Empirical Gas Laws: Boyle, Charles, Gay-Lussac

Squeeze a bicycle pump with your thumb over the nozzle, and the resistance you feel is Boyle's law made tangible. Heat a sealed can of hairspray, and the risk of it bursting is Gay-Lussac's law in action. These everyday encounters with gases obey precise, quantitatively reproducible mathematical relationships. Long before anyone knew that gases are swarms of colliding molecules, seventeenth- and eighteenth-century experimentalists mapped out how pressure, volume, and temperature trade off against each other. Their empirical laws matter because they are the experimental scaffolding on which the ideal gas law is built. Every term in that later equation traces back to a measured proportionality discovered here. This article presents the three foundational pressure-volume-temperature relationships. You will see how each law was measured. You will see the mathematical form each one takes. You will see how they combine into a single combined gas law. Worked numerical examples show the laws in action, and common misconceptions are flagged along the way.

Key takeaways

Boyle's law: at constant temperature and amount, $pV = \text{constant}$ , so pressure and volume are inversely proportional.
Charles's law: at constant pressure and amount, $V/T = \text{constant}$ , so volume grows linearly with absolute temperature.
Gay-Lussac's law: at constant volume and amount, $p/T = \text{constant}$ , so pressure grows linearly with absolute temperature.
All three are limiting laws, most accurate at low pressure and high temperature, where gases behave nearly ideally.
Combining them gives the combined gas law, $p_1V_1/T_1 = p_2V_2/T_2$ , valid for a fixed amount of gas.
Key idea: each law isolates one pair of state variables while holding the third fixed, and together they reveal that pressure, volume, and temperature are locked into a single underlying equation of state.

The core insight

Boyle's, Charles's, and Gay-Lussac's laws are three complementary slices through the same experimental reality: for a fixed amount of gas, pressure, volume, and absolute temperature are not independent, but are linked by one equation of state, and each classical law is what you see when you freeze one of those three variables and watch the other two move.

The physical picture

Imagine a gas trapped in a cylinder by a frictionless, movable piston. The state of that gas — how it behaves — is completely fixed once you specify three quantities: its pressure $p$ , its volume $V$ , and its temperature $T$ (assuming the amount of substance, $n$ , is fixed). As covered in the discussion of state variables and equations of state, these quantities are not all independent; nature imposes a functional constraint between them. The three empirical gas laws are the historical routes by which that constraint was mapped out piece by piece, each one holding one variable fixed and measuring how the other two respond. Robert Boyle, working in Oxford in 1662, trapped air in the short closed end of a J-shaped glass tube and poured mercury into the long open end. Adding mercury increased the pressure on the trapped air and compressed it. He found that doubling the pressure halved the volume, tripling it cut the volume to a third, and so on — an inverse proportionality, at constant temperature. A century and a half later, Jacques Charles (and later Joseph Louis Gay-Lussac, who published the systematic version in 1802) heated and cooled gas samples at constant pressure and found that volume increased linearly with temperature, vanishing — in extrapolation — at a common temperature of about $-273\,^\circ\text{C}$ for every gas tested. Gay-Lussac also examined gas heated in a sealed, rigid vessel and found that pressure, too, rises linearly with temperature. The unifying picture is a three-dimensional surface of allowed $(p, V, T)$ states. Boyle's law is the curve traced on that surface at fixed $T$ ; Charles's law is the line traced at fixed $p$ ; Gay-Lussac's law is the line traced at fixed $V$ . No single experiment reveals the whole surface, but the three together pin it down completely for a fixed quantity of gas, which is exactly what the ideal gas law later encodes in one compact equation.

Figure

Boyle's law: pressure-volume isotherms at three fixed temperatures

Figure

Charles's and Gay-Lussac's laws: linear extrapolation to absolute zero

The core relations

pV = \text{constant} \quad (n, T \text{ fixed})

where: $p$ = pressure, $V$ = volume — this is Boyle's law, an inverse proportionality between $p$ and $V$ .

\frac{V}{T} = \text{constant} \quad (n, p \text{ fixed})

where: $T$ = absolute (Kelvin) temperature — this is Charles's law, a direct proportionality between $V$ and $T$ .

\frac{p}{T} = \text{constant} \quad (n, V \text{ fixed})

where: this is Gay-Lussac's law, a direct proportionality between $p$ and $T$ .

\frac{p_1V_1}{T_1} = \frac{p_2V_2}{T_2} \quad (n \text{ fixed})

where: subscripts 1 and 2 label two different states of the same fixed amount of gas — the combined gas law.

Derivation AFrom two-variable laws to the combined gas law

Goal. Show that Boyle's, Charles's, and Gay-Lussac's laws, each valid only under one restriction, merge into a single relation valid for any two states of a fixed amount of gas.

Step 1. State each law as a proportionality. Boyle's law says that at fixed $T$ and $n$ ,

p \propto \frac{1}{V}, \qquad \text{i.e. } pV = k_1(T).

Charles's law says that at fixed $p$ and $n$ ,

V \propto T, \qquad \text{i.e. } \frac{V}{T} = k_2(p).

The constants $k_1$ and $k_2$ are not universal constants — they depend on whichever variable was held fixed, and also on $n$ .

Step 2. Track a gas through two sequential fixed-variable steps. Take a gas sample from state $(p_1, V_1, T_1)$ to an intermediate state $(p_2, V', T_1)$ by an isothermal (Boyle) compression at constant $T_1$ :

p_1 V_1 = p_2 V'.

Then take it from $(p_2, V', T_1)$ to the final state $(p_2, V_2, T_2)$ by an isobaric (Charles) heating at constant $p_2$ :

\frac{V'}{T_1} = \frac{V_2}{T_2}.

Step 3. Eliminate the intermediate volume $V'$ . From the Boyle step,

V' = \frac{p_1 V_1}{p_2}.

Substitute into the Charles step:

\frac{p_1 V_1}{p_2 T_1} = \frac{V_2}{T_2}.

Step 4. Rearrange to isolate the combined relation. Cross-multiplying,

p_1 V_1 T_2 = p_2 V_2 T_1,

and dividing both sides by $T_1 T_2$ gives

\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}.

Step 5. Check that Gay-Lussac's law is consistent with this result. Set $V_1 = V_2$ (constant volume) in the boxed relation: it reduces immediately to $p_1/T_1 = p_2/T_2$ , exactly Gay-Lussac's law. The three empirical laws are therefore not independent facts but three faces of one relation.

\boxed{\dfrac{p_1V_1}{T_1} = \dfrac{p_2V_2}{T_2}} \quad \text{for a fixed amount of gas, } n \text{ constant.}

Worked examples

Example 1 — Foundational: a diving bell descends (Boyle's law). A rigid diving bell contains an air pocket of volume $20.0\ \text{L}$ at the surface, where the pressure is $1.00\ \text{atm}$ and the temperature is $20.0\,^\circ\text{C}$ . It is lowered to a depth where the pressure rises to $4.00\ \text{atm}$ , and the temperature at that depth happens to also be $20.0\,^\circ\text{C}$ (so temperature is unchanged — Boyle's law applies directly). Step 1 — write Boyle's law.

p_1 V_1 = p_2 V_2.

Step 2 — solve for $V_2$ .

V_2 = \frac{p_1 V_1}{p_2} = \frac{(1.00\ \text{atm})(20.0\ \text{L})}{4.00\ \text{atm}} = 5.00\ \text{L}.

Sanity check. Pressure quadrupled, so volume should drop to one quarter: $20.0\ \text{L}/4 = 5.00\ \text{L}$ . Matches. The air pocket shrinks to a quarter of its surface volume — a real hazard for scuba divers, which is why divers are trained never to hold their breath while ascending.

Example 2 — Applied, harder: a sealed aerosol can heated, then a combined change. An aerosol can is manufactured and sealed at $T_1 = 25.0\,^\circ\text{C} = 298.15\ \text{K}$ with internal pressure $p_1 = 2.80\ \text{atm}$ and fixed volume (rigid can). Two questions: (a) Manufacturers rate the can to withstand at most $p_{\max} = 5.00\ \text{atm}$ before rupture. Using Gay-Lussac's law, at what temperature does the can reach that limit if it is left in a hot car?

\frac{p_1}{T_1} = \frac{p_{\max}}{T_2} \quad \Rightarrow \quad T_2 = T_1 \times \frac{p_{\max}}{p_1}.

T_2 = 298.15\ \text{K} \times \frac{5.00\ \text{atm}}{2.80\ \text{atm}} = 298.15\ \text{K} \times 1.7857 = 532.4\ \text{K}.

Converting: $T_2 = 532.4 - 273.15 = 259.2\,^\circ\text{C}$ . Sanity check. This is far above any car interior temperature (which tops out around $60$ – $70\,^\circ\text{C}$ ), so an intact, undamaged can is not expected to burst from heat alone under normal conditions — consistent with real-world aerosol safety margins, and a reminder that puncturing (which is a volume/pressure event, not purely thermal) is the more common real hazard. (b) Now suppose instead that before heating, the can is first punctured slightly, so a small piston of gas of volume $500\ \text{cm}^3$ at $p_1 = 2.80\ \text{atm}$ and $298.15\ \text{K}$ escapes into the surrounding air, expanding until its pressure drops to $1.00\ \text{atm}$ , while simultaneously warming to $310\ \text{K}$ (body-temperature air near a hand holding the can). Find the new volume of that escaped gas parcel using the combined gas law.

\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2} \quad \Rightarrow \quad V_2 = V_1 \times \frac{p_1}{p_2} \times \frac{T_2}{T_1}.

V_2 = 500\ \text{cm}^3 \times \frac{2.80\ \text{atm}}{1.00\ \text{atm}} \times \frac{310\ \text{K}}{298.15\ \text{K}}.

V_2 = 500\ \text{cm}^3 \times 2.80 \times 1.0397 = 500\ \text{cm}^3 \times 2.911 = 1.456\times 10^{3}\ \text{cm}^3.

Sanity check. Pressure dropped by a factor of 2.80, which alone would nearly triple the volume; the modest temperature rise adds a small further expansion (factor 1.04). The combined factor of 2.91 is dominated by the pressure drop, as expected since gases are far more sensitive to pressure changes than to modest temperature changes near room temperature.

Limiting cases & common traps

Situation	What happens	Common trap
Very low pressure, any of the three laws	Agreement with real gas data is excellent	Assuming the laws are exact at all pressures — they are limiting laws
High pressure or near condensation	Real gases deviate; $pV$ is no longer constant at fixed $T$	Extrapolating Boyle's law into the liquefaction regime
Temperature in Celsius substituted directly	Charles's and Gay-Lussac's proportionalities fail	Forgetting to convert to kelvin before using $V/T$ or $p/T$
Extrapolating $V \to 0$	Defines absolute zero operationally, $\approx -273.15\,^\circ\text{C}$	Thinking real gas volume truly reaches zero — gases liquefy first
Changing amount of gas, $n$	None of the three simple laws apply as stated	Applying Boyle's law across a process where gas leaks in or out

The chemist behind it

Jacques Alexandre César Charles was a French physicist and balloon pioneer who, in the 1780s, noticed that different gases (hydrogen, air, oxygen) expanded by seemingly the same fraction per degree of heating at constant pressure. He never published this finding formally. It was Joseph Louis Gay-Lussac who, in 1802, carried out careful, systematic measurements across many gases and published the quantitative law, generously naming it after Charles's earlier unpublished observation. Gay-Lussac's own name is more properly attached to the constant-volume pressure-temperature law, though usage varies by textbook and country — a small but telling reminder that scientific credit and naming conventions do not always track perfectly with who published what, when.

Origins · 1 minJoseph Louis Gay-Lussac and how this was discovered

Joseph Louis Gay-Lussac

1778-1850 · French chemist who quantified how gas volume and pressure scale with temperature, and co-discovered boron

Gay-Lussac combined daring experimentation (including a hot-air balloon ascent to nearly 7,000 meters to sample high-altitude air) with meticulous quantitative measurement, turning qualitative observations about heated gases into the precise linear laws still taught today.

Ask the tutor: How does Avogadro's principle combine with these three laws to produce the full ideal gas equation, $pV = nRT$ ?

Why must temperature be in kelvin for Charles's and Gay-Lussac's laws?

Both laws state a direct proportionality, $V \propto T$ or $p \propto T$ , which requires that $V$ or $p$ genuinely equal zero when $T$ equals zero on the scale used. The Celsius scale has an arbitrary zero point (the freezing point of water), so $V/T$ using Celsius values is not constant across an experiment. The kelvin scale sets its zero at the extrapolated point where volume or pressure would vanish, making the proportionality mathematically consistent. Always convert Celsius to kelvin, using $T(\text{K}) = T(^\circ\text{C}) + 273.15$ , before applying either law.

Are these laws exact, or only approximate?

They are limiting laws, exact only in the hypothetical limit of zero pressure, where gas molecules are far apart and rarely interact. At ordinary laboratory pressures they are excellent approximations for most gases, typically accurate to within a percent or two. Deviations grow as pressure increases or temperature drops toward condensation, because intermolecular forces and finite molecular volume — effects the empirical laws ignore — become significant. Real gas equations, such as the van der Waals equation, correct for these effects.

How do these three laws relate to the ideal gas law?

The ideal gas law, $pV = nRT$ , is the single equation of state that reproduces all three empirical laws as special cases. Fixing $T$ and $n$ recovers Boyle's law; fixing $p$ and $n$ recovers Charles's law; fixing $V$ and $n$ recovers Gay-Lussac's law. Historically, the empirical laws came first, established piecemeal through careful experiment; the ideal gas law came later, unifying them and adding the role of the amount of substance, $n$ , through Avogadro's principle.

What is the combined gas law used for in practice?

It lets you relate two arbitrary states of a fixed amount of gas without knowing $n$ or the gas constant explicitly, which is convenient whenever you only care about ratios. Typical applications include weather balloons rising to lower pressure and temperature, gas cylinders warming or cooling, and pistons undergoing simultaneous pressure and temperature changes. It is mathematically identical to the ideal gas law applied twice and divided, so it carries no extra physical content beyond $pV = nRT$ , only convenience.

Did Boyle, Charles, and Gay-Lussac know about molecules?

No. All three worked purely empirically, decades to over a century before the kinetic theory of gases and the atomic-molecular picture became widely accepted in chemistry. They measured macroscopic quantities — pressure with mercury columns, volume with graduated glassware, temperature with early thermometers — and fit mathematical relationships to the data. The molecular explanation, that pressure arises from molecular collisions with container walls, came later and provided the mechanistic "why" behind laws that were already established as empirical "what."