Boyle's, Charles's, and Avogadro's Laws

Gases obey remarkably simple rules. Squeeze a gas and its volume shrinks. This shrinkage is in exact inverse proportion to the pressure applied. Heat it and its volume swells. This swelling is in exact direct proportion to its absolute temperature. Add more molecules and the volume grows. This growth is in direct proportion to the amount of substance. These three statements were discovered separately over more than a century. They describe how pressure ( $P$ ), volume ( $V$ ), absolute temperature ( $T$ ), and amount of gas ( $n$ ) relate to one another. This holds whenever the fourth variable is held fixed. Each law was originally an experimental generalization. It was squeezed out of mercury columns, bladders, and balloons, rather than derived from theory. Robert Boyle worked with trapped air in a bent glass tube. Jacques Charles and Joseph Gay-Lussac tracked balloons and sealed gas samples through a range of temperatures. Amedeo Avogadro proposed, on comparing gas densities, that equal volumes of any gas at the same $P$ and $T$ contain equal numbers of particles. None of them needed to know that gases are made of molecules in ceaseless motion. That molecular picture, the domain of Kinetic Molecular Theory, came later. It explained why the laws hold. This article treats the three laws as independent, verifiable relationships. Each is tested by holding two of the four variables fixed. Understanding them individually is what makes the combined The Ideal Gas Law transparent rather than mysterious. $PV = nRT$ is simply the algebraic fusion of Boyle's, Charles's, and Avogadro's separate discoveries. It is built on the pressure concept and the Kelvin scale you already have in hand.

Key takeaways

Boyle's Law: at constant $n$ and $T$ , $P$ and $V$ are inversely proportional — $PV = \text{constant}$ .
Charles's Law: at constant $n$ and $P$ , $V$ and $T$ (in kelvin) are directly proportional — $V/T = \text{constant}$ .
Avogadro's Law: at constant $T$ and $P$ , $V$ and $n$ are directly proportional — $V/n = \text{constant}$ .
All temperatures in these laws must be absolute (kelvin); using °C gives nonsense, including negative volumes.
The three laws are special cases of one master relation, revealed by holding different pairs of variables fixed.
Key idea: each law isolates one cause-and-effect pair among $P$ , $V$ , $T$ , $n$ by freezing the other two variables, and together they force the existence of a single constant, $R$ , common to all gases.

The core insight

A gas law is a controlled experiment written as an equation: change one variable, hold two others fixed, and watch how the fourth responds.

The physical picture

Picture a fixed number of gas molecules, $n$ , sealed in a cylinder with a movable piston. Three quantities describe the state of this trapped gas: how hard it pushes on the walls ( $P$ ), how much space it occupies ( $V$ ), and how energetically its molecules move ( $T$ ). The three laws are three independent knobs you can turn:

Boyle's Law turns the piston in or out at constant temperature. Compressing the gas into half the volume doubles the collision rate on the walls, so the pressure doubles. Formally, at fixed $n$ and $T$ ,

P_1V_1 = P_2V_2

Charles's Law heats or cools the gas while letting the piston float freely so pressure stays constant. Warmer molecules strike the walls harder and more often; the only way to keep pressure unchanged is for the piston to move outward, increasing volume. At fixed $n$ and $P$ ,

\frac{V_1}{T_1} = \frac{V_2}{T_2}

Avogadro's Law injects more gas molecules at constant $T$ and $P$ . More molecules means more wall collisions per second; the only way to hold pressure steady is for volume to expand proportionally. At fixed $T$ and $P$ ,

\frac{V_1}{n_1} = \frac{V_2}{n_2}

Each law is an idealization — real gases deviate somewhat, especially at high pressure or low temperature, a story told separately in the treatment of real-gas corrections. At the pressures and temperatures common in an introductory laboratory, the deviations are too small to matter.

Figure

Boyle's Law: pressure and volume trade off at constant temperature and amount

Figure

Charles's Law: volume rises linearly with absolute temperature, and extrapolates to zero at 0 K

The core relations

P_1 V_1 = P_2 V_2 \qquad (\text{Boyle's Law; } n,\,T \text{ constant})

where: $P_1, V_1$ are the initial pressure and volume; $P_2, V_2$ are the final pressure and volume, in any consistent units of pressure and volume.

\frac{V_1}{T_1} = \frac{V_2}{T_2} \qquad (\text{Charles's Law; } n,\,P \text{ constant})

where: $T_1, T_2$ are absolute temperatures in kelvin — never Celsius or Fahrenheit.

\frac{V_1}{n_1} = \frac{V_2}{n_2} \qquad (\text{Avogadro's Law; } T,\,P \text{ constant})

where: $n_1, n_2$ are amounts of gas in moles.

\frac{P_1V_1}{n_1T_1} = \frac{P_2V_2}{n_2T_2} \qquad (\text{the combined/general gas relation})

where: this single equation contains all three laws as the special cases obtained by freezing two of $n$ , $P$ , $T$ at a time; it is the direct ancestor of $PV = nRT$ .

Derivation AFrom three separate laws to one combined relation

Goal. Show that Boyle's, Charles's, and Avogadro's Laws are not three unrelated facts but three faces of a single proportionality among $P$ , $V$ , $T$ , and $n$ , and that this proportionality forces $PV/(nT)$ to be a universal constant.

Step 1 — Restate each law as a proportionality. Boyle's Law says that at fixed $n, T$ : $V \propto 1/P$ . Charles's Law says that at fixed $n, P$ : $V \propto T$ . Avogadro's Law says that at fixed $T, P$ : $V \propto n$ .

Step 2 — Combine the three proportionalities. Because $V$ is separately proportional to $1/P$ , to $T$ , and to $n$ , and these are independent variables, standard rules for combining proportionalities give

V \propto \frac{nT}{P}

This step is the crux: each law was measured by holding two variables fixed, but nothing prevents all three proportionalities from holding simultaneously, since each was verified independently of the others.

Step 3 — Insert a proportionality constant. Turning $V \propto nT/P$ into an equation requires a constant of proportionality, call it $R$ :

V = R\,\frac{nT}{P}

Step 4 — Rearrange into the symmetric form. Multiplying both sides by $P$ and dividing by $nT$ ,

\frac{PV}{nT} = R

Because $R$ is the same number for one mole of any ideal gas regardless of its identity — a fact Avogadro's hypothesis guarantees, since it treats gas volume as depending only on particle count, not particle type — this constant is universal. It is the gas constant, and this equation is exactly $PV = nRT$ .

Step 5 — Recover each original law as a limiting case. Setting $n_1T_1/P_1 = n_2T_2/P_2$ (both equal to $V/R$ ) and specializing:

Hold $n, T$ fixed → $P_1V_1 = P_2V_2$ , Boyle's Law.
Hold $n, P$ fixed → $V_1/T_1 = V_2/T_2$ , Charles's Law.
Hold $T, P$ fixed → $V_1/n_1 = V_2/n_2$ , Avogadro's Law.

Each empirical law is thus a cross-section of the single surface described by $PV = nRT$ , sliced along a plane where two variables happen not to change.

Worked examples

Example 1 — Boyle's Law with a non-round compression. A sealed syringe holds 0.180 mol of argon gas at 1.35 atm in a volume of 4.20 L at constant temperature. The plunger is pushed in until the volume becomes 1.65 L. Find the new pressure. Since $n$ and $T$ are both unchanged, Boyle's Law applies directly:

P_1V_1 = P_2V_2

P_2 = \frac{P_1V_1}{V_2} = \frac{(1.35\ \text{atm})(4.20\ \text{L})}{1.65\ \text{L}}

P_2 = \frac{5.670\ \text{L}\,\cdot\,\text{atm}}{1.65\ \text{L}} = 3.44\ \text{atm}

The volume shrank by a factor of $4.20/1.65 = 2.545$ , and the pressure grew by very nearly that same factor ( $3.44/1.35 = 2.548$ , matching within rounding) — exactly the inverse proportionality Boyle's Law demands. Limiting-case check: as $V_2 \to V_1$ (no compression at all), the formula gives $P_2 \to P_1$ , correctly predicting no pressure change when the volume does not change.

Example 2 — Charles's Law with a realistic temperature swing. A weather balloon contains 12.0 L of helium at 22 °C at the moment it is released at constant atmospheric pressure. As it rises, the gas cools to $-48\,^{\circ}\text{C}$ . Find its new volume, assuming no gas escapes. First convert both temperatures to kelvin, since Charles's Law requires an absolute scale:

T_1 = 22 + 273.15 = 295.15\ \text{K}, \qquad T_2 = -48 + 273.15 = 225.15\ \text{K}

Apply Charles's Law ( $n, P$ constant):

\frac{V_1}{T_1} = \frac{V_2}{T_2}

V_2 = V_1 \cdot \frac{T_2}{T_1} = 12.0\ \text{L} \times \frac{225.15\ \text{K}}{295.15\ \text{K}}

V_2 = 12.0\ \text{L} \times 0.7628 = 9.15\ \text{L}

The balloon shrinks from 12.0 L to 9.15 L as it cools — volume tracks absolute temperature directly, exactly as the law predicts, and using Celsius values directly in the ratio (forgetting the $+273.15$ conversion) would have given a badly wrong, even negative-looking, result. Limiting-case check: as $T_2 \to 0\ \text{K}$ , the formula predicts $V_2 \to 0$ — a real gas would liquefy long before reaching absolute zero, which is precisely why the ideal gas laws are understood as extrapolated idealizations, not literal predictions near 0 K.

Limiting cases & common traps

Situation	What happens	Why
$V_2 \to V_1$ in Boyle's Law	$P_2 \to P_1$	No volume change means no pressure change; the proportionality is trivially satisfied
$T \to 0\ \text{K}$ in Charles's Law	$V \to 0$	Mathematical extrapolation; real gases liquefy first, so this limit is never physically reached
$n_2 = 2n_1$ in Avogadro's Law	$V_2 = 2V_1$	Doubling moles at fixed $T,P$ exactly doubles volume — this is the defining content of the law
Mixing up °C and K in Charles's or the combined law	Wildly wrong, sometimes negative, volumes	Only the kelvin scale has volume proportional to temperature through zero
Assuming Boyle's Law works at changing $T$	Wrong prediction	Boyle's Law explicitly requires constant temperature; a temperature change must be handled by the combined relation

The chemist behind it

Robert Boyle published his pressure–volume observations in 1662, using a J-shaped glass tube: trapping air at the closed end and pouring in mercury to compress it, he found that pressure and volume changed in exact inverse proportion. He was a founder of the experimental method in chemistry, one of the first to insist that reproducible, quantitative measurement — not qualitative alchemical description — should decide chemical questions. More than a century later, Jacques Charles noticed (though he never published it) that different gases all expand by the same fraction per degree of heating, an observation formalized and published by Joseph Gay-Lussac in 1802. Extrapolating this linear expansion backward, later scientists found that all gases' volumes appeared to approach zero at the same temperature, roughly $-273\,^{\circ}\text{C}$ — indirect early evidence for what would become the absolute (Kelvin) temperature scale. Amedeo Avogadro's 1811 hypothesis was the most conceptually daring of the three: he proposed that equal volumes of any gas, at the same temperature and pressure, contain equal numbers of particles, regardless of the gas's identity or the size of its molecules. This idea was controversial for decades — it required distinguishing atoms from molecules at a time when confusion between the two was rampant — but it ultimately became the basis for determining molecular formulas and atomic masses, and it is the reason the constant $R$ in $PV = nRT$ is the same for helium, nitrogen, or any other (ideal) gas.

Origins · 1 minAmedeo Avogadro and how this was discovered

Amedeo Avogadro

1776–1856 · Turin, Kingdom of Sardinia; proposed that gas volume depends on particle count, not particle identity, resolving early confusion between atoms and molecules

Ask the tutor: If I know a gas's pressure, volume, and temperature in one state, how do I predict any single one of those three quantities after two of the others change at once?

Do these three laws apply to any gas, or only specific ones?

They are idealizations that apply approximately to all gases under moderate conditions — roughly atmospheric pressure and temperatures well above the boiling point. Avogadro's Law is what guarantees this gas-independence: it says the relationship between $V$ and $n$ (at fixed $T, P$ ) does not depend on which gas you use, so the same constant $R$ works for helium, oxygen, or carbon dioxide, as long as behavior stays close to ideal.

What goes wrong for real gases, and when should I worry about it?

Real gas molecules have finite size and attract one another weakly, so at high pressure or low temperature — conditions that push molecules close together — actual volumes and pressures deviate from the simple proportionalities. At typical introductory-laboratory conditions (near 1 atm, room temperature), the deviations are usually under a percent or two, small enough that the ideal laws are excellent approximations.

How are these three laws related to the combined gas law taught for solving multi-variable problems?

The combined gas law,

\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}

holds $n$ fixed while letting $P$ , $V$ , and $T$ all change simultaneously. It is obtained from the general relation $PV/(nT) = \text{constant}$ by simply canceling the (unchanged) $n$ from both sides; Boyle's and Charles's Laws are the further special cases where, in addition, $T$ or $P$ respectively is also held fixed.

Why does Avogadro's Law imply that molar volume is the same for every ideal gas at a given T and P?

If $V/n$ is the same constant for any gas at fixed $T$ and $P$ , then one mole of any gas — no matter how heavy or light its molecules — occupies the identical volume under those conditions. At 0 °C and 1 atm, for example, this shared molar volume is close to 22.4 L, regardless of whether the gas is helium or carbon dioxide, precisely because the law says volume depends on the number of particles, not their mass or size.

Can I use any pressure and volume units in Boyle's Law, or do they need to be SI?

Any consistent units work, since Boyle's Law is a ratio statement: as long as $P_1$ and $P_2$ share the same units, and $V_1$ and $V_2$ share the same units, the constant $P_1V_1 = P_2V_2$ holds regardless of whether you use atm and liters, or torr and milliliters. This is different from the combined The Ideal Gas Law, where the gas constant $R$ 's numerical value depends on the unit system chosen.