Amedeo Avogadro

How to use this course

Amedeo Avogadro is a name on a constant, \(6.022\times10^{23}\) per mole, that he never measured and could not have imagined measuring. What he did, in a nineteen-page essay of 1811, was to propose that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules, and that the molecules of elements can be made of several atoms. With those two assumptions, Joseph Louis Gay-Lussac’s volume ratios became molecular formulas. Chemists ignored the idea for nearly fifty years. This course follows the hypothesis from its birth in Vercelli to its acceptance at Karlsruhe in 1860, its proof by the kinetic theory, and the measurement of the number that bears his name.

Each technical lesson opens with a paragraph for the general reader, then states the laws and arguments precisely, works through Avogadro’s own numbers where they survive, and ends with exercises whose solutions are folded. Avogadro’s vocabulary differs from ours, and the lessons translate it as they go.

Timeline

A Turin family of lawyers, 1776–1796

Lorenzo Romano Amedeo Carlo Avogadro was born in Turin on 9 August 1776. Turin was the capital of the Kingdom of Sardinia, ruled by the House of Savoy, which held Piedmont, Savoy and the island of Sardinia. The Avogadro family was an old noble family of Vercelli, east of Turin, and for generations its men had served the state as lawyers and judges. His father, Count Filippo Avogadro, was a magistrate who rose to high office in the Piedmontese judiciary. His mother was Anna Maria Vercellone. Amedeo inherited the family title in 1787, becoming Count of Quaregna and Cerreto.

He was expected to continue the tradition, and he did. He studied law at the University of Turin, became a bachelor of jurisprudence in 1792 at sixteen, and took a doctorate in ecclesiastical law in 1796 at twenty. He then began to practise.

The same year Napoleon Bonaparte’s Army of Italy defeated the Piedmontese. By 1798 the king had left the mainland, and in 1802 Piedmont was annexed to France and divided into French departments. Avogadro’s professional life began under this French administration, which brought the metric system, French law and access to French science.

From law to physics, 1800–1809

In 1801 Avogadro became secretary to the prefecture of the department of Eridano, the French department centred on Turin. From about 1800 he also began a private study of mathematics and physics, apparently on his own initiative. The decision changed the course of his life.

His first research, presented to the Academy of Sciences of Turin in 1803, was a study of electricity written with his brother Felice. On 5 July 1804 the Academy elected him a corresponding member. In 1806 he was appointed demonstrator at the college attached to the Academy, his first teaching post. That year and the next he published two memoirs in the Paris Journal de physique on the behaviour of a non-conducting layer placed between oppositely charged conductors, a question that later became the theory of dielectrics.

In 1809 he published Idées sur l’acidité et l’alcalinité, proposing that acidity and alkalinity are not absolute properties but positions on a continuous scale. A substance acts as an acid towards one partner and as a base towards another. In 1813 he engaged the electrochemical theory of Jöns Jacob Berzelius in a reply published in the Annales de chimie. The acidity paper shows the same habit of mind as the 1811 essay: a willingness to replace a fixed classification by a relation.

Vercelli and the 1811 essay, 1809–1820

On 7 October 1809 Avogadro became professor of natural philosophy (mathematics and physics) at the Royal College of Vercelli, the town where his family had its roots and property. He taught there for ten years. Vercelli was far from the centres of chemistry, and the 1811 essay contains no new measurements. Avogadro worked from published data, above all those of Gay-Lussac and John Dalton, and from the densities of gases measured in Paris.

In 1808 Dalton’s New System of Chemical Philosophy proposed that each element consists of identical atoms with a characteristic weight. The same year Gay-Lussac announced that gases combine in simple ratios by volume. Dalton could not reconcile the two results, and rejected Gay-Lussac’s. Avogadro saw a way to combine them. His essay appeared in 1811 in the Journal de physique, de chimie et d’histoire naturelle, edited by Jean-Claude Delamétherie. He followed it with a second memoir in 1814 and a longer one in 1821.

He married Felicita Mazzé; most sources give six children. On 21 November 1819 he was elected a full member of the Turin Academy. By then the French had gone, the House of Savoy had returned in 1814, and the restored government was reorganizing the university. In 1820 it created a chair of mathematical physics, the first in Italy, and gave it to Avogadro.

Equal volumes, equal numbers

For the general reader. If two boxes of the same size, at the same temperature and pressure, always held the same number of gas particles, you could weigh the particles against each other without ever seeing one. Weigh a litre of oxygen and a litre of hydrogen; the ratio is the ratio of the particle masses. Avogadro proposed exactly this in 1811. It turned gas densities into a scale of molecular weights.

The hypothesis

Avogadro began from Gay-Lussac’s law: gases combine in simple volume ratios, and the volume of a gaseous product is in a simple ratio to the reactants. The most natural explanation, he wrote, is that the number of molecules in a gas is proportional to its volume.

Avogadro used several words for particles. A molécule intégrante is the smallest free unit of a gas, our molecule. A molécule élémentaire is the smallest unit of an element, our atom. A molécule constituante is the free unit of an elementary gas, our molecule of an element such as O\(_2\). The ambiguity of “molecule” in his writing, and in everyone else’s, was one reason the idea was hard to absorb.

Densities to masses

Avogadro took gas densities relative to air: oxygen 1.10359, nitrogen 0.96913, and hydrogen 0.07321 (the value of Jean-Baptiste Biot and François Arago). Dividing by hydrogen gives the masses of the molecules relative to the hydrogen molecule:

The modern values on this scale are 15.87 and 13.90. His errors came from the density of hydrogen, which is hard to measure because the gas is so light and so easily contaminated.

Why not Dalton’s rule?

Dalton thought each particle of gas was surrounded by an atmosphere of caloric, and that particles of different gases had different sizes, so equal volumes need not contain equal numbers. Avogadro noted that nothing decided between the two assumptions in advance, but his own made Gay-Lussac’s simple ratios natural, while Dalton’s made them a coincidence. In the modern ideal-gas law \(pV=NkT\), the number \(N\) depends only on \(p\), \(V\) and \(T\), which is Avogadro’s hypothesis in one line (Lesson 07 derives it).

Half-molecules and formulas

For the general reader. Two volumes of hydrogen and one of oxygen make two volumes of steam, not one. If every volume holds the same number of particles, the oxygen particles must have split between two water particles. Avogadro concluded that a particle of oxygen gas is not a single atom but a pair. This was the step Dalton refused, and the step that made chemistry consistent.

The argument

Let one volume contain \(n\) molecules. Then 2 volumes of hydrogen (\(2n\) molecules) and 1 of oxygen (\(n\) molecules) yield 2 volumes of water vapour (\(2n\) molecules). Each of the \(2n\) water molecules contains oxygen, but only \(n\) oxygen molecules were supplied. Unless oxygen can be created, each oxygen molecule has split into at least two parts. Avogadro wrote that the integral molecule of water consists of a half-molecule of oxygen with one molecule, or two half-molecules, of hydrogen.

The modern equations show the counting directly:

Avogadro stated in words the formulas we write as H\(_2\)O, NH\(_3\), NO, N\(_2\)O, HCl and CO. Dalton, using the rule of greatest simplicity, had HO for water and NH for ammonia, and his atomic weights were wrong in consequence. Avogadro remarked that Dalton’s water was nearly right only through compensating errors, and that Dalton’s ammonia was entirely at fault.

Why were chemists unconvinced?

Avogadro offered no mechanism. In Dalton’s picture like particles repel, so two oxygen atoms should not bind. In Berzelius’s electrochemical theory, atoms combine because they carry opposite charges, so two identical atoms could not attract. Avogadro also allowed molecules to divide into halves, and in later work into smaller fractions, whenever the data seemed to demand it, which looked arbitrary. His essays were dense, written in French by a provincial teacher, and gave few new measurements. The hypothesis was cited, but rarely used.

Extending the method, 1814–1821

For the general reader. Most elements are not gases. To find their molecular weights Avogadro had to reason from compounds, using guesses about how many molecules combine and analogies between similar substances. Some of his results were right, such as silica as SiO₂; some were wrong. In the same years André-Marie Ampère reached the same hypothesis in Paris by a different route.

The memoir of 1814

In his second memoir Avogadro gave formulas, equivalent to modern ones, for carbon dioxide, carbon disulphide, sulphur dioxide and hydrogen sulphide. By analogy with carbonic acid he inferred that silica is SiO\(_2\), against Berzelius, who wrote SiO\(_3\). He proposed molecular weights for many metals, including mercury, iron, silver, gold, lead, copper, potassium and sodium. For these he had no gas densities, and he chose between alternatives by the simplicity of the resulting formulas of oxides and salts. Several values were off by factors of two.

Ampère’s version

In 1814 André-Marie Ampère published, as a letter to Claude-Louis Berthollet, a theory of chemical combination based on the same hypothesis. He reached it from the geometry of crystals: he imagined molecules as polyhedra of atoms and combined them according to the volume ratios. He did not cite Avogadro and probably did not know his paper. In France the hypothesis was often called Ampère’s; in Italy, Avogadro’s.

The memoir of 1821

In Nouvelles considérations sur la théorie des proportions déterminées dans les combinaisons (1821) Avogadro restated the hypothesis, reviewed the data, and extended it to organic compounds. He also argued for a link between molecular volume in the gas and the specific heat of the gas, a theme he pursued for the rest of his career (Lesson 10).

The kinetic theory proves the law

For the general reader. Why should equal volumes hold equal numbers? In the 1850s physicists pictured a gas as a swarm of tiny particles in rapid motion, bouncing off the walls. Their pressure comes from those impacts. If particles of every gas have the same average energy of motion at a given temperature, then equal pressures and volumes require equal numbers. Avogadro’s hypothesis became a theorem.

Pressure from impacts

Consider \(N\) molecules of mass \(m\) in a box of volume \(V\). A molecule with velocity component \(v_x\) striking a wall perpendicular to \(x\) transfers momentum \(2mv_x\). In a time \(\Delta t\), the molecules with \(v_x>0\) that hit an area \(A\) are those within distance \(v_x\Delta t\), numbering \(\tfrac NV A v_x\Delta t\) per velocity class. Averaging over directions, with half of the molecules moving towards the wall,

August Krönig (1856) and Rudolf Clausius (1857) gave this derivation. James Clerk Maxwell (1860) found the velocity distribution and showed that in a mixture of gases in equilibrium the average translational kinetic energy is the same for every species.

Clausius used the argument explicitly to defend Avogadro’s hypothesis. Writing \(N=nN_A\) and \(R=N_Ak\) gives the ideal-gas law \(pV=nRT\). The kinetic theory also explains Graham’s law of effusion: the rate at which gas escapes through a small hole is proportional to the mean speed, hence to \(1/\sqrt m\) at fixed temperature.

Cannizzaro and Karlsruhe

For the general reader. By 1860 chemists could not agree on the formula of water. Some wrote HO, some H₂O, some H₂O₂, and textbooks printed long lists of rival formulas for simple acids. In September 1860 more than a hundred chemists met at Karlsruhe to sort out the mess. At the end, Stanislao Cannizzaro’s pamphlet was handed out. It showed that Avogadro’s forgotten hypothesis, applied consistently, settled every atomic weight.

The Sunto (1858)

Cannizzaro, a professor at Genoa, wrote a course outline, the Sunto di un corso di filosofia chimica, published in 1858. He took Avogadro’s hypothesis as the foundation, used vapour densities to fix molecular weights, and drew a clear line between atoms and molecules. At Karlsruhe (3–5 September 1860) his friend Angelo Pavesi distributed copies. Lothar Meyer later wrote that reading it on his way home made everything clear.

Anomalous vapour densities

Some substances gave vapour densities half what their formulas predicted: ammonium chloride, phosphorus pentachloride, sulphuric acid. Opponents took these as refutations of Avogadro. Cannizzaro and Hermann Kopp argued that these vapours dissociate on heating. NH\(_4\)Cl vapour is a mixture of NH\(_3\) and HCl, with twice as many molecules and half the density. Leopold Pebal’s diffusion experiment of 1862 confirmed it for ammonium chloride. The other anomaly ran the opposite way: mercury vapour has a density matching a single atom per molecule, and sulphur vapour just above its boiling point is far denser than S\(_2\), approaching S\(_2\) only at high temperature. The elements need not be diatomic, but the hypothesis still holds.

Counting molecules: the Avogadro constant

For the general reader. Avogadro’s hypothesis says how many molecules a volume of gas contains relative to another, but not how many in absolute terms. That number, about 602 thousand billion billion per mole, was first estimated in 1865 and pinned down by Jean Perrin around 1909, who named it after Avogadro. Since 2019 it has been fixed by definition, and the mole is defined by counting.

Loschmidt (1865)

Josef Loschmidt combined two kinetic-theory facts. The mean free path \(\lambda=1/(\sqrt2\pi nd^2)\), obtained from viscosity, involves the number density \(n\) and the diameter \(d\). The volume of the liquid, where molecules nearly touch, gives the volume they occupy, roughly \(n\cdot\pi d^3/6\) per unit volume of gas. Two equations fix two unknowns. His estimate of the number of molecules per cubic centimetre of gas fell well below the modern value, but it was the first sound estimate of molecular scale; the number at 0 °C and 1 atm, \(2.687\times10^{19}\) per cm³, is now called the Loschmidt constant.

Perrin (1908–1909)

Jean Perrin prepared emulsions of uniform resin grains and counted them at different heights under a microscope. In equilibrium the number density follows the barometric law with the buoyancy-corrected weight of a grain:

Measuring the halving height gives \(N_A\). Perrin also used Albert Einstein’s 1905 theory of Brownian motion. In 1909 he proposed the name “Avogadro’s number” for the number of molecules in a gram-molecule (then 32 g of oxygen). He received the Nobel Prize in Physics for 1926.

Charge and crystals

Michael Faraday’s laws of electrolysis give the charge per mole of univalent ions, \(F\approx96\,485\) C/mol. Robert Millikan’s measurement of the electron charge \(e\) (1910 onwards) then gives \(N_A=F/e\). The most precise method uses a nearly perfect sphere of silicon-28: X-ray interferometry measures the lattice spacing \(a\), the sphere’s mass and volume give the density \(\rho\), and with 8 atoms per cubic cell, \(N_A=8M/(\rho a^3)\).

The physics of ponderable bodies

For the general reader. Avogadro thought of himself as a physicist. His four-volume Fisica de’ corpi ponderabili (1837–1841), on the physics of matter, was one of the first treatises of molecular physics. He studied specific heats, thermal expansion, vapour pressures and the volumes of atoms in solids, trying to find in them the same simple numerical relations that the gas volumes had shown.

Specific heats

In 1819 Pierre Louis Dulong and Alexis Thérèse Petit found that the specific heat of a solid element times its atomic weight is roughly constant. In modern units, the molar heat capacity of most solid elements near room temperature is close to \(3R\approx25\ \mathrm{J\,mol^{-1}K^{-1}}\). Avogadro worked on specific heats of compound gases (1816–1817) and of solids and liquids (1833), seeking a law that would fix atomic weights and formulas in the condensed state.

To force his data into simple laws, Avogadro assigned solids and liquids formulas with fractional atoms. He wrote liquid water, for example, as containing a quarter as many atoms as steam. These proposals did not survive, partly because the specific heats he relied on were inaccurate.

Atomic volumes and other work

Between 1843 and 1851 he published four memoirs on atomic volumes, the volume per mole of a solid or liquid. He compared them across families of compounds to estimate the volumes of individual atoms, a line of work that Hermann Kopp developed for organic liquids. His last memoir (1851) analysed Henri Victor Regnault’s measurements of the compressibility of gases, which showed that real gases deviate from Robert Boyle’s law. His early electrical memoirs on a non-conductor between charged plates anticipated the polarization picture of dielectrics later made precise by Michael Faraday.

The Turin chair, lost and regained, 1820–1850

The chair of mathematical physics created for Avogadro in 1820 was short-lived. In March 1821 liberal officers in Piedmont rose in favour of a constitution; the uprising was crushed within weeks, and the restored government purged the university. Avogadro’s chair was suppressed, and he lost it in July 1822. In 1823 he received the honorary title of professor emeritus. Some accounts link his dismissal to sympathy with the revolutionaries; the official explanation was that he needed rest from heavy teaching. He continued to research and publish, mostly through the Turin Academy.

In 1832 the government re-established the chair. It went first to the French mathematician Augustin-Louis Cauchy, who had left France after the July Revolution rather than swear allegiance to Louis-Philippe. Cauchy left Turin for Prague at the end of 1833, and on 28 November 1834 Avogadro was reappointed. He held the chair until he retired in 1850.

He also served the state. He sat on the government commission on statistics, presided over the commission on weights and measures, which introduced the metric system into Piedmont, and after 1848 served on the superior council of public instruction. He took part in the second congress of Italian scientists at Turin in 1840.

Last years, death and legacy, 1850–today

Avogadro retired from teaching in 1850 at 74 and continued to write; his memoir on Regnault’s gas measurements appeared in 1851. He died in Turin on 9 July 1856, aged 79. Recognition came only after his death. Four years later Cannizzaro’s campaign at Karlsruhe made the hypothesis the basis of chemistry, and within a decade Dmitri Mendeleev and Lothar Meyer built periodic tables on the atomic weights it produced.

Legacy

• Molecules: the distinction between atom and molecule, and the polyatomic molecules of the elements, which made consistent formulas and atomic weights possible.
• Physics: Avogadro’s law, derived from the kinetic theory by Clausius and Maxwell, is part of the ideal-gas law \(pV=nRT\).
• The constant: \(N_A\), named by Perrin in 1909 and fixed exactly in the SI since 2019, links the atomic and human scales. The mole, a base unit of the SI, is defined through it.
• Honours: in 1911 Turin marked the centenary of the essay with a celebration attended by King Victor Emmanuel III, and the Academy published his collected works. A lunar crater (named 1970) and the mineral avogadrite bear his name, and 23 October (from \(6.02\times10^{23}\)) is celebrated by chemistry teachers as Mole Day.

Sources and further reading

Biography

1. Amedeo Avogadro, Wikipedia.
2. M. P. Crosland, Avogadro, Amedeo, Dictionary of Scientific Biography, vol. 1, Scribner, 1970; online at Encyclopedia.com.
3. Amedeo Avogadro, biographical note, Department of Physics, University of Pisa.
4. M. Morselli, Amedeo Avogadro: A Scientific Biography, Reidel, 1984.

Science

1. A. Avogadro, Essai d’une manière de déterminer les masses relatives des molécules élémentaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons, Journal de physique 73 (1811), 58–76. English translation, Classic Chemistry, Le Moyne College.
2. A. Avogadro, Mémoire sur les masses relatives des molécules des corps simples, Journal de physique 78 (1814), 131–156.
3. A. Avogadro, Nouvelles considérations sur la théorie des proportions déterminées dans les combinaisons, Memorie della Reale Accademia delle Scienze di Torino 26 (1821).
4. A. Avogadro, Fisica de’ corpi ponderabili, ossia Trattato della costituzione materiale de’ corpi, 4 vols., Turin, 1837–1841.
5. A.-M. Ampère, Lettre de M. Ampère à M. le comte Berthollet, Annales de chimie 90 (1814).
6. S. Cannizzaro, Sunto di un corso di filosofia chimica, Il Nuovo Cimento 7 (1858); Alembic Club Reprint 18.
7. J. Perrin, Mouvement brownien et réalité moléculaire, Annales de chimie et de physique 18 (1909); Les Atomes, 1913.
8. Avogadro constant, Wikipedia; BIPM, The International System of Units, 9th edition, 2019.
9. A. J. Rocke, Chemical Atomism in the Nineteenth Century: From Dalton to Cannizzaro, Ohio State University Press, 1984.