Avogadro’s number, roughly 6.022 × 10²³, was not determined in a single eureka moment but through more than a century of increasingly precise experiments spanning at least half a dozen independent methods. The story runs from nineteenth-century kinetic theory through Brownian motion observations, electrochemistry, oil-drop experiments, radiation physics, and finally to ultra-precise measurements of near-perfect silicon spheres. Each method attacked the problem from a different angle, and the fact that they all converged on the same enormous figure is itself one of the strongest pieces of evidence that atoms and molecules are real, countable things.
Where the Idea Came From
In 1811, Amedeo Avogadro proposed that equal volumes of different gases, held at the same temperature and pressure, contain the same number of particles. He was trying to make sense of how gases combine in simple volume ratios during chemical reactions, and his hypothesis implied that those reactions could be understood as processes between individual atoms or molecules.1Reports on Progress in Physics. History and progress in the accurate determination of the Avogadro constant The idea was largely ignored for decades. Avogadro himself never tried to count how many particles were in a given amount of gas. He had no way to do so.
Things changed in 1860 when the Italian chemist Stanislao Cannizzaro revived Avogadro’s hypothesis at the first International Chemical Congress in Karlsruhe, Germany. The conference itself did not immediately settle the matter, but Cannizzaro distributed a pamphlet that proved persuasive. As the German chemist Lothar Meyer later wrote, reading it on the train home made “the scales fall from my eyes.”2Notes and Records of the Royal Society of London. Stanislao Cannizzaro, F. R. S. (1826-1910) and the first International Chemical Conference at Karlsruhe in 1860 Once chemists accepted consistent atomic weights, the question shifted from whether molecules existed to how many of them filled a mole of substance. Pinning down that number became one of the great experimental challenges of the next 150 years.
Planck and Black-Body Radiation
One of the earliest credible estimates came from an unexpected direction. Around 1900, Max Planck was working on the theory of thermal radiation, trying to explain the spectrum of light emitted by a hot object. His quantum theory of radiation introduced a tiny constant (now called the Boltzmann constant) that linked the average energy of a particle to the temperature of its surroundings. Planck realized that if you know the universal gas constant and the Boltzmann constant, you can simply divide one by the other to get the number of particles in a mole. His calculation yielded about 6.175 × 10²³, remarkably close to the modern value.3Revista Brasileira de Ensino de Física. Max Planck’s Determination of the Avogadro Constant
What made Planck’s estimate striking was that it came from radiation physics, not chemistry. He was studying the behavior of light, not counting molecules directly. The fact that his number agreed with chemical estimates reinforced the idea that the atomic picture of matter was consistent across different branches of physics.
Einstein, Brownian Motion, and Perrin’s Painstaking Counts
The determination that did the most to convince skeptics that atoms were real came from watching tiny particles jitter around in water. In 1905, Albert Einstein published a theoretical treatment showing that the random zigzag motion of microscopic particles suspended in a fluid (what we call Brownian motion) could be explained by collisions with surrounding molecules. He derived a mathematical relationship connecting the average distance a particle drifts over time to the temperature of the fluid, the viscosity, and the number of molecules involved.4The British Journal for the History of Science. The case of Brownian motion Crucially, that relationship contained Avogadro’s number as an unknown that could be solved for, if someone was patient enough to measure the motion directly.
That someone was Jean Perrin. Starting in 1908, Perrin used a microscope to track the positions of tiny particles of gamboge (a tree resin) suspended in water, recording their displacements frame by frame. It was extraordinarily tedious work. By measuring how far particles drifted on average in a known interval of time, he could plug the results into Einstein’s equation and calculate Avogadro’s number. His result landed in the range of about 6 × 10²³, close enough to confirm the theoretical prediction and earn him the 1926 Nobel Prize in Physics.5American Journal of Physics. Einstein, Perrin, and the reality of atoms: 1905 revisited
The significance here went beyond precision. Until Perrin’s experiments, a vocal minority of scientists (including the prominent physicist Ernst Mach) held that atoms were useful fictions, not physical realities. Perrin’s work, grounded in Einstein’s theory, made it very difficult to maintain that position. If you could watch the consequences of molecular collisions under a microscope, molecules had to be there.
The Electrochemical Route Through the Faraday Constant
A completely different strategy for reaching Avogadro’s number runs through electrochemistry. When you pass an electric current through a solution to deposit or dissolve a metal, there is a fixed relationship between the amount of charge used and the amount of substance deposited. The total charge needed to deposit one mole of a singly charged ion is called the Faraday constant. If you also know the charge on a single electron, you can divide the Faraday constant by the elementary charge and get the number of particles in a mole.
Getting the Faraday constant right demanded extremely careful experiments. At the U.S. National Bureau of Standards (now NIST), researchers measured it by dissolving ultra-pure silver in perchloric acid solutions, tracking exactly how much charge was passed and how much silver dissolved. Early work in 1960 yielded a Faraday value of about 96,490 coulombs per mole on the chemical scale.6PubMed Central. Determination of the Value of the Faraday With a Silver-Perchloric Acid Coulometer Later measurements refined this to 96,486.33 coulombs per mole with an uncertainty of just a few parts per million, using oxygen-free high-purity silver and comprehensive mass spectrometric impurity analysis.7PubMed Central. The Electrochemical Equivalent of Pure Silver-A Value of the Faraday
This electrochemical route was powerful because its systematic errors were completely different from those in Brownian motion or radiation experiments. You are measuring masses and electric currents, not tracking particle jitter or fitting radiation spectra. Agreement between such different methods was strong evidence that the answer was right. Refinements to this approach continued into the 2010s, with groups developing increasingly sophisticated cells for silver dissolution to push down uncertainty even further.8Measurement Science and Technology. A system for controlling electrical and chemical parameters in the Faraday constant determination by dissolution of silver
Oil Drops and the Elementary Charge
The electrochemical route depends on knowing the charge of a single electron, and that value has its own experimental history. Robert Millikan’s oil-drop experiment, conducted between 1909 and 1913, was the classic method. Millikan suspended tiny charged oil droplets between two metal plates, balanced them against gravity using an electric field, and from the balance point calculated the charge on each droplet. He found that the charges always came in whole-number multiples of a smallest unit, which he identified as the elementary charge. With that number in hand, dividing the Faraday constant by the elementary charge yielded Avogadro’s number directly. Millikan’s work provided one of the earliest high-precision routes to the constant, and he was awarded the Nobel Prize in 1923.
X-ray Crystallography and the Silicon Sphere
By the late twentieth century, the most precise approach to Avogadro’s number had moved to crystal physics. The idea is conceptually simple: if you know the spacing between atoms in a crystal and you know the volume and mass of a macroscopic piece of that crystal, you can count how many atoms are inside. In practice, this is ferociously difficult.
The international Avogadro project chose silicon as its crystal because silicon can be grown with extraordinary purity and structural perfection. The silicon used was not ordinary industrial silicon but material highly enriched in one isotope, silicon-28, to remove the complication of mixed isotopes muddying the mass measurements. Researchers ground these crystals into nearly perfect spheres, then measured their mass, volume, and lattice spacing with extreme care. The volume required knowing the shape to within a few nanometers across a sphere roughly the size of a tennis ball. Even the thin oxide layer that forms naturally on the silicon surface had to be characterized separately, because it differs in density from the crystal underneath and would throw off the calculation if ignored.9Metrologia. Surface layer determination for the Si spheres of the Avogadro project
The reward for this painstaking work was spectacular precision. By 2015, the x-ray crystal density method achieved a relative uncertainty of about two parts in a hundred million, the level needed to support a redefinition of the kilogram.10Metrologia. Improved measurement results for the Avogadro constant using a 28Si-enriched crystal This is like measuring the distance from New York to Los Angeles and being off by less than a millimeter. No previous method had come close to that kind of accuracy.
Why So Many Methods Matter
It is tempting to think of the silicon-sphere result as the “real” answer and everything that came before as stepping stones. But the history of these measurements matters for a deeper reason: each method probes a different physical phenomenon, and the fact that they all converge on the same number is itself a profound finding. Brownian motion measures the consequences of molecular bombardment. Electrochemistry measures how charge and mass relate at the atomic level. X-ray crystallography measures the geometry of atomic packing. Black-body radiation connects energy quanta to the particle count. If any one of these fundamental theories were wrong, its Avogadro estimate would diverge from the others. The convergence is really a consistency check on large swaths of physics and chemistry at once.
This is also why Perrin’s relatively crude 1908 measurements carried such philosophical weight, even though they were far less precise than what came later. The value did not need to be exact to the eighth decimal place to be revolutionary. It needed to be in the same ballpark as estimates from completely different physics. And it was.
From a Measured Constant to a Fixed Definition
For most of its history, Avogadro’s number was something you measured, and each new experiment slightly updated the best known value. That changed on May 20, 2019, when the International System of Units was revised. The mole was redefined so that it contains exactly 6.02214076 × 10²³ elementary entities, and the Avogadro constant was fixed at that precise value by international agreement.11Annalen der Physik. The Avogadro Constant for the Definition and Realization of the Mole Before this revision, the mole was defined via the mass of carbon-12, which tied it to the kilogram. The new definition broke that link, making the mole independent of any physical artifact.
This was possible only because decades of measurement had driven the uncertainty so low that the value could be frozen without disrupting practical chemistry or industry.12Metrologia. Amount of substance and the mole in the SI The fixed value, 6.02214076 × 10²³, is now part of the definition of the SI system, much as the speed of light was fixed at exactly 299,792,458 meters per second in 1983.13Measurement: Sensors. The SI unit mole and Avogadro constant
A subtle but important consequence: you can no longer “measure” Avogadro’s number and get a new, different answer. It is what it is, by definition. What you can do is use the silicon-sphere method (or any future technique) to realize the mole, meaning to create a physical sample that contains one mole of something with very high confidence. The direction of the logic has flipped: instead of measuring a perfect sphere to discover the number, you use the fixed number to check whether your sphere is as perfect as you think.
Common Misconceptions About the Number
One persistent misunderstanding is that Avogadro himself determined the number. He did not. He proposed the hypothesis that equal gas volumes hold equal particle counts, but he had no tools to measure how many. The number was not named after him until long after his death, as a tribute to the hypothesis that made counting meaningful in the first place.1Reports on Progress in Physics. History and progress in the accurate determination of the Avogadro constant
Another common confusion involves what the number refers to. Avogadro’s number is the count of entities in one mole of a substance, and those entities can be atoms, molecules, ions, or even electrons depending on context. It is not specifically “the number of atoms in 12 grams of carbon-12,” although that used to be the operational definition before the 2019 redefinition. Under the current SI, a mole is simply a fixed count, not tied to any particular substance.
People also sometimes assume that the number was chosen for some deep physical reason. It was not. It is an accident of the mass scale chemists happened to adopt. If the gram had been defined differently, or if chemists had built their system around a different reference element, the mole would contain a different number of particles, and we would have fixed a different value in 2019. The number is large because individual atoms are very light compared to the everyday mass units humans use. If you could somehow inflate an atom to the size of a blueberry, one mole of them would cover the Earth’s surface several kilometers deep.
Why the Silicon Spheres Are Still Being Refined
Even though the numerical value of Avogadro’s constant is now frozen, the experimental program that produced it has not stopped. The silicon sphere work continues because the same apparatus and techniques serve a new purpose: realizing the kilogram. Before 2019, the kilogram was defined by a platinum-iridium cylinder stored in a vault in Paris. Now it is defined through the Planck constant, and one way to realize it in practice is to use a silicon sphere whose atom count you know with extreme precision. The sphere’s mass can then be calculated from the known mass of individual silicon-28 atoms and the number of atoms it contains.
This means the research groups that spent decades chasing Avogadro’s number are now in the business of building the world’s most accurate mass standards. The two enriched silicon-28 spheres at the heart of the Avogadro project have become reference objects for national metrology labs around the world. Every refinement in measuring their surface layers, lattice spacing, or isotopic composition translates directly into a more accurate kilogram.9Metrologia. Surface layer determination for the Si spheres of the Avogadro project The surface oxide layer alone, just a nanometer or two thick, changes the sphere’s apparent mass enough to matter at the parts-per-hundred-million level. Characterizing that layer requires a battery of surface analysis techniques and remains an active area of research.10Metrologia. Improved measurement results for the Avogadro constant using a 28Si-enriched crystal
There is something satisfying about this arc. A number that began as a philosophical question in 1811, debated by chemists who could not even agree on the existence of atoms, ended up at the center of how we define fundamental units of measurement in the twenty-first century. The quest to count molecules turned out to be, in the end, a quest to weigh them.