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Boyle's Law Calculator

Find final pressure when volume changes at constant temperature

Parameters

atmⓘ
Lⓘ
Lⓘ
Show Trail

Controls

xⓘ

Calculated Values

Final Pressure P₂:
2.00;atm2.00;atm
P₁V₁ = P₂V₂:
4.00;atm⋅L4.00;atm·L

Examples

Cylinder compression

P₁=1 atm, V₁=4 L → V₂=2 L.

  • P₂: 2.002.00

Balloon expansion

P₁=2 bar, V₁=1 L → V₂=3 L.

  • P₂: 0.670.67

Visualization

Boyle's Law — Pressure–Volume Relation at Constant Temperature

Boyle's law (Robert Boyle, 1662; also Mariotte in France) states that for a fixed quantity of ideal gas at constant temperature, pressure is inversely proportional to volume: P ∝ 1/V, or equivalently P₁V₁ = P₂V₂.

This is a direct consequence of the ideal gas law PV = nRT when n and T are held constant: PV = constant = nRT. The constant has units of energy (Pa·m³ = J; atm·L is common in chemistry).

Compressing a gas at constant T (e.g. syringe with thumb on nozzle) decreases volume and increases pressure proportionally. Halving volume doubles pressure. Expanding into a larger container lowers pressure.

On a P–V graph at fixed T, Boyle's law traces a hyperbola (isotherm). The area under a reversible path ∫P dV gives work. Product P×V at any point equals nRT for that temperature.

Limitations: real gases deviate at high pressure (molecular volume, attractions — van der Waals: (P + a/V²)(V − b) = RT) and very low temperature near condensation. Boyle's law is excellent for dilute gases near room temperature and moderate pressure.

Boyle's law combines with Charles's law (V ∝ T at fixed P) and Gay-Lussac's law (P ∝ T at fixed V) to form the complete ideal gas law PV = nRT, unifying the gas laws discovered in the 17th–18th centuries.

Key Concepts

  • P₁V₁ = P₂V₂ at constant T and n
  • P ∝ 1/V — inverse proportionality
  • Special case of PV = nRT with T fixed
  • P–V isotherm is a hyperbola
  • P×V has units of energy (J or atm·L)
  • Fails for dense real gases — use van der Waals

Real-World Applications

  • Syringes, bicycle pumps, and medical inhalers
  • Scuba diving (Boyle's law — pressure/volume at depth)
  • Vacuum pumps and barometers
  • Balloons and sealed gas containers at ~room T
  • Class 11 gas laws and ideal gas law introduction
  • Breathing mechanics (lung volume and pressure changes)

Explore Further

More thermodynamics tools

Physics Equations

Boyle's Law:
P1V1=P2V2P_1 V_1 = P_2 V_2
Solve P₂:
P2=P1V1V2P_2 = \frac{P_1 V_1}{V_2}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Constant T and n

Isothermal process for fixed amount of gas.

Explanation:

Boyle's law applies to ideal gas at constant temperature.

2

Step 2: Boyle's Law

Equation:

P1V1=P2V2P_1 V_1 = P_2 V_2

Explanation:

Pressure inversely proportional to volume.

3

Step 3: Solve for P₂

Calculation:

P2=P1V1V2=1×42=2.0000P_2 = \frac{P_1 V_1}{V_2} = \frac{1 \times 4}{2} = 2.0000

Result:

P2=2.0000P₂ = 2.0000
4

Step 4: Check Trend

Volume doubled → pressure halved.

Explanation:

V₂ > V₁ ⇒ P₂ < P₁.

5

Step 5: Verify Product

Calculation:

P1V1=4.0000,P2V2=4.0000P_1 V_1 = 4.0000, \quad P_2 V_2 = 4.0000

Explanation:

Products should match (ideal gas).

6

Step 6: Units

Keep P and V in consistent units.

Explanation:

e.g. Pa·m³ or atm·L — product has units of energy.

Frequently Asked Questions (FAQ)

Can I use Celsius?

Boyle uses only P and V ratios; T must stay constant in kelvin if temperature could drift.

Why product P×V has energy units?

Pa·m³ = J; useful in work calculations.

Real gas deviation?

At high P, molecular volume and attractions matter — van der Waals correction.

Isothermal vs Boyle?

Boyle is the P–V relation at fixed T; isothermal is the process type.

Vacuum pumps?

Expanding volume lowers pressure per Boyle at ~constant T.

Practice MCQs

  1. Volume triples at constant T, pressure becomes:
  2. Boyle's law requires constant:
  3. P₁V₁ equals:
  4. Graph of P vs V at constant T:
  5. Doubling pressure changes volume by:
  6. Boyle's law is part of:

Related Calculators

These tools connect to the same physics concepts used in this calculator.

First Law of ThermodynamicsCalculate internal energy change ΔU = Q − W from heat and work.Adiabatic ProcessFind P₂ and T₂ for reversible adiabatic ideal gas expansion (Q = 0).Isothermal ProcessCalculate work W = nRT ln(V₂/V₁) and heat for isothermal ideal gas.Charles's LawCalculate volume change with temperature at constant pressure.Error PropagationCombine measurement uncertainties for sums, products, quotients, and powers in experimental formulas.Related Measurement & Uncertainty topic.Maxwell-Boltzmann DistributionCalculate velocity distributions and characteristic velocities for ideal gases using Maxwell-Boltzmann statistics.Related Statistical Physics topic.Projectile MotionCalculate and visualize the path of objects under the influence of gravity.Related Mechanics topic.