Combined Gas Law Calculator — Solve p₁V₁/T₁ = p₂V₂/T₂ with Thermodynamic Processes

Combined Gas Law Calculator

This Combined Gas Law calculator lets you analyze ideal gas behavior across five thermodynamic processes — combined, isochoric, isobaric, isothermal, and adiabatic. Enter the known parameters and instantly solve for the missing pressure, volume, or temperature, plus get full thermodynamic results: work (W), heat (Q), and change in internal energy (ΔU).

What Is the Combined Gas Law?

The Combined Gas Law merges Boyle's Law, Charles's Law, and Gay-Lussac's Law into a single equation that relates pressure, volume, and temperature for a fixed amount of ideal gas:

p₁V₁ / T₁ = p₂V₂ / T₂

where:

• p₁, p₂ — initial and final absolute pressure
• V₁, V₂ — initial and final volume
• T₁, T₂ — initial and final absolute temperature (in Kelvin)

Solving for any one of the six variables is straightforward when the other five are known. The combined gas law is valid for ideal gases undergoing processes where the amount of substance (n) stays constant.

The Five Thermodynamic Processes

1. Combined Gas Law (General)

No constraint on pressure, volume, or temperature. Solve for any of the six state variables using p₁V₁/T₁ = p₂V₂/T₂. Thermodynamic quantities (W, Q, ΔU) require additional information and are not calculated in the general mode.

2. Isochoric Process (Constant Volume)

Volume is held constant: V₁ = V₂. The simplified law is Gay-Lussac's Law: p₁/T₁ = p₂/T₂.

• p₂ = p₁ × T₂ / T₁
• W = 0 — no work done (volume doesn't change)
• ΔU = Cᵥ · n · ΔT — all heat goes into internal energy
• Q = ΔU

3. Isobaric Process (Constant Pressure)

Pressure is held constant: p₁ = p₂. The simplified law is Charles's Law: V₁/T₁ = V₂/T₂.

• V₂ = V₁ × T₂ / T₁
• W = p · (V₂ − V₁) — work done by expansion or compression
• ΔU = Cᵥ · n · ΔT
• Q = Cₚ · n · ΔT

4. Isothermal Process (Constant Temperature)

Temperature is held constant: T₁ = T₂. The simplified law is Boyle's Law: p₁V₁ = p₂V₂.

• p₂ = p₁V₁ / V₂
• ΔU = 0 — internal energy of an ideal gas depends only on temperature
• W = n · R · T · ln(V₂/V₁)
• Q = W

5. Adiabatic Process (No Heat Exchange)

No heat is exchanged with surroundings: Q = 0. Uses the adiabatic relations with heat capacity ratio γ.

• T₂ = T₁ · (V₁/V₂)^γ−1
• p₂ = p₁ · (V₁/V₂)^γ
• W = (p₁V₁ − p₂V₂) / (γ − 1)
• ΔU = −W
• Q = 0

Gas Types and Heat Capacity Ratios

The thermodynamic calculations depend on the type of gas through the molar heat capacities Cᵥ (at constant volume) and Cₚ (at constant pressure), and the ratio γ = Cₚ/Cᵥ:

R = 8.3144598 J/(mol·K) is the universal gas constant.

First Law of Thermodynamics

The thermodynamic results are governed by the First Law of Thermodynamics:

ΔU = Q − W

where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system. This calculator computes W, Q, and ΔU in joules (J), kilojoules (kJ), and BTU.

Supported Units

Pressure Units

• kPa — Kilopascal (metric, default)
• Pa — Pascal (SI base unit)
• MPa — Megapascal
• atm — Standard atmosphere
• bar — Bar
• mmHg — Millimetre of mercury (Torr)
• psi — Pound per square inch (US customary)
• inHg — Inch of mercury (US customary)

Volume Units

• L — Liter (metric, default)
• mL — Milliliter
• m³ — Cubic meter (SI base unit)
• cm³ — Cubic centimeter
• ft³ — Cubic foot (US customary)
• in³ — Cubic inch (US customary)
• gal — US gallon

Temperature Units

• K — Kelvin (absolute, metric, default)
• °C — Celsius (metric)
• °F — Fahrenheit (US customary)
• °R — Rankine (absolute, US customary)

Energy Output

Thermodynamic quantities (W, Q, ΔU) are always displayed in J, kJ, and BTU simultaneously, covering both metric and US customary usage without requiring a separate unit selector.

Combined Gas Law Examples

Example 1: Solving for Final Pressure (Combined Law)

A gas starts at p₁ = 100 kPa, V₁ = 2 L, T₁ = 300 K. It is compressed to V₂ = 1 L and heated to T₂ = 450 K. Find p₂.

p₂ = p₁V₁T₂ / (T₁V₂) = 100 × 2 × 450 / (300 × 1) = 300 kPa

Example 2: Isochoric Process (Heating at Constant Volume)

A rigid container holds 1 mol of diatomic gas at p₁ = 100 kPa, V = 24.8 L, T₁ = 300 K. It is heated to T₂ = 600 K.

p₂ = 100 × 600/300 = 200 kPa, W = 0, ΔU = Q = Cᵥ · 1 · 300 ≈ 6236 J

Example 3: Isobaric Process (Expansion at Constant Pressure)

1 mol of diatomic gas expands at constant pressure p = 101.325 kPa from T₁ = 300 K (V₁ ≈ 24.79 L) to T₂ = 600 K.

V₂ = V₁ × T₂/T₁ ≈ 49.58 L, W = p·ΔV ≈ 2494 J, Q ≈ 8730 J, ΔU ≈ 6236 J

Example 4: Adiabatic Compression (Tire Pump)

Air (diatomic, γ = 1.4) at p₁ = 101.325 kPa, T₁ = 293 K, V₁ = 1 L is compressed adiabatically to V₂ = 0.1 L.

p₂ = 101.325 × 10^1.4 ≈ 2541 kPa, T₂ = 293 × 10^0.4 ≈ 736 K

This explains why a bicycle pump gets hot during rapid inflation — adiabatic compression raises the air temperature significantly.

Real-World Applications

• Internal combustion engines: The power stroke approximates adiabatic expansion; the compression stroke approximates adiabatic compression.
• Refrigeration cycles: Refrigerants undergo isobaric condensation and evaporation, and adiabatic compression/expansion.
• Weather and atmosphere: Rising air parcels cool adiabatically, causing clouds to form at the lifting condensation level.
• Pressure cookers: Steam is heated at near-constant volume (isochoric) until pressure builds, then at constant pressure.
• Gas cylinders: Compressed gas stored in rigid tanks follows isochoric behavior as temperature changes.
• Scuba diving: The change in pressure as a diver descends affects gas volume in tanks and buoyancy compensators (Boyle's Law / isothermal approximation).
• Balloon physics: A balloon expanding at constant atmospheric pressure follows an isobaric process.

Frequently Asked Questions

What is the Combined Gas Law?

The Combined Gas Law combines Boyle's, Charles's, and Gay-Lussac's laws into one equation: p₁V₁/T₁ = p₂V₂/T₂. It describes how pressure, volume, and temperature of a fixed amount of ideal gas are related when the gas changes state.

What is the difference between the Combined Gas Law and the Ideal Gas Law?

The Ideal Gas Law (pV = nRT) gives an absolute relationship between pressure, volume, temperature, and amount. The Combined Gas Law eliminates the constant R and n, comparing two states of the same gas. The Ideal Gas Law includes Avogadro's Law, making it more general.

Why does temperature need to be in Kelvin?

Gas law calculations require absolute temperature because gas pressure and volume are proportional to the average kinetic energy of molecules, which equals zero at absolute zero (0 K). Using °C or °F directly would give incorrect results due to their arbitrary zero points.

What is an adiabatic process?

An adiabatic process occurs when no heat is exchanged between the gas and its surroundings (Q = 0). In practice, this happens in rapid compressions or expansions where there is insufficient time for heat transfer. Work done on or by the gas entirely changes the internal energy: ΔU = −W.

What does γ (gamma) represent?

γ (gamma) is the heat capacity ratio Cₚ/Cᵥ. It determines how sharply pressure and temperature change during adiabatic processes. Monoatomic gases (like helium) have γ = 5/3 ≈ 1.667; diatomic gases (like air, nitrogen) have γ = 7/5 = 1.4; polyatomic gases have γ = 4/3 ≈ 1.333.

What units can I use for pressure and volume?

This calculator supports all common metric and US customary (imperial) units: pressure in Pa, kPa, MPa, atm, bar, mmHg, psi, and inHg; volume in m³, L, mL, cm³, ft³, in³, and US gallons. Temperature can be entered in K, °C, °F, or °R — all conversions are handled automatically.