The Mole: The Story of a Number

In 1646, a physician named Jean Chrysostomus Magnenus posed an interesting puzzle in his work Democritus reviviscens sive de atomis. He asked readers to imagine burning a single grain of frankincense in a vast cathedral. The aromatic smoke would expand to fill the entire space, a volume perhaps millions of times larger than the tiny grain itself. His question: Could you determine how many individual particles of incense now filled that cathedral?

For Magnenus, this was purely philosophical speculation. The experimental and theoretical tools needed to answer his question would not be developed for another two centuries. His thought experiment does, however, illustrate the fundamental challenge that eventually drove the development of the mole concept. We will return to this problem at the end of this article.

The Crisis in Early Chemistry

By the early 1800s, chemistry faced a fundamental measurement problem. John Dalton had revived atomic theory in 1803, proposing that each element consisted of identical atoms with characteristic weights. He assembled the first table of atomic weights by analyzing the mass ratios in chemical compounds. If 8 grams of oxygen always combined with 1 gram of hydrogen to make water, and you assumed one atom of each, then oxygen atoms must be eight times heavier than hydrogen atoms.

This approach worked for establishing relative weights, but it had a critical limitation. Chemists had a table of relative atomic masses but no absolute reference point. They knew oxygen atoms were 16 times heavier than hydrogen atoms, but they had no way to determine the actual mass of either atom. More problematically, they could not determine how many atoms were present in any weighable laboratory sample. To use an analogy, it was like having a recipe that only specified proportions: “Use flour that weighs 15 times your butter and sugar that weighs 10 times your butter.” You could maintain the ratios, but you would have no way to scale the recipe appropriately.

The situation became more complicated in 1808 when Joseph Louis Gay-Lussac published his studies on gas reactions. He found that gases combined in simple whole-number ratios by volume. Two volumes of hydrogen reacted with exactly one volume of oxygen to produce water. Two volumes of carbon monoxide combined with one volume of oxygen to make two volumes of carbon dioxide. The patterns appeared systematic, but they seemed to contradict Dalton’s atomic weights.

The fundamental problem was molecular formulas. Was water HO or H₂O? Was the air we breathe composed of single atoms of nitrogen and oxygen, or diatomic molecules? Without knowing the number of particles in a measurable volume, chemists could not determine molecular formulas. Without molecular formulas, they could not correctly assign atomic weights. The field was stuck in circular reasoning.

Avogadro’s Hypothesis and Fifty Years of Silence

In 1811, an Italian physicist named Amedeo Avogadro published a hypothesis ([Original]; [English]) that would eventually resolve the confusion: equal volumes of all gases, measured at the same temperature and pressure, contain the same number of particles.

Consider the implications. If one liter of hydrogen and one liter of chlorine at room temperature and pressure contained the same number of molecules (despite chlorine being much heavier), then Gay-Lussac’s volume ratios became interpretable at the molecular level. When two volumes of hydrogen reacted with one volume of oxygen to make water, it meant two particles of hydrogen combined with one particle of oxygen. However, this required that gases exist as diatomic molecules (H₂, O₂), not individual atoms. The reaction was 2 H₂ + O₂ → 2 H₂O.

Avogadro had essentially provided a counting tool. If you could measure the volume of a gas, you knew something about the number of particles it contained, even if you did not know the absolute value of that number. One liter of any gas contained some fixed number N of molecules. Two liters contained 2N. The absolute value of N remained unknown, but the concept was significant.

The chemistry community, however, was not prepared to accept this hypothesis. Dalton himself rejected the idea of diatomic elements, finding no compelling reason why two identical atoms would bind together. The influential Swedish chemist Jöns Jacob Berzelius developed his own competing system. For nearly fifty years, Avogadro’s work received little attention while chemistry remained fragmented. Different chemists used different atomic weights and different molecular formulas. The same compound might be written three different ways in three different textbooks.

The resolution came in 1860 at the first International Chemical Congress in Karlsruhe, Germany. Stanislao Cannizzaro, an Italian chemist, presented a systematic analysis showing how Avogadro’s hypothesis resolved contradictions in atomic weight determinations. He distributed a pamphlet that methodically worked through the logic. Water was H₂O, not HO. Oxygen gas was O₂. The atomic weight of carbon was 12, not 6.

The German chemist Lothar Meyer later described reading Cannizzaro’s pamphlet: “It was as though scales fell from my eyes.” Within a decade, Avogadro’s hypothesis gained general acceptance and became known as Avogadro’s law. Chemistry finally had a consistent system of atomic weights and molecular formulas. However, the number itself (the actual count of particles in a given volume or mass) remained experimentally undetermined.

The Hunt for the Number

Once chemistry had a reliable system of atomic weights, the question became quantitatively important: how many atoms are actually in 12 grams of carbon, or 16 grams of oxygen, or 1 gram of hydrogen? Avogadro’s law guaranteed these amounts all contained the same number of particles, but that number remained unknown.

The first person to calculate a numerical value was an Austrian physicist named Johann Josef Loschmidt. In 1865, using the kinetic theory of gases, he estimated the number of molecules in a cubic centimeter of gas at standard temperature and pressure (now called the Loschmidt constant). His approach combined measurements of gas viscosity and diffusion rates with theoretical calculations about molecular speeds and mean free paths (the average distance a molecule travels between collisions). When converted using modern definitions, his estimate corresponds to approximately 6 × 10²³ particles per mole, remarkably close to the accepted value. This was the first concrete estimate of the magnitude of atomic-scale counting.

However, many scientists remained skeptical. Atoms were still theoretical constructs to some researchers, useful for explaining chemical reactions but not necessarily real physical objects. The decisive experimental evidence came from studies of Brownian motion.

In 1827, the botanist Robert Brown had observed that tiny pollen grains suspended in water exhibited continuous random motion. The particles appeared to move spontaneously with no external driving force. For decades, this phenomenon lacked a satisfactory explanation. Then in 1905, Albert Einstein published a theoretical paper showing that Brownian motion results from collisions with individual water molecules. The random thermal motion of molecules produces a measurable statistical pattern in the movement of suspended particles. Importantly, his equations connected observable particle displacement to molecular bombardment, and the mathematics contained Avogadro’s number as a parameter.

Jean Baptiste Perrin, a French physicist, recognized that Einstein’s theory provided an experimental method for determining Avogadro’s number. Between 1908 and 1913, Perrin performed a series of careful experiments. He prepared suspensions of resin spheres with uniform diameter (approximately 0.5 μm). Using a microscope, he tracked individual spheres, recording their positions at regular intervals to measure their displacement. The motion was random (a particle might move left, then right, then up), but Einstein had predicted that the average displacement over many observations would follow a statistical distribution that depended on Avogadro’s number.

Perrin’s measurements from this method gave a value of approximately 6.0 × 10²³. He then applied an entirely different experimental approach based on gravitational settling. In a column of water, the resin spheres settle toward the bottom due to gravity, but thermal motion drives them upward. At equilibrium, the concentration of spheres decreases with height following a relationship analogous to the barometric equation for atmospheric pressure. Avogadro’s number again appeared as a parameter in the mathematical expression. From concentration gradient measurements, Perrin calculated a value of 6.5 × 10²³.

He applied yet another method based on light scattering from colloidal suspensions, obtaining a value near 6 × 10²³. The convergence of these independent experimental techniques on the same value provided compelling evidence. In 1926, Perrin was awarded the Nobel Prize in Physics for this work. His experiments both determined Avogadro’s number with reasonable precision and established the physical reality of atoms and molecules. It was Perrin who proposed the name “Avogadro’s number” (N_A) to honor the Italian physicist whose hypothesis had made such determinations possible.

Why This Specific Number?

The value of Avogadro’s constant (6.022 × 10²³ mol⁻¹) often appears arbitrary. Why not a round number like 1 × 10²⁴? The answer lies in historical definitions of mass units.

(A note on terminology: “Avogadro’s number” typically refers to the dimensionless numerical value 6.022 × 10²³, while “Avogadro’s constant” (N_A) is the physical constant with units of mol⁻¹. In practice, the terms are often used interchangeably.)

The number emerges from the definition of atomic mass units. In the early 20th century, chemists chose carbon-12 as their reference standard and defined its atomic mass as exactly 12 atomic mass units (amu). One atomic mass unit was therefore defined as exactly 1/12 the mass of a single carbon-12 atom. This choice was practical: carbon is abundant, carbon-12 is the most common isotope, and setting its mass to 12 preserved the approximate integer values that chemists had been using.

The connection to Avogadro’s constant follows from how the mole was defined: the number of atoms in exactly 12 grams of carbon-12. Since each carbon-12 atom has a mass of 12 amu, the ratio between grams and atomic mass units is fixed. One gram equals exactly N_A atomic mass units, where N_A is Avogadro’s number.

To see why the value is what it is, consider that one atomic mass unit equals approximately 1.66054 × 10⁻²⁴ g. To convert one gram into atomic mass units: 1 g ÷ (1.66054 × 10⁻²⁴ g amu⁻¹) ≈ 6.022 × 10²³ amu. This is Avogadro’s constant. The value is simply the conversion factor between two mass scales: the macroscopic gram and the atomic mass unit.

If chemists had originally chosen a different reference (for example, defining hydrogen as exactly 1.0000 rather than carbon as 12.0000), or if the gram had been defined differently, Avogadro’s number would have a different value. There is nothing physically fundamental about 6.022 × 10²³. It is a consequence of unit definitions. What matters is the concept: the mole provides a conversion between the atomic scale and the laboratory scale.

The 2019 Redefinition: From Measurement to Declaration

For most of the 20th century, the mole was defined in terms of a physical substance: the amount of substance containing as many particles as there are atoms in exactly 12 grams of carbon-12. This definition worked well in practice and provided a tangible reference. When Avogadro’s number was needed, it was measured using techniques such as X-ray crystallography (counting atoms in precisely machined silicon spheres) or by measuring fundamental constants that contained N_A as a parameter. By the early 2000s, the value was known to about nine significant figures.

However, there was a conceptual issue. The mole’s definition depended on the kilogram, which itself was defined by a physical artifact: a platinum-iridium cylinder maintained in a vault near Paris. If that cylinder’s mass changed (even by micrograms due to contamination or cleaning), the definition of the mole would shift. More fundamentally, basing units on physical objects seemed increasingly problematic compared to defining them in terms of fundamental constants.

In 2019, the International System of Units underwent a major revision. The scientific community, coordinated through CODATA (the Committee on Data for Science and Technology), inverted the logic of several base units. Instead of defining the mole and then measuring Avogadro’s constant, the approach was reversed: fix Avogadro’s constant by definition and let the mole follow from it.

The process involved making the most precise measurements of Avogadro’s constant possible using all available techniques, yielding a value with high precision: 6.02214076 × 10²³ mol⁻¹. On May 20, 2019, this number was declared exact by definition. The Avogadro constant is now a defined constant, like the speed of light.

Under the new system:

Avogadro’s constant is exactly 6.02214076 × 10²³ mol⁻¹ (defined, not measured)
The mole is defined as the amount of substance containing exactly 6.02214076 × 10²³ elementary entities
The kilogram is now defined in terms of Planck’s constant (another fixed value), eliminating the circular dependence

From a practical laboratory standpoint, nothing changed. The numerical value remained the same within experimental uncertainty. However, the conceptual change was significant. The mole is no longer tied to a physical sample of carbon-12. It is a fixed numerical scaling factor between the atomic world and the macroscopic world. Even if every sample of carbon-12 were destroyed, the mole would remain defined.

The Scale of a Mole

The magnitude of Avogadro’s number (6.022 × 10²³) is difficult to grasp intuitively. Some comparisons may help illustrate the scale:

One mole of water molecules has a mass of 18 grams (approximately one tablespoon). This tablespoon contains roughly as many molecules as there are stars in the observable universe (current estimates range from 10²³ to 10²⁴ stars).

If you had a mole of rice grains and spread them evenly over Earth’s surface, they would form a layer approximately 120 kilometers deep.

If you could count one atom per second without stopping, it would require nearly 2 × 10¹⁶ years to count one mole, which is over a million times the current age of the universe.

Despite this enormous magnitude, chemists work with mole quantities routinely. Measuring out 58.5 grams of table salt (NaCl) gives you one mole, or 6.022 × 10²³ formula units. Pipetting a 1.0 M solution means dispensing 6.022 × 10²³ particles per liter. The mole concept provides the practical link between individual particles and laboratory-scale quantities.

Returning to the Cathedral

We can now return to Jean Chrysostomus Magnenus and his frankincense thought experiment. In 1646, his question was unanswerable. How many particles filled the cathedral? He had no experimental or theoretical tools to approach the problem.

With the concepts developed over the subsequent two centuries, the problem becomes tractable. Suppose a grain of frankincense has a mass of 50 mg (0.050 g). The primary aromatic compounds in frankincense are terpenes and boswellic acids. A representative molecule, boswellic acid, has the formula C₃₀H₄₈O₃ and a molar mass of approximately 456 g mol⁻¹.

The calculation proceeds as follows: 0.050 g ÷ 456 g mol⁻¹ ≈ 1.1 × 10⁻⁴ mol. Multiplying by Avogadro’s constant gives the number of molecules: (1.1 × 10⁻⁴ mol) × (6.022 × 10²³ molecules mol⁻¹) ≈ 6.6 × 10¹⁹ molecules.

Magnenus’s cathedral contained approximately 66 quintillion aromatic molecules from that single grain of incense. His thought experiment, posed when atoms were purely speculative, now has a quantitative answer. The path from his question to this answer required the contributions of Dalton, Gay-Lussac, Avogadro, Cannizzaro, Perrin, and many others. The mole represents the culmination of this historical development.

Why This History Matters

This historical development has direct practical relevance. Every time you use a periodic table, you are applying the framework built by these historical contributions. When you see that carbon has a molar mass of 12.011 g mol⁻¹, you are using a value that connects a single atom (mass ≈ 12 amu) to a laboratory quantity (12.011 grams) through Avogadro’s constant. This connection allows translation between molecular-level mechanisms and macroscopic measurements.

The mole concept is distinctive to chemistry. Physics primarily deals with forces and fields. Biology focuses on living systems. Chemistry operates simultaneously at two scales: the molecular level where reactions occur through individual collisions, and the macroscopic level where measurements are made in grams and liters. The mole provides the quantitative link between these scales. It exists because generations of scientists developed both the theoretical framework and experimental methods needed to bridge the atomic and laboratory worlds. What seemed uncountable in 1646 became countable by the early 20th century, and we now use these particle counts routinely in chemical calculations.