# How was Avogadro's number first determined?

I read on Wikipedia how the numerical value of Avogadro's number can be found by doing an experiment, provided you have the numerical value of Faraday's constant; but it seems to me that Faraday's constant could not be known before Avogadro's number was as it's the electric charge per mole. (How could we know the charge of a single electron just by knowing the charge of a mole of electrons, without knowing the ratio of the number of particles in both?)

I just want to know the method physically used, and the reasoning and calculations done by the first person who found the number $6.0221417930\times10^{23}$ (or however accurate it was first discovered to be).

Note: I see on the Wikipedia page for Avogadro constant that the numerical value was first obtained by "Johann Josef Loschmidt who, in 1865, estimated the average diameter of the molecules in air by a method that is equivalent to calculating the number of particles in a given volume of gas;" but I can't access any of the original sources that are cited. Can somebody explain it to me, or else give an accessible link so I can read about what exactly Loschmidt did?

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"how could you measure the charge per mole before you knew you had a mole?" You certainly do not need to know Avogadro's number to know that you have a mole of something!! If you want 1 mole of hydrogen gas, just measure out 1 gram of it. If you want 1 mole of water, measure out 18 grams of it. Etc. etc. – Steve B Jul 16 '12 at 15:23
@SteveB Exactly! I think what the OP meant was "How could we know the charge of a single electron just by knowing the charge of a mole of electrons without knowing the ratio of the number of particles in both?". Presumably, he didn't know that that was exactly how Avogadro's number was first discovered. – YatharthROCK Nov 17 '13 at 13:38
Pete, if you're still around, perhaps you could consider accepting another answer? The formerly accepted answer was deleted for being little more than a link. – David Z Sep 13 '14 at 23:38

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 $N_A = 6.022 \times 10^{23}$.

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Wow, Magnenus was awesome! thanks Felix for a most interesting bit of info. – WetSavannaAnimal aka Rod Vance Sep 5 '13 at 4:45
Do you have a citation for that last figure for Loschmidt's calculation? Everything else I read indicates he was only accurate to within about an order of magnitude. – Wedge Jun 11 at 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.

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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.

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There's also a (maybe small) push to just define Avogadro's number exactly as a fundamental constant, which would, if I understand correctly, also get rid of the problem of Le Grand K as a reference mass. See americanscientist.org/issues/pub/… – Willie Wong Aug 20 '11 at 14:34
This things were the work of Agnes Pockels! en.wikipedia.org/wiki/Agnes_Pockels – Georg Oct 22 '11 at 10:31

The Avogadro Number was discovered by Sir Michael Faraday but its importance and significance was realized much later by Avogadro while dealing with industrial synthesis and chemical reactions. In those days the chemists weren't aware of law of equal proportions which led to wastage of chemicals in industrial synthesis.

Faraday passed 96480 C of electricity throush hydrogen cations and found that 1gram hydrogen was formed. Then he analysed that when 1 electron with the charge of 1.6 X 10 to the power -19 coulombs gave 1 hydrogen atom then 96480C must give 6.023 X 10 to the power 23 atoms of hydrogen.

By this research scientists started calculating relative atomic masses of other atoms with respect to hydrogen. Later hydrogen became difficult for experiment, so C-12 was chosen for the determination of relative atomic masses.

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Interesting, did Faraday really know the electron charge at that time? – Nanite Sep 5 '13 at 6:46

In 1811, Avogadro states that equal volumes of different gases at the same temperature contain equal numbers of molecules.

Hydrogen gas is found to be 2 gram at 1 atm, 273 kelvin and 22.4 litre. At that time it is already known that I mole of hydrogen gas actually has two hydrogen atoms. So as a standard, one mole is defined as the number of atoms contained in 1 gram of hydrogen (or 2 grams of hydrogen gas).

To find the number of atoms in one mole, we need to find a relationship between the macroscopic (volume, pressure, temperature) data and the microscopic (no. of molecules) data. This is accomplished by the kinetic molecular theory and the ideal gas law. Kinetic molecular theory gives us a relationship between the kinetic energy of a molecule from the temperature. The collision of the molecules with the wall of the container is what gives us the pressure. Hence there is a relationship between no.of molecules and the pressure. We know all ideal gas have the same numbers of molecules in a constant pressure and volume, and we can substitute the conditions for our standard 1 gram hydrogen to find Avogadro’s constant.

From the ideal gas law

$PV = NK_bT\tag{1}$

where $K_b$ is the Boltzmann constant and $T$ the absolute temperature,

$$N= 101325 \times 0.0224 / (273 \times 1.3806 \times 10^{-23})= 6.022 \times 10^{23}$$

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