The Gay-Lussac's law calculator above handles the branch of the gas laws that most people meet first in real life and last in a textbook: what happens to the pressure inside a sealed rigid container when you change its temperature. A tyre on a hot motorway, a gas cylinder in the sun, an autoclave coming up to temperature, a sealed vessel being steam-cleaned — all of them are this one relation.
Arb Digital builds free tools that pick one job and finish it properly. The law itself is a single ratio, but two things around it cause almost every wrong answer: the temperatures have to be absolute, and the pressures have to be absolute too. This page converts both for you and shows the kelvin values it actually used, so you can see whether the arithmetic did what you expected.
What This Gay-Lussac's Law Calculator Does
It solves P₁ ÷ T₁ = P₂ ÷ T₂ for whichever of the four quantities you leave unknown. Pressure can be entered in pascals, kilopascals, bar, atmospheres, psi or millimetres of mercury, and temperature in Celsius, kelvin or Fahrenheit. Internally everything becomes pascals and kelvin before the ratio is taken.
The result grid shows the temperature ratio, the percentage change in pressure, and both temperatures converted to kelvin. That last pair is not padding. Seeing that 20 °C and 100 °C are 293.15 K and 373.15 K makes it immediately obvious why a fivefold rise in Celsius produces only a 27 per cent rise in pressure, which is the single most common source of confusion with this law.
The conditions attached to the law are strict. The volume must be constant, which means a genuinely rigid container. The amount of gas must be constant, which means no leaks and no chemistry. And the gas must be far enough from condensing that ideal behaviour is a reasonable approximation.
How to Use It
- Choose which value you want. The selector decides which box is treated as the output; the other three are read as inputs.
- Convert gauge pressure to absolute before entering it. Add atmospheric pressure — about 101.3 kPa or 14.7 psi at sea level — to any reading taken from an ordinary pressure gauge.
- Enter temperatures in whatever scale you have. The conversion to kelvin is done for you and shown in the grid so you can check it.
- Confirm the container is genuinely rigid. A steel cylinder qualifies. A tyre, a balloon or a plastic bottle only approximately does, and the softer the container the less this law applies.
- Read the percentage change, not just the final figure. It tells you at a glance whether the effect is small enough to ignore or large enough to matter for a pressure rating.
The Formula: How Gay-Lussac's Law Is Calculated
At constant volume and constant amount of gas, pressure is directly proportional to absolute temperature: P ÷ T is a constant, so P₁ ÷ T₁ = P₂ ÷ T₂. Rearranged, P₂ = P₁ × (T₂ ÷ T₁) and T₂ = T₁ × (P₂ ÷ P₁). OpenStax Chemistry 2e, section 9.2, states that the pressure of a given amount of gas is directly proportional to its temperature on the kelvin scale when the volume is held constant, and notes that Amontons established the relationship around 1700 before Gay-Lussac refined it around 1800. Both names attach to it, which is why some textbooks call this Amontons's law.
The law is a special case of the ideal gas equation PV = nRT. Hold V and n constant and the equation says P is proportional to T with a constant of nR ÷ V. The gas constant R itself is a defined value; the NIST CODATA value of the molar gas constant is exactly 8.314462618 J per mole per kelvin under the 2019 SI redefinition.
Kelvin conversion is K = °C + 273.15, or K = (°F + 459.67) × 5 ÷ 9. The absolute scale is not a stylistic preference. The law says pressure is proportional to temperature, and proportionality requires a scale whose zero means zero — zero thermal energy, zero pressure. Celsius zero is the freezing point of water and has no physical significance for a gas.
Work the defaults. Heating a sealed rigid vessel from 20 °C to 100 °C means 293.15 K to 373.15 K, a ratio of 1.2729. Starting from 101.325 kPa absolute, the final pressure is 101.325 × 1.2729 = 128.98 kPa, a rise of 27.29 per cent. Had you used Celsius directly the ratio would have come out as 5, giving a badly wrong 506.6 kPa.
Gauge Pressure Is the Trap That Catches Everyone
Absolute temperature gets all the attention, but absolute pressure causes just as many errors and is mentioned far less often. Ordinary pressure gauges — on tyres, compressors, boilers, cylinders — read zero when open to the atmosphere. They are measuring the difference from atmospheric, not the true pressure of the gas.
Gay-Lussac's law needs the true pressure. Feed it gauge values and the proportionality breaks, because a gauge reading of zero does not correspond to a temperature of absolute zero. The correction is simply to add atmospheric pressure, about 101.3 kPa, 1.013 bar or 14.7 psi at sea level, before entering the figure, and to subtract it again if you want the answer as a gauge reading.
Work a tyre example to see the size of the error. A tyre at 32 psi gauge and 15 °C that heats to 50 °C: correctly, that is 46.7 psi absolute at 288.15 K rising to 323.15 K, giving 52.4 psi absolute, which is 37.7 psi gauge — a rise of 5.7 psi. Done wrongly with gauge pressures, you get 32 × (323.15 ÷ 288.15) = 35.9 psi, a rise of only 3.9 psi. The wrong method understates the increase by a third.
Which Gas Law Holds What Constant
The three two-variable gas laws are constantly confused because they look alike, and the only thing distinguishing them is which variable is pinned. Gay-Lussac's law holds volume constant and relates pressure to temperature: P₁/T₁ = P₂/T₂. Charles's law holds pressure constant and relates volume to temperature: V₁/T₁ = V₂/T₂. Boyle's law holds temperature constant and relates pressure to volume: P₁V₁ = P₂V₂.
The quickest way to pick the right one is to ask what the container does. A rigid sealed vessel cannot change volume, so volume is the constant and Gay-Lussac applies. A piston free to move against the atmosphere holds pressure constant, so Charles applies. A syringe being compressed slowly enough to stay at room temperature holds temperature constant, so Boyle applies.
When two or more change at once, none of the pair laws work and you need the combined form, P₁V₁/T₁ = P₂V₂/T₂, or the full ideal gas equation if the amount of gas changes too. The ideal gas law calculator covers PV = nRT directly and is the right page whenever more than two quantities are in play, or whenever you need the number of moles.
Where the Ideal Assumption Runs Out
Gay-Lussac's law is exact for an ideal gas and approximate for a real one. The approximation is excellent for air, nitrogen, oxygen and similar gases at ordinary temperatures and pressures up to several atmospheres, typically within a per cent or so.
It degrades in two directions. At high pressure the molecules occupy a meaningful fraction of the container volume and attract each other, so real gases deviate; above roughly ten atmospheres the error becomes worth accounting for and a compressibility factor is normally used. Near the condensation point the law fails completely, because some of the substance is no longer a gas. That is what makes a propane cylinder behave nothing like this calculation: it contains a liquid in equilibrium with its vapour, and the pressure follows the vapour pressure curve, which is exponential in temperature rather than linear.
The other silent assumption is that no gas enters or leaves. A vessel with a relief valve is not a constant-amount system once the valve lifts. Nor is a container whose seal leaks slightly as pressure rises, which is common enough that a measured pressure lower than the calculated one is usually a leak rather than a failure of the physics. The pressure converter and the temperature converter handle unit changes if you need to move a result into different units afterwards.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using Celsius or Fahrenheit in the ratio — the law needs an absolute scale. Doubling a Celsius figure does not double the absolute temperature, and 0 °C in the denominator produces an infinite answer.
- Entering gauge pressure — add atmospheric pressure first, about 101.3 kPa or 14.7 psi at sea level, or the proportionality is being applied to the wrong quantity.
- Applying it to a flexible container — a balloon or a soft bottle changes volume as pressure rises, which is Charles's law territory, not this one.
- Using it for a liquefied gas cylinder — propane and butane cylinders hold liquid in equilibrium with vapour, and their pressure follows a vapour pressure curve, not a linear law.
- Forgetting the gas can escape — once a relief valve lifts or a seal weeps, the amount of gas is no longer constant and the calculation no longer describes the system.
Related Free Tools From Arb Digital
When more than two quantities change, or when you need moles rather than a ratio, the ideal gas law calculator solves PV = nRT directly and is the general case this page is a slice of. For mixtures, the partial pressure calculator splits a total pressure between components. Unit handling is covered by the pressure converter for pascals, bar, psi and atmospheres, and the temperature converter for Celsius, Fahrenheit and kelvin. If the heating itself is the question rather than the resulting pressure, the specific heat calculator gives the energy required to make the temperature change happen, and the vapor pressure calculator covers the liquid-and-vapour case where this law does not apply. Everything Arb Digital publishes is listed in the free online tools hub.
Frequently Asked Questions
At constant volume and constant amount of gas, pressure is directly proportional to absolute temperature, so P1 divided by T1 equals P2 divided by T2. It is also called Amontons's law, because Amontons established the relationship around 1700 before Gay-Lussac refined it.
Volume, along with the amount of gas. Charles's law holds pressure constant and relates volume to temperature instead, while Boyle's law holds temperature constant and relates pressure to volume.
Because the law states a proportionality, which requires a scale whose zero means zero. Celsius zero is the freezing point of water and has no physical meaning for a gas, so using it produces ratios that are simply wrong.
Absolute. A gauge reads the difference from atmospheric pressure, so add about 101.3 kPa or 14.7 psi at sea level before entering the figure, and subtract it again if you want the answer as a gauge reading.
Roughly 1 psi for every 5 degrees Celsius, working in absolute pressure. A tyre at 32 psi gauge and 15 degrees Celsius reaches about 37.7 psi gauge at 50 degrees, an increase of 5.7 psi rather than the 3.9 psi you get from using gauge pressures wrongly.
No. A propane cylinder holds liquid in equilibrium with its vapour, so the pressure follows the vapour pressure curve of the liquid, which rises far more steeply with temperature than this linear relationship.
Very good for air, nitrogen and similar gases at ordinary temperatures and up to a few atmospheres, typically within a per cent. Above roughly ten atmospheres, or near the condensation point, real gas deviations become significant.
This tool is provided for educational use. Pressurised systems carry real hazards and vessel ratings depend on factors this calculation does not include, so nothing on this page is engineering or safety guidance.
