Physics

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

Solve the combined gas law for any unknown pressure, volume, or temperature. Also handles Boyle's Law (constant T), Charles's Law (constant P), and Gay-Lussac's Law (constant V) as special cases. Supports all major pressure and volume units.

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Gas Law

P₁V₁/T₁ = P₂V₂/T₂

Quick Presets

Solve For

Initial State (1)

P₁ pressure

V₁ volume

T₁ temperature

Final State (2)

P₂ pressure

V₂ volume

T₂ temperature

Enter calculate · Esc reset

What Is the Combined Gas Law Calculator | P₁V₁/T₁ = P₂V₂/T₂?

The combined gas law unifies three classic relationships between the pressure, volume, and temperature of a fixed amount of ideal gas. Enter any five of the six variables and this calculator instantly solves for the sixth, with full unit conversion across all common pressure, volume, and temperature units.

›Four law modes, switch between the full combined law and each of the three special cases (Boyle, Charles, Gay-Lussac) with a single click.
›Solve for any variable, in the combined law you can solve for any of P₁, V₁, T₁, P₂, V₂, or T₂. The field you're solving for is automatically greyed out.
›Full unit support, pressure in atm, Pa, kPa, bar, mmHg, or psi; volume in L, mL, m³, or cm³; temperature in K, °C, or °F. All conversions happen internally in SI (Pa, m³, K).
›Condition comparison table, for the combined law, see how V changes if T doubles, if P halves, or if both change at once.
›STP and SATP reference, standard conditions are shown alongside every result so you can compare your gas state to the textbook benchmarks.
›Step-by-step working, the exact substitution and algebra for your inputs is shown in a monospaced panel for easy verification.

Formula

Combined Gas Law

P₁V₁ / T₁ = P₂V₂ / T₂

Special Cases

Boyle's Law (T constant): P₁V₁ = P₂V₂

Charles's Law (P constant): V₁/T₁ = V₂/T₂

Gay-Lussac's Law (V constant): P₁/T₁ = P₂/T₂

Solve for any unknown (Combined)

P₂ = P₁V₁T₂ / (T₁V₂)

V₂ = P₁V₁T₂ / (T₁P₂)

T₂ = P₂V₂T₁ / (P₁V₁)

Symbol	Name	Description
P₁	Initial pressure	Pressure at the initial state, in any unit; auto-converted to Pa internally
V₁	Initial volume	Volume at the initial state, in L, mL, m³, or cm³
T₁	Initial temperature	Temperature at initial state, must be > 0 K; converted from °C or °F automatically
P₂	Final pressure	Pressure at the final state
V₂	Final volume	Volume at the final state
T₂	Final temperature	Temperature at final state, must be > 0 K

How to Use

1
Select the gas law: Choose Combined Gas Law to work with all six variables, or one of the three special cases if a variable is held constant.
2
Select solve for: Click the variable you want to find (P₁, V₁, T₁, P₂, V₂, or T₂). That field becomes greyed out, leave it blank.
3
Enter known values: Type values for the other five variables. Use the unit dropdown next to each field to select your preferred unit.
4
Try a preset: Load a pre-built example, Boyle's demo, a balloon cooling, a scuba tank release, or Gay-Lussac's doubling, to see a worked case.
5
Press Enter or Calculate: The unknown appears in large type. Its value is also converted to every other unit for that quantity.
6
Change the display unit: Click the unit selector inside the result box to instantly switch the displayed unit without recalculating.
7
Review the steps: Expand the step-by-step panel to see the formula, SI conversions, substitution, and final answer in one clean block.

Example Calculation

Scuba tank: P₁ = 200 atm, V₁ = 12 L, T₁ = 293 K → P₂ = 1 atm, T₂ = 310 K. Find V₂.

Given: P₁=200 atm, V₁=12 L, T₁=293 K, P₂=1 atm, T₂=310 K

Step 1: Convert all to SI

P₁ = 200 × 101 325 = 20 265 000 Pa

V₁ = 12 × 0.001 = 0.012 m³

T₁ = 293 K

P₂ = 1 × 101 325 = 101 325 Pa

T₂ = 310 K

Step 2: Apply formula

P₁V₁/T₁ = P₂V₂/T₂ → V₂ = P₁V₁T₂ / (T₁P₂)

Step 3: Substitute

V₂ = (20 265 000 × 0.012 × 310) / (293 × 101 325)

= 75 381 600 / 29 688 225

= 2.5390... m³

V₂ ≈ 2.539 m³ = 2 539 L

What this means physically

A 12-litre tank at 200 atm expands to 2 539 litres when released to 1 atm, accounting for the slight temperature rise from 293 K to 310 K. This is why a single scuba tank can supply hundreds of breaths, the gas is compressed to a tiny fraction of its "breathing volume".

Understanding Combined Gas Law | P₁V₁/T₁ = P₂V₂/T₂

The Three Individual Gas Laws

The combined gas law is built from three relationships discovered independently in the 17th–19th centuries, each holding one variable constant:

Law	Formula	Constant	Statement
Boyle's (1662)	P₁V₁ = P₂V₂	Temperature	Pressure and volume are inversely proportional at constant T
Charles's (1787)	V₁/T₁ = V₂/T₂	Pressure	Volume is directly proportional to temperature at constant P
Gay-Lussac's (1808)	P₁/T₁ = P₂/T₂	Volume	Pressure is directly proportional to temperature at constant V

Each law is a special case of the combined gas law obtained by cancelling the constant variable from both sides. If T is constant, T₁ = T₂ and the combined law reduces to P₁V₁ = P₂V₂, Boyle's Law.

The Combined Gas Law

The combined gas law merges all three into a single equation that handles the general case where pressure, volume, and temperature all change simultaneously:

P₁V₁ / T₁ = P₂V₂ / T₂

Equivalently: P₁V₁T₂ = P₂V₂T₁

Both sides of the equation represent the same quantity, the product PV/T, evaluated at two different states of the same fixed sample of gas. This ratio remains constant as long as the amount of gas (moles) does not change.

›To solve for any variable: rearrange by cross-multiplication. For V₂: V₂ = P₁V₁T₂ / (T₁P₂).
›The combined law applies to a closed system (no gas added or removed).
›It assumes ideal gas behaviour, accurate for most gases at moderate pressures and temperatures far from condensation.
›For absolute accuracy with real gases (high P or near boiling), use the van der Waals equation or compressibility charts.

Important: Temperature Must Be in Kelvin

The gas laws require absolute temperature, temperature measured in Kelvin (K), not Celsius or Fahrenheit. This is because the gas law equations are proportionalities: at 0°C, a gas still has kinetic energy and exerts pressure; only at 0 K (absolute zero) would it theoretically stop.

Temperature conversion reminders

K = °C + 273.15

K = (°F − 32) × 5/9 + 273.15

0°C = 273.15 K

25°C = 298.15 K (SATP)

100°C = 373.15 K (boiling water at sea level)

Using Celsius or Fahrenheit directly in the formula gives completely wrong answers. This calculator automatically converts °C and °F to Kelvin before computing.

STP and SATP Reference Conditions

Two sets of "standard conditions" appear throughout chemistry and physics:

Condition	Temperature	Pressure	Molar volume (ideal gas)
STP (IUPAC 1982–2014)	0°C (273.15 K)	1 atm (101 325 Pa)	22.414 L/mol
SATP (current IUPAC)	25°C (298.15 K)	1 bar (100 000 Pa)	24.789 L/mol
NTP (US engineering)	20°C (293.15 K)	1 atm (101 325 Pa)	24.055 L/mol

Always confirm which standard conditions your textbook or problem uses. Many chemistry problems quote "STP" but use the older 1 atm definition; IUPAC now uses 1 bar for SATP.

Real-World Applications of Gas Laws

›Scuba diving. Tank compressed to 200 atm; at 1 atm body pressure the gas expands ~200-fold. The combined law predicts volume at any depth-pressure combination.
›Weather balloons. As altitude increases, pressure drops and the balloon expands (Boyle). Temperature also drops, slightly compressing it (Charles). The combined law predicts the final volume at cruising altitude.
›Automotive tyres. After a long drive, tyres heat up (Gay-Lussac). Pressure rises, this is why manufacturers specify cold-tyre pressure and why over-inflating on a cold day can lead to proper pressure when hot.
›Breathing physiology. The diaphragm increases lung volume, which by Boyle's Law drops pressure below atmospheric, drawing in air. Exhaling compresses the lungs, raising pressure above atmospheric.
›Spray cans. Aerosol propellants follow Boyle: as gas expands out of the can, the remaining gas occupies more volume at lower pressure, the spray weakens as the can empties.
›Refrigeration. Refrigerant gas is compressed (pressure rises, temperature rises), then cooled and expanded through a valve. The combined law governs each phase of the cycle.

Frequently Asked Questions

What is the combined gas law and when do I use it?

The combined gas law handles the most general case. Choose a simpler law when one variable is truly constant:

›Constant T → Boyle's Law (isothermal process)
›Constant P → Charles's Law (isobaric process)
›Constant V → Gay-Lussac's Law (isochoric/isovolumetric process)
›All three change → Combined Gas Law

Why must temperature be in Kelvin?

Converting to Kelvin is mandatory. This calculator does it automatically from °C or °F. Always verify that T is in Kelvin before substituting into any gas law formula.