Boyle's Law

General Chemistry • Gases

Written by STEM Calculators Team Published October 19, 2025 Updated February 24, 2026

Boyle’s Law

Isothermal (constant T, constant n) gas compression/expansion. Use the two-state form \(P_iV_i=P_fV_f\) or compute the single-state constant \(a=PV\). Units are chemistry-friendly (Pa, kPa, bar, atm, Torr/mmHg, psi; L, mL, m³, cm³).

Mode Solve for Significant figures

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Paste one line with values (units optional). Examples: Two-state (solving \(P_f\)): 1 atm, 2 L, 1 L • All four: 1 atm, 2 L, 2 atm, 1 L • Single state: 1 atm, 2 L

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Initial state (i)

Pressure \(P_i\)

Volume \(V_i\)

Final state (f)

Pressure \(P_f\)

Volume \(V_f\)

Constraints: \(T=\) const, \(n=\) const. All pressures and volumes must be positive.

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Boyle’s Law — Pressure–Volume Relationship

For a fixed amount of gas at constant temperature, the pressure is inversely proportional to the volume.

\[ P \propto \frac{1}{V} \qquad\Longleftrightarrow\qquad P V = a \ (\text{constant}) \] Two-state (problem-solving) form: \[ \boxed{\,P_i V_i = P_f V_f\,} \]

Subscripts \(i\) and \(f\) denote the initial and final states of the same gas sample at the same temperature.

Where it comes from

The ideal-gas equation \(P V = n R T\) reduces to Boyle’s law when the amount of gas and the temperature are fixed:

\[ P V = n R T \quad\Rightarrow\quad \underbrace{nRT}_{\text{constant}} = a \quad\Rightarrow\quad P V = a \]

Units that work well in chemistry

Pressure: Pa, kPa, bar, atm, Torr/mmHg, psi.
Volume: m³, L, mL (= cm³).
You may keep consistent pairs on both sides, e.g. atm with L, or kPa with mL. If you want the numerical constant \(a\), convert to SI (Pa·m³) where \(1\ \text{Pa·m}^3 = 1\ \text{J}\).

Common conversions

\(1\ \text{atm}=101325\ \text{Pa}=101.325\ \text{kPa}=760\ \text{Torr}\approx760\ \text{mmHg}\)
\(1\ \text{L}=10^{-3}\ \text{m}^3\), \(1\ \text{mL}=1\ \text{cm}^3=10^{-6}\ \text{m}^3\)

The shape of the \(P\)–\(V\) graph

At constant \(T\) the curve is a rectangular hyperbola \(P=a/V\). As volume decreases, pressure rises; the axes are asymptotes (the curve never touches them). The slope at a point is negative: \(\displaystyle \frac{dP}{dV} = -\frac{a}{V^2}\).

Schematic P=a/V at fixed T and n. Your calculator page draws an exact curve using your inputs.

Worked examples

1) Finding a new pressure after compression

A gas at \(P_i=1.00\ \text{atm}\) and \(V_i=2.00\ \text{L}\) is compressed isothermally to \(V_f=1.00\ \text{L}\). What is \(P_f\)?

\[ \begin{aligned} P_i V_i &= P_f V_f \\ P_f &= \frac{P_i V_i}{V_f} = \frac{(1.00\ \text{atm})(2.00\ \text{L})}{1.00\ \text{L}} = \boxed{2.00\ \text{atm}} \end{aligned} \]

2) Finding a new volume when the pressure changes

A sample at \(P_i=98.0\ \text{kPa}\), \(V_i=550\ \text{mL}\) is brought to \(P_f=125\ \text{kPa}\) at the same temperature. What is \(V_f\)?

\[ \begin{aligned} V_f &= \frac{P_i V_i}{P_f} = \frac{(98.0\ \text{kPa})(550\ \text{mL})}{125\ \text{kPa}} = \boxed{431\ \text{mL}} \end{aligned} \]

Assumptions & Pitfalls

Temperature and amount fixed. If \(T\) or \(n\) change, use the combined gas law or the ideal-gas equation directly.
Use absolute pressure. Gas laws assume absolute, not gauge, pressure.
Don’t mix units across sides. Keep a consistent pressure unit on both sides and a consistent volume unit on both sides.
Real-gas deviations. At very high pressure or very low temperature, real gases deviate from \(PV=a\); Boyle’s law is best near ambient conditions.

Practice (self-check)

A gas occupies \(750\ \text{mL}\) at \(0.980\ \text{atm}\). What volume will it occupy at \(1.20\ \text{atm}\) (isothermal)?
Answer
\(V_f=(0.980\times750)/1.20=612.5\ \text{mL}\approx \mathbf{613\ \text{mL}}\).
A 2.50-L sample at \(125\ \text{kPa}\) is expanded to \(4.00\ \text{L}\). What is the new pressure?
Answer
\(P_f=(125\times2.50)/4.00=78.1\ \text{kPa}\).

When you’re ready, switch to the calculator to compute and see the exact \(P\)–\(V\) curve for your numbers.

Frequently Asked Questions

What is Boyle's law and what is the formula?

Boyle's law states that for a fixed amount of gas at constant temperature, pressure is inversely proportional to volume. The standard form is P1 x V1 = P2 x V2.

How do I solve for P2 in Boyle's law?

Rearrange P1 x V1 = P2 x V2 to get P2 = (P1 x V1) / V2. Use consistent units for pressure and volume (or let the calculator convert them).

Do I need to use absolute pressure for Boyle's law?

Yes, Boyle's law is defined for absolute pressure. If a problem gives gauge pressure, convert to absolute pressure by adding atmospheric pressure before applying the formula.

When does Boyle's law not work well?

It becomes less accurate when temperature is not constant or when gases deviate strongly from ideal behavior, such as at very high pressures or very low temperatures.

Boyle's Law

Boyle’s Law

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Initial state (i)

Final state (f)

P–V graph (isothermal)

Calculation steps

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Frequently Asked Questions

Boyle’s Law

Quick fill (paste / CSV)

Initial state (i)

Final state (f)

P–V graph (isothermal)

Calculation steps

Rate this calculator

Frequently Asked Questions

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