Combined Gas Law Calculator — P₁V₁/T₁ = P₂V₂/T₂

Background.

This combined gas law calculator solves the two-state equation P₁V₁/T₁ = P₂V₂/T₂ for whichever one of the six quantities you do not know. It is the calculator you want when a fixed amount of gas moves from one set of conditions to another: a balloon rising through the atmosphere, a syringe being compressed, an aerosol can warming in a parked car, a diving cylinder cooling overnight, a sample being transferred from a warm bench to a cold room.

It is deliberately a different tool from the ideal gas law calculator on this site. That one solves the single-state equation PV = nRT and needs to know how much gas you have. This one relates two states of the same gas sample, and the amount of substance cancels between them — which is why you never have to enter moles, grams or the gas constant here. If the amount of gas changes between your two states, because gas was added, vented, leaked, condensed or consumed by a reaction, this equation does not apply and you need PV = nRT for each state separately.

The page also replaces three separate calculators, because Boyle's law, Charles's law and Gay-Lussac's law are not three equations — they are one equation with one variable held fixed. Hold the temperature constant and P₁V₁/T₁ = P₂V₂/T₂ collapses to P₁V₁ = P₂V₂, which Robert Boyle published in 1662. Hold the pressure constant and it collapses to V₁/T₁ = V₂/T₂, which Jacques Charles found around 1787 and Joseph Louis Gay-Lussac published in 1802. Hold the volume constant and it collapses to P₁/T₁ = P₂/T₂, Gay-Lussac's 1809 result and the reason gas cylinders carry temperature warnings. Enter equal values for whichever quantity is fixed and the calculator will tell you, in the output panel, which named law your numbers actually correspond to.

Three conventions govern every number on this page and getting any of them wrong will silently ruin the answer. **Pressure is absolute, not gauge.** A tyre gauge reading 220 kPa on a sea-level forecourt corresponds to 321.325 kPa absolute, and it is the absolute value that appears in the equation. **Temperature is absolute, in kelvin.** K = °C + 273.15. Entering 20 instead of 293.15 does not produce a slightly wrong answer, it produces a meaningless one, because the proportionality is measured from absolute zero rather than from the freezing point of water. **Volume and pressure units only have to be self-consistent.** Because only the ratios P₁/P₂ and V₁/V₂ enter the equation, you can work in atmospheres and cubic feet as long as both states use the same units; the calculator labels its fields kPa and litres because those are the SI-coherent choices, and 1 kPa × 1 L = 1 J exactly.

A fourth thing to fix before you start is which "standard conditions" you mean, because the governing bodies disagree. IUPAC has defined standard conditions for gases as 273.15 K and 10⁵ Pa since 1982, having previously used 273.15 K and 101 325 Pa; the two give the familiar molar volumes 22.711 L/mol and 22.414 L/mol respectively. The US Environmental Protection Agency, in 40 CFR §60.2, defines standard conditions for emissions testing as 293 K and 101.3 kPa — a twenty-degree difference that changes a reported gas volume by about seven percent. SATP, used in thermochemistry, is 298.15 K and 10⁵ Pa. Nothing on this page assumes any of them; you supply both states explicitly. But if you are converting a measurement to "standard" conditions, check which standard your data sheet, exam paper or regulator means before you type a temperature.

Finally, the model. This is the ideal-gas approximation: point-like molecules with no volume of their own and no forces between them except elastic collisions. For dry air, nitrogen, oxygen, helium and the noble gases at or below a few atmospheres and well above their boiling points, it is accurate to better than about 0.1 percent. It degrades progressively at high pressure, where the molecules' own volume stops being negligible, and near condensation, where intermolecular attraction becomes comparable to the thermal energy — a scuba cylinder at 200 bar, carbon dioxide near room temperature, or steam near its saturation line all need a real-gas equation of state instead. On significant figures: the solver carries every intermediate step at 30-digit precision and rounds once, at the end, to ten decimal places. That is far more precision than your inputs justify. Round the displayed answer yourself to the number of significant figures in your least precise measurement.

What is combined gas law calculator (boyle, charles & gay-lussac)?

The combined gas law states that for a fixed amount of an ideal gas, the quantity PV/T has the same value in any two states: P₁V₁/T₁ = P₂V₂/T₂. P is absolute pressure, V is volume, and T is absolute (thermodynamic) temperature in kelvin. Subscript 1 labels the initial state and subscript 2 the final state; the law says nothing about the path between them, only that the two endpoints share a value of PV/T. That shared value is not arbitrary — it equals nR, the amount of substance multiplied by the molar gas constant R = 8.314462618 J mol⁻¹ K⁻¹ (CODATA 2022, exact). Because n and R are both fixed, they cancel when you write the equation for two states of the same sample, which is the whole reason the combined law is useful: it answers real questions without requiring you to know how much gas is in the container. The law is the union of three named results, each of which is the combined law with one variable frozen. Boyle's law (1662) is the isothermal case T₂ = T₁, giving P₁V₁ = P₂V₂: squeeze a gas at constant temperature and its pressure rises in inverse proportion. Charles's law (Charles c. 1787, published by Gay-Lussac 1802) is the isobaric case P₂ = P₁, giving V₁/T₁ = V₂/T₂: heat a gas at constant pressure and it expands in direct proportion to absolute temperature. Gay-Lussac's law (1809) is the isochoric case V₂ = V₁, giving P₁/T₁ = P₂/T₂: heat a gas in a rigid sealed container and its pressure rises in direct proportion to absolute temperature. Avogadro's law, V ∝ n at fixed P and T, is the fourth member of the family, but it changes the amount of gas and therefore sits outside the two-state combined law; it is what you need PV = nRT for. Two failure modes account for almost every wrong answer. The first is gauge pressure: pressure gauges read the excess over ambient, so a reading of zero means about 101.325 kPa absolute at sea level, and a gauge value entered directly into this equation understates the true pressure. The second is Celsius: because the law is a proportionality to absolute temperature, a Celsius value is not merely offset, it is the wrong quantity, and it can even be negative — which the calculator rejects outright.

How to use this calculator.

1. 01Choose the one quantity you do not know from the 'Solve for' menu. Everything else must be supplied.
2. 02Enter the initial state: P₁ in kilopascals (absolute), V₁ in litres, T₁ in kelvin. Convert first — 1 atm = 101.325 kPa, 1 bar = 100 kPa, 1 psi = 6.894757 kPa, 1 mmHg = 0.1333224 kPa; K = °C + 273.15 and K = (°F + 459.67) × 5/9.
3. 03Enter the final state the same way. Leave the field you are solving for at any value — it is ignored in that mode.
4. 04To model Boyle's law, set T₂ equal to T₁. To model Charles's law, set P₂ equal to P₁. To model Gay-Lussac's law, set V₂ equal to V₁. The 'Which gas law this is' output will confirm which one you have entered.
5. 05Convert any gauge pressure to absolute before entering it: absolute = gauge + local atmospheric pressure (about 101.325 kPa at sea level, roughly 90 kPa at 1000 m, about 84 kPa in Nairobi).
6. 06Read the answer, then cross-check it against the PV/T output: recompute P₂V₂/T₂ by hand and it must reproduce that same number.
7. 07Round the answer yourself. The calculator shows ten decimal places of internal precision; report only as many significant figures as your least precise input carries.
8. 08If gas was added, vented, leaked or produced between the two states, stop — the combined law does not apply. Use the ideal gas law calculator on each state separately instead.

The formula.

Start from the ideal gas equation of state PV = nRT and write it for two states of the same sealed sample. Because n (the amount of substance) and R (the molar gas constant, 8.314462618 J mol⁻¹ K⁻¹, exact under CODATA 2022) are identical in both states, P₁V₁/T₁ = nR = P₂V₂/T₂. That is the combined gas law, and the calculator reports nR back to you as the 'PV/T' output — in kPa·L/K, which equals J/K because 1 kPa × 1 L = 1 J exactly.

Rearranged for each unknown:

V₂ = P₁V₁T₂ / (T₁P₂) P₂ = P₁V₁T₂ / (T₁V₂) T₂ = T₁P₂V₂ / (P₁V₁) V₁ = P₂V₂T₁ / (T₂P₁) P₁ = P₂V₂T₁ / (T₂V₁) T₁ = T₂P₁V₁ / (P₂V₂)

Rounding stage: FINAL ONLY. Every multiplication and division above is carried out in Decimal.js at 30-digit working precision and a single rounding to ten decimal places is applied at the return boundary. No intermediate quantity is rounded, so the six rearrangements are exact inverses of one another to within that final rounding — solve for V₂, feed the answer back in and solve for P₂, and you recover the pressure you started with.

Directional behaviour, read off the equations above rather than from memory. In the worked example below, dropping the pressure from 101.325 kPa to 25.0 kPa while the temperature falls from 293.15 K to 223.15 K takes 5.00 L to 15.4260097220 L. The volume grew, even though the gas got colder, because the pressure factor P₁/P₂ = 4.053 is far larger than the temperature factor T₂/T₁ = 0.7612; the product 4.053 × 0.7612 = 3.0852 is the net expansion. Cooling alone would have shrunk the balloon to 3.81 L. Pressure and temperature pull in opposite directions here and pressure wins — which is why weather balloons are launched slack and burst high in the stratosphere rather than being crushed by the cold.

The special cases are not separate formulas; they are these same expressions with one ratio equal to 1. Set T₂ = T₁ and the temperature factor drops out, leaving V₂ = V₁ × P₁/P₂ — Boyle. Set P₂ = P₁ and the pressure factor drops out, leaving V₂ = V₁ × T₂/T₁ — Charles. Set V₂ = V₁ in the pressure rearrangement and it leaves P₂ = P₁ × T₂/T₁ — Gay-Lussac. The calculator detects which case you have entered by comparing each pair to a relative tolerance of one part in 10⁹, and reports it in the 'Which gas law this is' output.

Invalid domain: all six quantities must be strictly greater than zero. A zero or negative absolute pressure or absolute temperature is not physical, and a zero in any of P₁, V₁, T₁, P₂, V₂, T₂ produces a division by zero in at least one rearrangement. Rather than returning an infinity or a NaN, the calculator refuses the input and names the offending field. A negative temperature is the usual symptom of a Celsius value being entered where kelvin was required.

A worked example.

Frequently asked questions.

References& sources.

1. [1]International Union of Pure and Applied Chemistry. Compendium of Chemical Terminology (the 'Gold Book'), 5th edition, online version 5.0.0 (2025), entry 'standard conditions for gases' (S05910): 'Temperature, 273.15 K (0 °C) and pressure of 10⁵ pascals. IUPAC recommends that the former use of the pressure of 1 atm as standard pressure (equivalent to 1.01325 × 10⁵ Pa) should be discontinued.' Source document: Pure Appl. Chem. 1990, 62, 2167 (Glossary of atmospheric chemistry terms), p. 2216. Independent standards body; open access; retrieved 2026-07-29.
2. [2]International Union of Pure and Applied Chemistry. Compendium of Chemical Terminology (the 'Gold Book'), 5th edition, online version 5.0.0 (2025), entry 'standard pressure' (S05921): 'Chosen value of pressure denoted by p°. In 1982 IUPAC recommended the value 10⁵ Pa, but prior to 1982 the value 101 325 Pa (= 1 atm) was usually used.' Source documents: Green Book, 2nd ed., p. 54; Pure Appl. Chem. 1994, 66, 533, p. 536. This is the citation behind the 22.414 L/mol versus 22.711 L/mol discrepancy discussed in the FAQ. Open access; retrieved 2026-07-29.
3. [3]Tiesinga, E., Mohr, P. J., Newell, D. B. & Taylor, B. N. CODATA Recommended Values of the Fundamental Physical Constants: 2022. NIST Standard Reference Database 121, 'molar gas constant' — R = 8.314 462 618… J mol⁻¹ K⁻¹, listed with standard uncertainty '(exact)' following the 2019 SI redefinition. R is used on this page only to interpret the PV/T output as nR; it does not enter the combined-gas-law arithmetic. Independent national metrology institute; open access; retrieved 2026-07-29.
4. [4]US Environmental Protection Agency. 40 CFR §60.2, Definitions (Standards of Performance for New Stationary Sources, subpart A): 'Standard conditions means a temperature of 293 K (68 °F) and a pressure of 101.3 kilopascals (29.92 in Hg).' Cited as the SECOND, independent authority on reference conditions — a US federal regulator whose standard temperature is 20 K above IUPAC's, which is why this page never assumes a standard state. Current as of the eCFR text retrieved 2026-07-29.
5. [5]International Union of Pure and Applied Chemistry. Quantities, Units and Symbols in Physical Chemistry (the 'Green Book'), 3rd edition, 2nd printing 2012, §2.10 'Chemical thermodynamics', p. 48 — symbols and coherent SI units for pressure p (Pa), volume V (m³), thermodynamic temperature T (K), and the definition of partial pressure pB = yB p. Establishes the unit and symbol conventions used throughout this page. Open access PDF; retrieved 2026-07-29.
6. [6]National Institute of Standards and Technology. NIST Chemistry WebBook, SRD 69 — Thermophysical Properties of Fluid Systems. Reference-quality equations of state (Span-Wagner for CO₂, IAPWS-95 for water, Lemmon et al. for air and nitrogen) that supersede the ideal-gas approximation at high pressure and near saturation. Consult it when you need to know how far the combined gas law is from the truth for a specific fluid at specific conditions. Independent national metrology institute; open access; retrieved 2026-07-29.
7. [7]Boyle, R. A Defence of the Doctrine Touching the Spring and Weight of the Air, against the Objections of Franciscus Linus (London, 1662). The appendix 'A Table of the Condensation of the Air' is the first published statement of the inverse pressure–volume relation now called Boyle's law — the isothermal special case of the equation on this page. PRINT / BIBLIOGRAPHIC REFERENCE ONLY: no stable open URL is given here because the 1662 edition's digitised copies live behind institution-specific handles that change; consult a rare-books catalogue or Early English Books Online.
8. [8]Gay-Lussac, J. L. 'Recherches sur la dilatation des gaz et des vapeurs.' Annales de Chimie 1802, 43, 137–175. The paper that published Jacques Charles's earlier unpublished constant-pressure measurements and established that all gases expand by the same fraction per degree — the isobaric special case. Gay-Lussac's own constant-volume result (pressure proportional to absolute temperature at fixed volume) followed in 1809. PRINT / BIBLIOGRAPHIC REFERENCE ONLY to a 19th-century journal; scanned volumes exist in several national-library collections under unstable identifiers.