Gay-Lussac's Law Calculator
Calculate how gas pressure changes with temperature in a sealed, rigid container using Gay-Lussac's law. Enter the initial pressure and temperature plus one final value to find the other.
Find Pressure or Temperature at Constant Volume
Gay-Lussac's Law Formula
Where:
- \(P_1\) — initial pressure of the gas
- \(T_1\) — initial absolute temperature (in kelvin)
- \(P_2\) — final pressure of the gas
- \(T_2\) — final absolute temperature (in kelvin)
The formula expresses that at constant volume, the pressure of a fixed amount of gas is directly proportional to its absolute temperature: the ratio \(P/T\) stays the same before and after any change.
Gay-Lussac's Law – Calculation Example
Example: a rigid steel tank holds a gas at 2 atm and 300 K. The tank is heated until the gas reaches 450 K. What is the final pressure?
\[ P_2 = P_1 \cdot \frac{T_2}{T_1} = 2\ \text{atm} \times \frac{450\ \text{K}}{300\ \text{K}} = 3\ \text{atm} \]
Result: the final pressure is 3 atm. The check gives \(P_1/T_1 = 0.006667\) atm/K and \(P_2/T_2 = 0.006667\) atm/K — the constant ratio confirms the calculation.
How to Use the Calculator
- Enter the Initial Pressure (P₁) and pick its unit — atm, kPa, mmHg, bar or psi.
- Enter the Initial Temperature (T₁) and its unit. Values in °C or °F are converted to kelvin automatically, because the law requires absolute temperature.
- In the Solve for dropdown, choose whether the unknown is the final pressure or the final temperature.
- Fill in the one final value you know: the Final Temperature (T₂) when solving for pressure, or the Final Pressure (P₂) when solving for temperature. Select the units you want the result displayed in.
- Read the results: the unknown final value, the constant \(P/T\) ratio, and a detailed breakdown showing every conversion and substitution.
What Is Gay-Lussac's Law?
Gay-Lussac's law states that the pressure exerted by a fixed mass of gas held at constant volume is directly proportional to its absolute temperature. Doubling the kelvin temperature of a gas in a sealed, rigid container doubles its pressure; halving the temperature halves the pressure. The law was formulated by the French chemist Joseph Louis Gay-Lussac in the early 1800s, based on experimental measurements of how gases behave when heated in closed vessels.
The physical reason is straightforward. In a rigid container the volume and the number of molecules never change, so the only way temperature can affect the pressure is through molecular motion. Heating the gas makes its molecules move faster, so they collide with the walls more often and more forcefully — which raises the pressure in exact proportion to the temperature in kelvin.
The key to applying the law correctly is absolute temperature. Celsius or Fahrenheit scales are arbitrary: 20 °C is not "twice as hot" as 10 °C. Kelvin starts at absolute zero, so 300 K genuinely is twice the thermal energy of 150 K. If you enter a temperature in °C or °F, this calculator converts it to kelvin before applying the formula, then converts the answer back to your chosen unit.
The result tells you what pressure (or temperature) the gas reaches after the change. If the calculated final pressure exceeds the rating of the actual container, the vessel may rupture — a practical concern for aerosol cans, gas cylinders and sealed pressure vessels, and the reason "do not incinerate" warnings exist.
Assumptions, Limitations and Tips
- Constant volume and amount of gas: the law applies only to a sealed, rigid container. If gas escapes or the container expands, use a more general relation such as the Van der Waals equation instead.
- Ideal gas behaviour: the law is exact for an ideal gas. Real gases deviate at very high pressures or near their condensation point, where intermolecular forces and molecular volume matter. For most everyday conditions (moderate pressures, temperatures well above boiling) the deviation is small.
- No phase change: the gas must remain a gas. If the temperature drops enough for the substance to condense or freeze, the proportionality breaks down.
- Use absolute temperature: never plug Celsius values into the formula by hand — a negative or zero °C value makes the ratio meaningless. This calculator handles the conversion, but keep it in mind on paper.
- Mixed units are fine for pressure: since the formula uses a ratio of pressures, \(P_1\) and \(P_2\) may be in different units as long as the calculator converts them (which it does internally), but temperatures must always be absolute.
Gay-Lussac's law is one of several gas-law relationships that each hold one variable fixed. If you're studying gas behaviour, you may also find the Graham's law calculator for effusion rates and the Van 't Hoff equation calculator for temperature effects on equilibrium useful companions.
Frequently Asked Questions
How do I calculate Gay-Lussac's law by hand?
Convert both temperatures to kelvin (add 273.15 to °C, or use \(K = (°F - 32) \times 5/9 + 273.15\)), then rearrange \(P_1/T_1 = P_2/T_2\). For the final pressure use \(P_2 = P_1 \times T_2/T_1\); for the final temperature use \(T_2 = T_1 \times P_2/P_1\). Keep pressures in the same unit on both sides — any pressure unit works as long as it is consistent.
Why must temperature be in kelvin for Gay-Lussac's law?
The law relates pressure to absolute thermal energy. Celsius and Fahrenheit have arbitrary zero points: 20 °C is not twice as hot as 10 °C, so their ratios are meaningless in the formula. Kelvin starts at absolute zero, where molecular motion stops, so 300 K genuinely has twice the thermal energy of 150 K. If you plug in Celsius values, the answer will be wrong — often wildly so.
What happens if I use °C in the formula anyway?
You get a wrong answer, and sometimes an impossible one. Example: gas at 2 atm is heated from 25 °C to 50 °C. The correct result is \(2 \times 323.15/298.15 \approx 2.17\) atm. Using raw Celsius gives \(2 \times 50/25 = 4\) atm — nearly double the true value. Worse, a temperature below 0 °C would produce a negative pressure. Always convert to kelvin first.
What are the units of the constant P/T ratio?
The ratio \(P/T\) carries units of pressure per kelvin, such as atm/K or kPa/K. Its numerical value depends on the pressure unit you choose — 0.006667 atm/K equals about 0.6757 kPa/K — but for one fixed gas sample in a rigid container it never changes as the gas is heated or cooled. Comparing ratios is only meaningful when both use the same pressure unit.
Does Gay-Lussac's law work for aerosol cans, tires and gas cylinders?
As a first approximation, yes — these are roughly sealed, rigid volumes. A tire at 35 psi (absolute) heated from 293 K to 323 K rises to about 38.6 psi. Real containers stretch slightly and gas can leak, so measured values differ a little. Safety caution: heating sealed vessels can raise pressure far above their rating; never incinerate aerosol cans or expose gas cylinders to high temperatures.
How is Gay-Lussac's law different from Charles's and Boyle's laws?
All three hold one variable of a fixed gas sample constant. Boyle's law fixes temperature and relates pressure to volume (\(PV = \) constant); Charles's law fixes pressure and relates volume to temperature (\(V/T = \) constant); Gay-Lussac's law fixes volume and relates pressure to temperature (\(P/T = \) constant). Together they combine into the combined gas law and, with the amount of gas, into the ideal gas law \(PV = nRT\).
Can the pressure or temperature reach zero or go negative?
Pressure falls toward zero only as temperature approaches absolute zero (0 K = −273.15 °C), because \(P\) is proportional to \(T\). No gas can be colder than that, so the law never yields a negative result when applied correctly. In practice the gas condenses or freezes long before 0 K, so the proportionality — and the gas itself — stops existing. The calculator rejects any input below absolute zero.