Gay-Lussac's law of combining volumes

Take 20 cm³ of hydrogen gas and 20 cm³ of chlorine gas, mix them at the same temperature and pressure, and let them react. The product — hydrogen chloride gas — does not come out to some odd number like 27 cm³ or 33 cm³. It comes out to exactly 40 cm³, a clean multiple of the starting volumes. This is not a coincidence of this one reaction; every gaseous reaction chemists have measured shows the same pattern: reacting and product gas volumes stand in simple whole-number ratios to one another. Joseph Louis Gay-Lussac reported this in 1808, years before anyone could count molecules directly, and it became one of the load-bearing clues that let chemists work out how atoms combine. This article states the law precisely, shows why it holds only under a specific condition, and uses it to read off unknown gas volumes from known reactions — the same skill tested throughout the mole concept chapter.

The core insight

When gases react, the volumes of reactants and gaseous products — measured at the same temperature and pressure — are always in a simple whole-number ratio.

The core idea

Statement. Gay-Lussac's Law of Combining Volumes says: when gases react with one another, they do so in volumes that bear a simple whole-number ratio to each other and to the volume of any gaseous product formed, provided all volumes are measured at the same temperature and pressure. The condition in italics is not decoration — it is the whole reason the ratio comes out "simple." Volume depends on temperature and pressure (a fixed mass of gas expands when heated), so unless every measurement is taken under identical conditions, the numbers you record are not comparable and the neat ratio disappears.

Reading the ratio off a reaction. For hydrogen and chlorine combining to give hydrogen chloride,

H2(g)+Cl2(g)→2HCl(g)\text{H}_2(g) + \text{Cl}_2(g) \rightarrow 2\text{HCl}(g)

where: the coefficients 1, 1 and 2 are the numbers of gas particles reacting and forming — one volume of hydrogen reacts with one volume of chlorine to give two volumes of hydrogen chloride, a ratio of 1:1:21:1:2. For nitrogen and hydrogen combining to give ammonia,

N2(g)+3H2(g)→2NH3(g)\text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g)

where: one volume of nitrogen reacts with three volumes of hydrogen to give two volumes of ammonia — a ratio of 1:3:21:3:2. Notice that four volumes of reactant gas (1 + 3) shrink into two volumes of product; the total volume of gas is not conserved, only the whole-number ratio is.

Figure
Bar diagram comparing reactant and product gas volumes for two combining volume reactions
Combining volumes are whole-number ratios H2 + Cl2 to 2HCl ratio 1:1:2 H2 = 1 Cl2 = 1 HCl = 2 N2 + 3H2 to 2NH3 ratio 1:3:2 N2 = 1 H2 = 3 NH3 = 2 Bar height stands for volume; the ratio of heights is always a small whole number valid only when all gases are measured at the same temperature and pressure Avogadro: equal volumes at the same T and P hold equal numbers of molecules, so the bar-height ratio is also the molecule ratio, i.e. the equation coefficients

Why it links to atomicity and formulae. Gay-Lussac found the pattern by experiment alone, with no explanation for why volumes should behave this way. The explanation arrived through Avogadro's Law: equal volumes of gases at the same temperature and pressure contain equal numbers of molecules. Combine the two laws and the volume ratio becomes a molecule ratio, which is exactly the set of coefficients in the balanced equation. That chain of reasoning — combining-volume data, plus Avogadro's Law, plus counting atoms — is how chemists first pinned down that hydrogen and chlorine are diatomic and that ammonia is NH₃, without ever weighing a single molecule directly.

Derivation ATurning a combining-volume ratio into an unknown gas volume

The goal is to show, using nothing beyond Avogadro's Law and the balanced equation, why the volume ratio observed in the laboratory must equal the ratio of coefficients in the equation, and therefore why an unknown reactant or product volume can be found by simple scaling once any one volume is known — this is the calculation skill examined throughout the chapter.

Step 1. Start from Avogadro's Law: at the same temperature and pressure, equal volumes of any gas contain equal numbers of molecules. Call that fixed number of molecules per "unit volume" NN, the same NN for every gas being compared, since all are held at identical temperature and pressure.

Step 2. A gas occupying volume VV therefore contains a number of molecules directly proportional to VV; if a reactant occupies V1V_1 and another occupies V2V_2 under the same conditions, the ratio of molecules reacting is V1:V2V_1 : V_2, exactly the ratio of volumes measured.

Step 3. A balanced equation such as aA(g)+bB(g)→cC(g)a\text{A}(g) + b\text{B}(g) \rightarrow c\text{C}(g) states that molecules of A, B and C react and form in the fixed ratio a:b:ca : b : c. Since molecule ratio equals volume ratio (Step 2), the volumes of A, B and C measured in the flask must also stand in the ratio a:b:ca : b : c — this is Gay-Lussac's Law, now explained rather than merely observed.

Step 4. This gives a direct calculation rule. If one volume, say VAV_\text{A}, corresponds to "aa units" of the ratio, then one ratio-unit equals VA/aV_\text{A}/a, and any other volume is found by multiplying its own coefficient by that unit: VC=c×(VA/a)V_\text{C} = c \times (V_\text{A}/a). The condition throughout is that every volume in the problem is measured at the same temperature and pressure — if it is not, the ratio no longer applies and no scaling is valid.

Worked example

Problem: 20 cm³ of hydrogen gas reacts completely with 20 cm³ of chlorine gas, both measured at the same temperature and pressure, to form hydrogen chloride gas. Find the volume of hydrogen chloride formed, and check the result against a second, extended case: what if 15 cm³ of hydrogen had been used instead, with excess chlorine available?

Step 1. Note the reactant volume ratio as given. Volume of H₂ : Volume of Cl₂ =20:20=1:1= 20 : 20 = 1 : 1, matching the known reaction H2(g)+Cl2(g)→2HCl(g)\text{H}_2(g) + \text{Cl}_2(g) \rightarrow 2\text{HCl}(g), whose combining-volume ratio is 1:1:21 : 1 : 2.

Step 2. Scale to the given volume. Here 20 cm³ of H₂ corresponds to "1 volume" in the ratio, so the product volume is 2×20=402 \times 20 = 40 cm³ of HCl — the coefficient of HCl (2) times the ratio-unit (20 cm³).

Step 3 — extended case. If only 15 cm³ of H₂ reacts (with enough Cl₂ present so H₂ is the limiting gas), the ratio-unit becomes 1515 cm³ instead of 2020 cm³, since 1 volume of H₂ still equals "1 unit." The chlorine consumed is 1×15=151 \times 15 = 15 cm³, and the HCl formed is 2×15=302 \times 15 = 30 cm³. The whole-number ratio 1:1:21:1:2 never changes — only the size of one "unit" changes with the amount of gas actually reacted.

Answer: 40 cm³ of HCl forms from 20 cm³ of H₂ and 20 cm³ of Cl₂; scaling the same 1:1:21:1:2 ratio, 15 cm³ of H₂ would give 30 cm³ of HCl.

Common traps

  • Forgetting the condition "same temperature and pressure" — without it, the volumes recorded are not directly comparable and the simple ratio breaks down.
  • Confusing this chemical combining-volume law with the physics gas laws (Boyle's, Charles') that relate pressure, volume and temperature for a single fixed gas sample — Gay-Lussac's Law here is about ratios between different reacting gases.
  • Applying the law to solids or liquids in a reaction — it is stated only for gaseous reactants and gaseous products; a solid or aqueous species in the equation is simply left out of the volume ratio.
  • Treating "simple whole-number ratio" as automatically meaning total volume is conserved — in N2+3H2→2NH3\text{N}_2 + 3\text{H}_2 \rightarrow 2\text{NH}_3, four volumes of reactant give only two volumes of product; the ratio is whole-number, not equal.

Ask the tutor: Want to practice scaling combining-volume ratios for a new reaction, or see how Gay-Lussac's and Avogadro's Laws together pin down a gas's molecular formula? Ask the AI tutor for a guided example.

Why does the law need same temperature and pressure to work?

Volume is not a fixed property of a fixed amount of gas — it changes with temperature and pressure, because a gas expands when heated or when the pressure on it drops. If you measured hydrogen's volume warm and chlorine's volume cold, their volumes would no longer be proportional to the number of molecules present, and the neat ratio would be lost. Holding temperature and pressure the same for every gas in the problem is what makes volume a fair stand-in for molecule count, via Avogadro's Law.

Is Gay-Lussac's Law of Combining Volumes the same as the pressure-temperature gas law also named after Gay-Lussac?

No, and mixing them up is a common exam slip. The law in this topic is a chemistry law about the ratio of gas volumes in a reaction — it needs a balanced equation and gives whole-number ratios like 1:1:2. The other law, sometimes also credited to Gay-Lussac, is a physics relationship between the pressure and temperature of one fixed sample of gas at constant volume. They share a name but answer completely different questions.

Can I use this law to find the volume of a solid or liquid product?

No. The law is defined only for volumes of gases, because only gas volumes are directly proportional to the number of molecules present at fixed temperature and pressure (via Avogadro's Law). If a reaction produces a solid or liquid alongside a gas, you read off the gas volume from the whole-number ratio as usual, but the solid or liquid product is handled separately, using mass and the mole concept, not this volume ratio.

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