The combined gas law calculator above handles the case where a fixed quantity of gas moves from one state to another and more than one property changes on the way. Compress air while it warms, let a balloon rise into thinner and colder air, heat a partly flexible container — in each case pressure, volume and temperature are all in motion at once, and the single-variable gas laws cannot describe it.
Arb Digital builds free calculators that state their own boundaries. This is a two-state relation between six quantities, with the amount of gas cancelling out because it never changes. That is a genuinely different tool from the ideal gas law calculator, which describes a single state and includes the number of moles and the gas constant explicitly. Use this page when you are comparing before and after; use that one when you need moles, mass or density at one moment.
What This Combined Gas Law Calculator Does
It solves P₁V₁/T₁ = P₂V₂/T₂ in all six directions. Give it five of the quantities and it returns the sixth, converting pressures to pascals and temperatures to kelvin internally so that mixed unit systems cause no trouble. Volumes are compared directly, since the ratio cancels the unit as long as both use the same one.
Alongside the answer it reports the three ratios separately. That breakdown is what makes a result interpretable: seeing that the pressure rose by a factor of 2.47 while the temperature rose only by a factor of 1.25 tells you immediately why the volume had to fall, and by roughly how much. A single output number hides that reasoning; three ratios expose it.
The fourth grid cell shows the value of PV/T itself, computed from both states. Because the law says that quantity is unchanged, the two figures agree by construction once the unknown has been solved. What the cell is really for is checking your inputs when you have measured all six quantities independently and want to know whether they are consistent.
How to Use It
- Choose the unknown. The hero label names it, so there is no ambiguity about which figure was computed and which were supplied.
- Enter the two pressures as absolute values. Gauge pressure must have atmospheric pressure added to it first — this is the most frequent source of wrong answers here.
- Enter the two volumes in a single unit. Litres, millilitres, cubic metres or cubic feet, but the same one for both.
- Enter the temperatures and pick their scale. Everything is converted to kelvin before the ratio is taken.
- Check the three ratios. They show which change dominated, which is usually the insight the question was after.
The Formula: How the Combined Gas Law Is Calculated
Start from the ideal gas equation pV = nRT, where n is the amount of substance and R is the molar gas constant, given as 8.314 462 618 J mol−1 K−1 and exact in the NIST CODATA value for the molar gas constant. Divide both sides by T and you get pV/T = nR. If the amount of gas does not change between the two states, then nR is the same number in both, so P₁V₁/T₁ must equal P₂V₂/T₂.
That derivation is why n and R are absent from this page. They have not been ignored; they have cancelled. OpenStax University Physics Volume 2, section 2.1 on the molecular model of an ideal gas, presents pV = nRT together with the constituent proportionalities and states that the temperature must be in kelvin and the pressure must be absolute.
Work the defaults. Two litres at 101.325 kPa and 25 °C, taken to 250 kPa and 100 °C. In kelvin the temperatures are 298.15 and 373.15. Rearranging, V₂ = P₁V₁T₂ ÷ (T₁P₂) = 101.325 × 2 × 373.15 ÷ (298.15 × 250) = 1.0145 L. The pressure rose by a factor of 2.467 and the temperature by 1.2516, so the volume fell by 1.2516 ÷ 2.467 = 0.5073 — almost exactly half.
Where It Sits Among the Other Gas Laws
Each of the classical gas laws is this equation with one quantity held fixed. Hold temperature constant and the T terms cancel, leaving P₁V₁ = P₂V₂ — that is Boyle's law, an inverse relationship between pressure and volume. Hold pressure constant and the P terms cancel, leaving V₁/T₁ = V₂/T₂, which is Charles's law. Hold volume constant and you get P₁/T₁ = P₂/T₂, the pressure–temperature law.
So the combined law is not a fourth law to memorise; it is the parent from which the other three are special cases. If a problem really does hold one quantity fixed, the narrower page is quicker and its explanation is more focused. If two or three quantities move, this is the only one of the four that applies.
The boundary against the ideal gas law calculator is different in kind. That equation is about a single state and involves the amount of gas explicitly, so it answers questions such as how many moles are in a cylinder or what the density of a gas is at given conditions. This page never sees n at all, and cannot answer those questions — but it also does not need to know them.
Absolute Pressure, and Why Gauge Readings Break the Law
Nearly every pressure instrument in ordinary use reads gauge pressure: the amount by which the pressure exceeds the surrounding atmosphere. A tyre gauge reading 220 kPa means the air inside is at about 321 kPa absolute, because roughly 101 kPa of atmosphere is already there and the gauge subtracts it.
Feed gauge values into a ratio and the result is wrong, sometimes dramatically. Using 220 and 250 instead of 321 and 351 changes the pressure ratio from 1.136 to 1.094 — a small error here, but the discrepancy grows enormous at low pressures, where a gauge reading near zero implies a ratio near infinity. The rule is simple: add atmospheric pressure to any gauge reading before it enters this calculation, and the pressure converter handles the unit rescaling once you have done so.
Vacuum work is the mirror image. A vacuum gauge reading is a pressure below atmospheric, so it must be subtracted rather than added, and the resulting absolute pressure can be very small. In that regime the ratios become large and sensitive, which is exactly when correct conversion matters most.
When the Amount of Gas Is Not Actually Constant
The whole derivation rests on n cancelling, which requires the same quantity of gas in both states. That assumption fails more often than people notice. A container that leaks between measurements has lost moles. A chemical reaction that produces or consumes gas changes them by design. A closed vessel containing liquid water will have more water vapour in the headspace at higher temperature, because evaporation adds molecules to the gas phase.
That last case is a particularly quiet trap in laboratory work. Heat a sealed flask containing a little water and the measured pressure rises faster than the combined gas law predicts, not because the law is wrong but because the sample is no longer the same sample. If you suspect this, switch to the full ideal gas equation with an explicit mole count, and use the partial pressure calculator to separate the contributions of the components in a mixture.
Real Gases and the Limits of the Ideal Model
The law assumes molecules with no volume of their own and no attraction between them. Real gases satisfy that well at ordinary temperatures and moderate pressures, and progressively less well as they are compressed or cooled toward condensation. Attraction between molecules pulls the real volume below the prediction; the finite size of the molecules pushes it above at high density. The two effects work in opposite directions and cancel at one particular temperature for each gas.
In practice the model is reliable for air at room conditions, for laboratory gas handling at a few atmospheres, and for atmospheric problems such as a balloon rising through the troposphere. It is unreliable for gases near their boiling points, for steam close to saturation, and for anything at tens of atmospheres, where a compressibility factor or a real-gas equation of state is needed instead. Prepare the inputs with the volume converter and the temperature converter, and remember that the model, not the arithmetic, is what limits accuracy.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using gauge pressure — add atmospheric pressure first, because the ratio only means something between absolute values.
- Leaving temperatures in Celsius — the ratio requires an absolute scale, and this error alone accounts for most wrong answers in gas problems.
- Assuming the amount of gas is constant — leaks, reactions and evaporation into a headspace all change the mole count and invalidate the comparison.
- Mixing units between the two states — pressure and volume units cancel only when both states use the same one.
- Reaching for it when one quantity is fixed — Boyle's, Charles's and the pressure–temperature law are the same equation simplified, and are quicker for those cases.
Related Free Tools From Arb Digital
For the fixed-temperature case use the Boyle's law calculator and for the fixed-pressure case the Charles's law calculator. When you need moles, mass or density at a single state, the ideal gas law calculator is the right tool, with the molar mass calculator to convert grams to moles. Mixtures are handled by the partial pressure calculator. Rescale inputs with the pressure converter, the volume converter and the temperature converter. The full free online tools hub lists everything.
Frequently Asked Questions
The statement that PV/T is unchanged for a fixed amount of gas moving between two states, so P₁V₁/T₁ equals P₂V₂/T₂. It covers processes where pressure, volume and temperature all change together.
The ideal gas law describes a single state and includes the amount of gas and the gas constant explicitly. The combined law compares two states of the same sample, so those two terms cancel and never appear.
Because they are. Hold temperature fixed and the T terms cancel to give Boyle's law; hold pressure fixed and the P terms cancel to give Charles's law; hold volume fixed and you get the pressure–temperature relation.
Yes. Most instruments read gauge pressure, which is the excess over atmospheric, so atmospheric pressure must be added before the value is used. Using a gauge reading in a ratio gives an answer that is quietly wrong.
Only that both states use the same one, since the law is a ratio and the unit cancels. Temperature is the exception: it must be converted to an absolute scale, which the tool does for you.
Then the law does not apply, because the derivation assumes the mole count is identical in both states. Leaks, gas-producing reactions and evaporation into a sealed headspace all break that assumption.
Good at ordinary temperatures and moderate pressures, and progressively worse as a gas approaches condensation or very high density, where molecular volume and intermolecular attraction stop being negligible.
This tool is provided for educational and estimating use. It assumes ideal gas behaviour and a fixed amount of gas, and does not model real-gas deviations, phase change, chemical reaction or leakage.
