases in hot sun, so tanks include safety relief valves.
Autoclaves
Sealed sterilizers use heated, pressurized steam to reach high temperatures.
Sealed containers in a fire
A closed can or bottle can burst when heat raises the internal pressure past what the walls can hold.
When Can You Use Gay-Lussac's Law?
Use Gay-Lussac's law when all of the following are true:
Constant volume
The process is isochoric, such as gas in a rigid, sealed container.
Constant amount of gas
No gas is added, removed, or lost, and nothing reacts.
A gas sample
The law applies to gases, not liquids or solids.
Absolute temperature
Both temperatures are expressed in Kelvin.
Gay-Lussac's Law vs Other Gas Laws
| Gas Law | Formula | Held Constant | Relationship |
|---|---|---|---|
| Gay-Lussac's law | P1/T1 = P2/T2 | Volume, moles | Pressure and temperature are directly related |
| Boyle's law | P1V1 = P2V2 | Temperature, moles | Pressure and volume are inversely related |
| Charles's law | V1/T1 = V2/T2 | Pressure, moles | Volume and temperature are directly related |
| Avogadro's law | V1/n1 = V2/n2 | Pressure, temperature | Volume and moles are directly related |
| Combined gas law | P1V1/T1 = P2V2/T2 | Moles | All three change together |
| Ideal gas law | PV = nRT | None | Links P, V, T, and moles |
Common Gay-Lussac's Law Mistakes
| Mistake | Why It Fails | Fix |
|---|---|---|
| Using °C or °F | Gas laws need absolute temperature | Convert with K = °C + 273.15 |
| Letting volume change | Gay-Lussac's law needs constant volume | Use the combined gas law instead |
| Using gauge pressure | Ignores atmospheric pressure | Add atmospheric pressure to get absolute |
| Swapping P1 and P2 | Puts the initial pressure in the final slot | Label all four values before solving |
| Swapping T1 and T2 | Inverts the temperature ratio | Pair each pressure with its temperature |
| Unit mismatch for pressure | atm and kPa in the same equation break the ratio | Convert to one unit first |
| Treating the relationship as inverse | Pressure and temperature move in the same direction | Remember: hotter means higher pressure |
| Confusing it with Charles's law | Charles's law links volume and temperature | Check which variable is held constant |
| Changing the amount of gas | Moles must stay constant | Use the ideal gas law |
| Rounding too early | Small errors grow through the calculation | Keep extra digits and round at the end |
Gay-Lussac's Law FAQs
Frequently asked questions about isochoric gas behavior, pressure-temperature relationships, pressure cookers, and units.
What is Gay-Lussac's law?
Gay-Lussac's law states that the pressure of a fixed amount of gas is directly proportional to its absolute temperature at constant volume. The formula is P1/T1 = P2/T2.
What is the formula for Gay-Lussac's law?
The formula is P1/T1 = P2/T2, where P1 and T1 are the initial pressure and temperature, and P2 and T2 are the final pressure and temperature in Kelvin.
How do you use a Gay-Lussac's law calculator?
Enter any three of the four values (P1, T1, P2, T2), choose 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 for Gay-Lussac's law?
Kelvin is an absolute scale that starts at absolute zero. Gas pressure is proportional to absolute temperature, so Celsius and Fahrenheit give incorrect results.
What does directly proportional mean in Gay-Lussac's law?
It means that when one quantity increases, the other increases by the same factor. Doubling the Kelvin temperature doubles the pressure.
Who discovered Gay-Lussac's law?
Guillaume Amontons observed the pressure-temperature relationship around 1702. Joseph Louis Gay-Lussac studied it further in the early 1800s, and the law now carries his name. It is also called Amontons's law.
What is an isochoric process?
An isochoric process is one in which volume stays constant. Gay-Lussac's law describes gas behavior during an isochoric change.
What is the difference between Gay-Lussac's law and Charles's law?
Gay-Lussac's law links pressure and temperature at constant volume. Charles's law links volume and temperature at constant pressure. Some textbooks mix up the names, so check which variable is held constant.
What units can I use for Gay-Lussac's law?
Any pressure unit works if it matches between the initial and final states. Common choices are atm, kPa, mmHg, torr, bar, and psi. Temperature must be in Kelvin, though the calculator accepts Celsius and Fahrenheit and converts them.
Why does pressure increase when temperature increases in a sealed container?
Heating makes gas particles move faster, so they hit the container walls harder and more often. With volume fixed, this raises the pressure.
Is Gay-Lussac's law accurate for real gases?
It is accurate for real gases at low to moderate pressure and temperatures well above their boiling points. Near condensation or at very high pressure, real gases deviate from ideal behavior.
How is Gay-Lussac's law related to the combined gas law?
Gay-Lussac's law is the special case of the combined gas law where volume is constant. Setting V1 = V2 in P1V1/T1 = P2V2/T2 gives P1/T1 = P2/T2.