How was Avogadro's number first determined?

The first estimate of Avogadro's number was made by a monk named Chrysostomus Magnenus in 1646. He burned a grain of incense in an abandoned church and assumed that there was one 'atom' of incense in his nose at soon as he could faintly smell it; He then compared the volume of the cavity of his nose with the volume of the church. In modern language, the result of his experiment was $N_A \ge 10^{22}$ ... quite amazing given the primitive setup.

Please remember that the year is 1646; the 'atoms' refer to Demokrit's ancient theory of indivisible units, not to atoms in our modern sense. I have this information from a physical chemistry lecture by Martin Quack at the ETH Zurich. Here are further references (see notes to page 4, in German): http://edoc.bbaw.de/volltexte/2007/477/pdf/23uFBK9ncwM.pdf

The first modern estimate was made by Loschmidt in 1865. He compared the mean free path of molecules in the gas phase to their liquid phase. He obtained the mean free path by measuring the viscosity of the gas and assumed that the liquid consists of densely packed spheres. He obtained $N_A \approx 4.7 \times 10^{23}$ compared to the modern value $N_A = 6.022 \times 10^{23}$.

The first undeniably reliable measurements of Avogadro's number came right at the turn of the twentieth century, with Millikan's measurement of the charge of the electron, Planck's blackbody radiation law, and Einstein's theory of Brownian motion.

Earlier measurements of Avogadro's number were really only estimates, they depended on the detailed model for atomic forces, and this was unknown. These three methods were the first model-independent ones, in that the answer they got was limited only by the experimental error, not by theoretical errors in the model. When it was observed that these methods gave the same answer three times, the existence of atoms became an established experimental fact.

Millikan

Faraday discovered the law of electrodeposition. When you run a current through a wire suspended in an ionic solution, as the current flows, material will deposit on the cathode and on the anode. what Faraday discovered is that the number of moles of the material is strictly proportional to the total charge that passes from one end to the other. Faraday's constant is the number of moles deposited per unit of charge. This law is not always right, sometimes you get half of the expected number of moles of material deposited.

When the electron was discovered in 1899, the explanation of Faraday's effect was obvious--- the ions in solution were missing electrons, and the current flowed from the negative cathode by depositing electrons on the ions in solution, thereby removing them from the solution and depositing them on the electrode. Then Faraday's constant is the charge on the electron times Avogadro's number. The reason that you sometimes get half the expected number of moles is that sometimes the ions are doubly-ionized, they need two electrons to become uncharged.

Millikan's experiment found the charge on the electron directly, by measuring the discreteness of the force on a droplet suspended in an electric field. This determined Avogadro's number.

Planck's blackbody law

Following Boltzmann, Planck found the statistical distribution of electromagnetic energy in a cavity using Boltzmann's distribution law: the probability of having energy E was $\exp(-E/kT)$. Planck also introduced Planck's constant to describe the discreteness of the energy of the electromagnetic oscillators. Both constants, k and h, could be extracted by fitting the known blackbody curves.

But Boltzmann's constant times Avogadro's number has a statistical interpretation, it is the "Gas constant" R you learn about in high school. So measuring Boltzmann's constant produces a theoretical value for Avogadro's number with no adjustible model parameters.

Einstein's diffusion law

A macroscopic particle in a solution obeys a statistical law--- it diffuses in space so that its average square distance from the starting point grows linearly with time. The coefficient of this linear growth is called the diffusion constant, and it seems hopeless to determine this constant theoretically, because it is determined by innumerable atomic collisions in the liquid.

But Einstein in 1905 discovered a fantastic law: that the diffusion constant can be understood immediately from the amount of friction force per unit velocity. The equation of motion for the Brownian particle is: $ m{d^2x\over dt^2} + \gamma {dx\over dt} + C\eta(t) $ = 0

Where m is the mass, $\gamma$ is the friction force per unit velocity, and $C\eta$ is a random noise that describes the molecular collisions. The random molecular collisions at macroscopic time scales must obey the law that they are independent Gaussian random variables at each time, because they are really the sum of many independent collisions which have a central limit theorem.

Einstein knew that the probability distribution of the velocity of the particle must be the Maxwell-Boltzmann distribution, by general laws of statistical thermodynamics:

$p(v) \propto e({-v^2\over 2mkT})$.

Ensuring that this is unchanged by the molecular noise force determines C in terms of m and kT.

Einstein noticed that the $d^2x\over dt^2$ term is irrelevant at long times. Ignoring the higher derivative term is called the "Smoluchowski approximation", although it is not really an approximation by a long-time exact description. It is explained here: Cross-field diffusion from Smoluchowski approximation, so the equation of motion for x is

$\gamma {dx\over dt} + C\eta = 0$,

and this gives the diffusion constant for x. The result is that if you know the macroscopic quantities $m,\gamma,T$, and you measure the diffusion constant to determine C, you find Boltzmann's constant k, and therefore Avogadro's number. This method required no photon assumption and no electron theory, it was based only on mechanics. The measurements on Brownian motion were carried out by Perrin a few years later, and earned Perrin the Nobel prize.

Avogadro's number was estimated at first only to order of magnitude precision, and then over the years by better and better techniques. Ben Franklin investigated thin layers of oil on water, but it was only realized later by Rayleigh that Franklin had made a monolayer: http://en.wikipedia.org/wiki/Langmuir%E2%80%93Blodgett_film If you know it's a monolayer, you can estimate the linear dimensions of a molecule and then get an order of magnitude estimate of Avogadro's number (or something equivalent to it). Some of the early estimates of the sizes and masses of molecules were based on viscosity. E.g., the viscosity of a dilute gas can be derived theoretically, and the theoretical expression depends on the scale of its atoms or molecules. Textbooks and popularizations often present a decades-long experimental program as a single experiment. Googling shows that Loschmidt did a whole bunch of different work on gases, including studies of diffusion, deviations from the ideal gas law, and liquified air. He seems to have studied these questions by multiple techniques, but it sounds like he got his best estimate of Avogadro's number from rates of diffusion of gases. It seems obvious to us now that setting the scale of atomic phenomena is an intrinsically interesting thing to do, but it was not always considered mainstream, important science in that era, and it didn't receive the kind of attention you'd expect. Many chemists considered atoms a mathematical model, not real objects. For insight into the science culture's attitudes, take a look at the story of Boltzmann's suicide. But this attitude doesn't seem to have been monolithic, since Loschmidt seems to have built a successful scientific career.