Boyle's Law Calculator (P₁V₁ = P₂V₂)
Solve pressure–volume problems for a fixed amount of gas at constant temperature. Enter any three of P₁, V₁, P₂, V₂ to find the fourth, or work out what a percent change in one variable does to the other.
Background
Boyle's Law describes an inverse relationship between pressure and volume for a fixed amount of gas held at constant temperature: squeeze the volume down and the pressure rises by the same factor, and vice versa. Mathematically, P₁V₁ = P₂V₂. This calculator solves the classic four-variable version with full unit support, and also handles the common "what if" question — if volume (or pressure) changes by a given percent, what happens to the other one?
How to use this calculator
- Choose Solve P₁V₁ = P₂V₂ for the classic problem: enter any three of P₁, V₁, P₂, V₂ and find the fourth.
- Choose Percent Change to answer "what if" questions — e.g. if volume drops by 30%, how much does pressure rise?
- Pick your preferred units for pressure (atm, kPa, mmHg, Pa, bar, psi) and volume (L, mL, cm³, m³) — everything converts automatically.
- Click Calculate to see the visual plus a full step-by-step explanation and a callout on what the result actually means.
How Boyle's Law works
For a fixed amount of gas at constant temperature, pressure and volume are inversely proportional: P₁V₁ = P₂V₂. Squeeze the gas into half the volume and the pressure doubles; let it expand to double the volume and the pressure halves.
"Constant temperature and amount of gas" is the key assumption — this is why it's called an isothermal process. If temperature or the number of gas molecules changes too, Boyle's Law alone isn't enough (that's when the combined or ideal gas law takes over).
Because the relationship is a fixed product (P×V = constant), a percent change in one variable maps to a specific, calculable percent change in the other — but the two percentages are not equal and opposite except for small changes; the relationship is multiplicative, not additive.
Units must be converted to a common basis before applying the formula. This calculator converts every pressure to atmospheres and every volume to liters internally, then converts the answer back to whatever unit you asked for.
Boyle's Law assumes ideal gas behavior. Real gases deviate somewhat at very high pressure or very low temperature, but the ideal approximation is accurate enough for the vast majority of classroom and everyday problems.
Formulas & Equations Used
Boyle's Law: P₁V₁ = P₂V₂
Solve for P₂: P₂ = P₁V₁ / V₂ Solve for V₂: V₂ = P₁V₁ / P₂
Solve for P₁: P₁ = P₂V₂ / V₁ Solve for V₁: V₁ = P₂V₂ / P₁
Percent change (volume given): P₂ = P₁ / (1 + %ΔV/100)
Percent change (pressure given): V₂ = V₁ / (1 + %ΔP/100)
Example Problems & Step-by-Step Solutions
Example 1 — Why divers must exhale on ascent
A diver's lungs hold 1.5 L of gas at 3.0 atm at depth. What's the volume at the surface (1.0 atm) if they hold their breath?
Step: V₂ = P₁V₁/P₂ = (3.0 × 1.5) / 1.0.
Result: V₂ = 4.5 L — the lungs would try to triple in volume, which is exactly why divers are trained to exhale continuously while ascending.
Example 2 — Compressing air with a pump
A bike pump starts with 600 mL of air at 1 atm and is compressed down to 150 mL. What's the new pressure?
Step: P₂ = P₁V₁/V₂ = (1 × 600) / 150.
Result: P₂ = 4 atm — quartering the volume quadruples the pressure.
Example 3 — Unit conversion in action
A gas at 760 mmHg and 2 L is compressed to 1 L. Find the new pressure.
Step: Convert 760 mmHg to 1.0 atm. Then P₂ = (1.0 × 2) / 1 = 2.0 atm.
Result: P₂ = 2.0 atm, or 1,520 mmHg converted back.
Example 4 — Volume up 300%, what happens to pressure?
A weather balloon's gas volume increases by 300% (quadruples) as it rises. How does the pressure change?
Step: New volume factor = 1 + 300/100 = 4. P₂ = P₁/4, a 75% decrease.
Result: Pressure drops by 75% — not 300%. Percent changes in P and V are related by reciprocals, not by simple addition or matching magnitudes.
Example 5 — Volume down 20%, what happens to pressure?
A sealed gas syringe's volume is reduced by 20%.
Step: New volume factor = 1 − 0.20 = 0.80. P₂ = P₁/0.80 = 1.25P₁.
Result: Pressure rises by 25%, not 20% — a smaller percent decrease in volume causes a larger percent increase in pressure.
Example 6 — Solving backward for the starting pressure
Gas ends at 2 atm and 5 mL after starting at 10 mL. What was the initial pressure?
Step: P₁ = P₂V₂/V₁ = (2 × 5) / 10.
Result: P₁ = 1 atm — the same equation works in reverse just as easily.
Frequently Asked Questions
When can I actually use Boyle's Law?
Only when temperature and the amount of gas (moles) stay constant — a closed, isothermal system. If temperature changes too, you need the combined gas law or ideal gas law instead.
Why don't the percent changes in P and V just cancel out?
Because P×V is a constant product, not a constant sum. A 20% decrease in V multiplies P by 1/0.80, which is a 25% increase — not exactly 20% — and the asymmetry grows for larger changes.
Which units does this calculator support?
Pressure: atm, kPa, mmHg, Pa, bar, and psi. Volume: L, mL, cm³, and m³. Every value is converted internally to atm and liters, then converted back to your chosen display unit.
Does this work for real (non-ideal) gases?
This calculator assumes ideal gas behavior, which holds very well at ordinary temperatures and pressures. Real gases deviate noticeably only under extreme compression or near their condensation point.
Can pressure or volume ever be zero or negative?
No — both are physical quantities that must stay strictly positive. A calculated percent-change scenario that would push either one to zero or below (a decrease of 100% or more) isn't physically possible.
How is this different from Charles's Law or Gay-Lussac's Law?
Boyle's Law holds temperature constant and relates pressure to volume. Charles's Law holds pressure constant and relates volume to temperature. Gay-Lussac's Law holds volume constant and relates pressure to temperature.
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