Skills to Develop

Make sure you thoroughly understand the following essential ideas:

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• Define Avogadro's number and explain why it is important to know.
• Define the mole. Be able to calculate the number of moles in a given mass of a substance, or the mass corresponding to a given number of moles.
• Define molecular weight, formula weight, and molar mass; explain how the latter differs from the first two.
• Be able to find the number of atoms or molecules in a given weight of a substance.
• Find the molar volume of a solid or liquid, given its density and molar mass.
• Explain how the molar volume of a metallic solid can lead to an estimate of atomic diameter.

The chemical changes we observe always involve discrete numbers of atoms that rearrange themselves into new configurations. These numbers are HUGE— far too large in magnitude for us to count or even visualize, but they are still numbers, and we need to have a way to deal with them. We also need a bridge between these numbers, which we are unable to measure directly, and the weights of substances, which we do measure and observe. The mole concept provides this bridge, and is central to all of quantitative chemistry.

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2. They determined “Avogadro’s number”, which is defined as the number of molecules contained in one mole of the substance. The Loschmidt constant is barely used, but Avogadro's number (6.022×10 23 ) finds wide application in physics and chemistry.

Counting Atoms: Avogadro's Number

Owing to their tiny size, atoms and molecules cannot be counted by direct observation. But much as we do when 'counting' beans in a jar, we can estimate the number of particles in a sample of an element or compound if we have some idea of the volume occupied by each particle and the volume of the container. Once this has been done, we know the number of formula units (to use the most general term for any combination of atoms we wish to define) in any arbitrary weight of the substance. The number will of course depend both on the formula of the substance and on the weight of the sample. However, if we consider a weight of substance that is the same as its formula (molecular) weight expressed in grams, we have only one number to know: Avogadro's number.

Avogadro's number

How Was Avogadro's Number Found

Avogadro's number is known to ten significant digits:

[N_A = 6.022141527 times 10^{23}.]

However, you only need to know it to three significant figures:

[N_A approx 6.02 times 10^{23}. label{3.2.1}]

So (6.02 times 10^{23}) of what? Well, of anything you like: apples, stars in the sky, burritos. However, the only practical use for (N_A) is to have a more convenient way of expressing the huge numbers of the tiny particles such as atoms or molecules that we deal with in chemistry. Avogadro's number is a collective number, just like a dozen. Students can think of (6.02 times 10^{23}) as the 'chemist's dozen'.

Before getting into the use of Avogadro's number in problems, take a moment to convince yourself of the reasoning embodied in the following examples.

Example (PageIndex{1}): Mass ratio from atomic weights

The atomic weights of oxygen and carbon are 16.0 and 12.0 atomic mass units ((u)), respectively. How much heavier is the oxygen atom in relation to carbon?

Solution

Atomic weights represent the relative masses of different kinds of atoms. This means that the atom of oxygen has a mass that is

[dfrac{16, cancel{u}}{12, cancel{u}} = dfrac{4}{3} ≈ 1.33 nonumber]

as great as the mass of a carbon atom.

Example (PageIndex{2}): Mass of a single atom

The absolute mass of a carbon atom is 12.0 unified atomic mass units ((u)). How many grams will a single oxygen atom weigh?

Solution

The absolute mass of a carbon atom is 12.0 (u) or

[12,cancel{u} times dfrac{1.6605 times 10^{–24}, g}{1 ,cancel{u}} = 1.99 times 10^{–23} , g text{ (per carbon atom)} nonumber]

The mass of the oxygen atom will be 4/3 greater (from Example (PageIndex{1})):

[ left( dfrac{4}{3} right) 1.99 times 10^{–23} , g = 2.66 times 10^{–23} , g text{ (per oxygen atom)} nonumber]

Alternatively we can do the calculation directly like with carbon:

[16,cancel{u} times dfrac{1.6605 times 10^{–24}, g}{1 ,cancel{u}} = 2.66 times 10^{–23} , g text{ (per oxygen atom)} nonumber]

Example (PageIndex{3}): Relative masses from atomic weights

Suppose that we have (N) carbon atoms, where (N) is a number large enough to give us a pile of carbon atoms whose mass is 12.0 grams. How much would the same number, (N), of oxygen atoms weigh?

Solution

We use the results from Example (PageIndex{1}) again. The collection of (N) oxygen atoms would have a mass of

[dfrac{4}{3} times 12, g = 16.0, g. nonumber]

Exercise (PageIndex{1})

What is the numerical value of (N) in Example (PageIndex{3})?

Using the results of Examples (PageIndex{2}) and (PageIndex{3}).

[N times 1.99 times 10^{–23} , g text{ (per carbon atom)} = 12, g nonumber]

or

[N = dfrac{12, cancel{g}}{1.99 times 10^{–23} , cancel{g} text{ (per carbon atom)}} = 6.03 times 10^{23} text{atoms} nonumber ]

There are a lot of atoms in 12 g of carbon.

Things to understand about Avogadro's number

• It is a number, just as is 'dozen', and thus is dimensionless.
• It is a huge number, far greater in magnitude than we can visualize
• Its practical use is limited to counting tiny things like atoms, molecules, 'formula units', electrons, or photons.
• The value of N_A can be known only to the precision that the number of atoms in a measurable weight of a substance can be estimated. Because large numbers of atoms cannot be counted directly, a variety of ingenious indirect measurements have been made involving such things as Brownian motion and X-ray scattering.
• The current value was determined by measuring the distances between the atoms of silicon in an ultrapure crystal of this element that was shaped into a perfect sphere. (The measurement was made by X-ray scattering.) When combined with the measured mass of this sphere, it yields Avogadro's number. However, there are two problems with this:
  • The silicon sphere is an artifact, rather than being something that occurs in nature, and thus may not be perfectly reproducible.
  • The standard of mass, the kilogram, is not precisely known, and its value appears to be changing. For these reasons, there are proposals to revise the definitions of both N_A and the kilogram.

Moles and their Uses

The mole (abbreviated mol) is the the SI measure of quantity of a 'chemical entity', which can be an atom, molecule, formula unit, electron or photon. One mole of anything is just Avogadro's number of that something. Or, if you think like a lawyer, you might prefer the official SI definition:

Definition: The Mole

The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12

Avogadro's number (Equation ref{3.2.1}) like any pure number, is dimensionless. However, it also defines the mole, so we can also express N_A as 6.02 × 10²³ mol^–1; in this form, it is properly known as Avogadro's constant. This construction emphasizes the role of Avogadro's number as a conversion factor between number of moles and number of 'entities'.

Example (PageIndex{4}): number of moles in N particles

How many moles of nickel atoms are there in 80 nickel atoms?

Solution

[dfrac{80 ;atoms}{6.02 times 10^{23} ; atoms; mol^{-1}} = 1.33 times 10^{-22} mol nonumber]

Is this answer reasonable? Yes, because 80 is an extremely small fraction of (N_A).

Molar Mass

The atomic weight, molecular weight, or formula weight of one mole of the fundamental units (atoms, molecules, or groups of atoms that correspond to the formula of a pure substance) is the ratio of its mass to 1/12 the mass of one mole of C¹² atoms, and being a ratio, is dimensionless. But at the same time, this molar mass (as many now prefer to call it) is also the observable mass of one mole (N_A) of the substance, so we frequently emphasize this by stating it explicitly as so many grams (or kilograms) per mole: g mol^–1.

It is important always to bear in mind that the mole is a number and not a mass. But each individual particle has a mass of its own, so a mole of any specific substance will always correspond to a certain mass of that substance.

Example (PageIndex{5}): Boron content of borax

Borax is the common name of sodium tetraborate, (ce{Na2B4O7}).

1. how many moles of boron are present in 20.0 g of borax?
2. how many grams of boron are present in 20.0 g of borax?

Solution

The formula weight of (ce{Na2B4O7}) so the molecular weight is:

[(2 times 23.0) + (4 times 10.8) + (7 times 16.0) = 201.2 nonumber]

1. 20 g of borax contains (20.0 g) ÷ (201 g mol^–1) = 0.10 mol of borax, and thus 0.40 mol of B.
2. 0.40 mol of boron has a mass of (0.40 mol) × (10.8 g mol^–1) = 4.3 g.

Example (PageIndex{6}): Magnesium in chlorophyll

The plant photosynthetic pigment chlorophyll contains 2.68 percent magnesium by weight. How many atoms of Mg will there be in 1.00 g of chlorophyll?

Solution

Each gram of chlorophyll contains 0.0268 g of Mg, atomic weight 24.3.

• Number of moles in this weight of Mg: (.0268 g) / (24.2 g mol^–1) = 0.00110 mol
• Number of atoms: (0.00110 mol) × (6.02E23 mol^–1) = (6.64 times 10^{20})

Is this answer reasonable? (Always be suspicious of huge-number answers!) Yes, because we would expect to have huge numbers of atoms in any observable quantity of a substance.

Molar Volume

This is the volume occupied by one mole of a pure substance. Molar volume depends on the density of a substance and, like density, varies with temperature owing to thermal expansion, and also with the pressure. For solids and liquids, these variables ordinarily have little practical effect, so the values quoted for 1 atm pressure and 25°C are generally useful over a fairly wide range of conditions. This is definitely not the case with gases, whose molar volumes must be calculated for a specific temperature and pressure.

Example (PageIndex{7}): Molar Volume of a Liquid

Methanol, CH₃OH, is a liquid having a density of 0.79 g per milliliter. Calculate the molar volume of methanol.

Solution

The molar volume will be the volume occupied by one molar mass (32 g) of the liquid. Expressing the density in liters instead of mL, we have

[V_M = dfrac{32; g; mol^{–1}}{790; g; L^{–1}}= 0.0405 ;L ;mol^{–1} nonumber]

The molar volume of a metallic element allows one to estimate the size of the atom. The idea is to mentally divide a piece of the metal into as many little cubic boxes as there are atoms, and then calculate the length of each box. Assuming that an atom sits in the center of each box and that each atom is in direct contact with its six neighbors (two along each dimension), this gives the diameter of the atom. The manner in which atoms pack together in actual metallic crystals is usually more complicated than this and it varies from metal to metal, so this calculation only provides an approximate value.

Example (PageIndex{8}): Radius of a Strontium Atom

The density of metallic strontium is 2.60 g cm^–3. Use this value to estimate the radius of the atom of Sr, whose atomic weight is 87.6.

Solution

The molar volume of Sr is:

[dfrac{87.6 ; g ; mol^{-1}}{2.60; g; cm^{-3}} = 33.7; cm^3; mol^{–1}]

The volume of each 'box' is'

[dfrac{33.7; cm^3 mol^{–1}} {6.02 times 10^{23}; mol^{–1}} = 5.48 times 10^{-23}; cm^3]

The side length of each box will be the cube root of this value, (3.79 times 10^{–8}; cm). The atomic radius will be half this value, or

[1.9 times 10^{–8}; cm = 1.9 times 10^{–10}; m = 190 pm]

Note: Your calculator probably has no cube root button, but you are expected to be able to find cube roots; you can usually use the x^y button with y=0.333. You should also be able estimate the magnitude of this value for checking. The easiest way is to express the number so that the exponent is a multiple of 3. Take (54 times 10^{-24}), for example. Since 3³=27 and 4³ = 64, you know that the cube root of 55 will be between 3 and 4, so the cube root should be a bit less than 4 × 10^–8.

So how good is our atomic radius? Standard tables give the atomic radius of strontium is in the range 192-220 pm.

Contributors

• Stephen Lower, Professor Emeritus (Simon Fraser U.) Chem1 Virtual Textbook

Avogadro's number, N_A, is defined to be exactly^[1]N_A = 6. 022 140 76 x 10²³.

Previously, N_A had been defined as the number of atoms in 12 grams of carbon-12 atoms in their ground state at rest. With the redefinition in 2019 of the kilogram in terms of fundamental constants, Avogadro's number was set at the above value.

The SI definition of Avogadro's constant (also designated N_A) is: the number of entities (such as atoms, ions, or molecules) per mole. In this definition N_A has dimension mol⁻¹. The numeric value of Avogadro's constant is N_A = 6.022 140 76 x 10²³ mol⁻¹.

Avogadro's constant and number have by definition the same numerical value. In practice the two terms are used interchangeably.

The symbol L, for Loschmidt constant, is also a recognized SI symbol with the same value as N_A.

Avogadro's constant N_A is named after the Italian physicist Amedeo Avogadro, and the Loschmidt constant L is named after the Czech-Austrian physicist Josef Loschmidt.

Remark: It has been argued that the genitive in 'Avogadro's constant' is incorrect, because its value was first estimated 10 years after Avogadro's death and the name was given half a century after Avogadro's death. If preferred, one may read 'Avogadro constant' and 'Avogadro number' everywhere, instead of the corresponding genitive forms used in this article.

• 1History of Avogadro's number

History of Avogadro's number

Since 1811, when Amedeo Avogadro put forward his law stating that equal volumes of gas (strictly ideal gas) contain equal number of particles,^[2] increasingly sophisticated methods of determining Avogadro’s constant have been developed. These include the kinetic theory of gases, properties of liquid solutions, measurement of the electron charge, black-body radiation, alpha particle emission, and X-ray measurements of crystals.

Without the belief that a macroscopic substance consists of minute particles (initially called atoms, later also molecules), it does not make sense to speak of Avogadro's number.This belief—called atomism— was born in antiquity and grew further in importance with the developments of chemistry early in the 19th century. An important milestone was John Dalton’s law of multiple proportions published in 1804 that gave rise to the first table of the relative weights of atoms. In 1808 Joseph-Louis Gay-Lussac published his law for the combining volumes of gases, namely that gases combine among themselves in very simple proportions of their volumes, and if the products are gases, their volumes are also in simple proportions. For instance, 1 liter of oxygen gas combines with 2 liters of hydrogen gas to form 2 liter of gaseous water. Especially Gay-Lussac's law was of great influence on Avogadro's historical publication of 1811 in which he enunciated his law.

In his 1811 paper Avogadro discusses Gay-Lussac's law and Dalton’s atomic theory. He calculates from gas densities that the molecular weight of nitrogen is 13.238 times the molecular weight of hydrogen (the modern value is 14). Avogadro was the first to propose that the gaseous elements, hydrogen, oxygen, and nitrogen, are diatomic molecules. He deduces that a molecule of water contains a molecule of oxygen and two molecules of hydrogen. Dalton, who had assumed earlier that water is formed from a molecule each of oxygen and hydrogen, rejects Avogadro's and Gay-Lussac's laws.

There are no testimonials that Avogadro ever speculated on the number of molecules in a given gas volume and his law went for a long time largely unnoticed, not in the least because it was not recognized that the law holds strictly only for ideal gases, which many dissociating and associating organic compounds are not. Four years after Avogadro's death, at the historic (1860) chemistry conference in Karlsruhe, his fellow-countryman Stanislao Cannizaro explained why the exceptions to Avogadro's law happen, why the law is important, and why Avogadro deserves the credit for it.

In the beginning of the twentieth century some scientists (the most notable ones being Friedrich Wilhelm Ostwald and Ernst Mach) still denied the existence of molecules. As discussed by Pais,^[3] the large number of measurements, based on completely different phenomena, that all led to basically the same value of Avogadro's constant, finally convinced Ostwald of the reality of molecules. Mach died in 1916 as disbeliever. The different experiments for determining N_A will be briefly reviewed.

Estimates from kinetic gas theory

The first estimate of Avogadro’s constant was performed by Loschmidt (1865, 1866).^[4] He gave the value 0.969 nm for the diameter of 'air molecules' and an equation that relates this diameter to the number L of molecules in one cm³ at standard temperature and pressure. The number L is called Loschmidt’s number; the Avogadro equivalent of Loschmidt's estimate is: N_A = 0.4×10²³. (Note that at present the SI definitions of L and N_A are the same, i.e., the different symbols stand for the same physical quantity). Loschmidt obtained this number by applying the kinetic gas theory of James Clerk Maxwell and Rudolf Clausius, together with experimental data on mean lengths of free paths of molecules in gases and molecular volumes of nitrogen-oxygen compounds. From the latter he estimated the size of the air molecules oxygen and nitrogen. His value 0.969 nm was a factor three too high, and since his equation for L has an inverse-square dependence on the diameter, his estimate for L was an order of magnitude too low. But in any case, Loschmidt was the first to show that Avogadro’s constant is very large and molecules are very small. In 1873 Maxwell used his own kinetic theory of the diffusion coefficient of a gas to obtain a ten times larger value: N_A = 4.2×10²³.

A simple method for getting the actual volume of molecules is by using the Van der Waals equation (1873) that contains a parameter b, the volume of a single molecule. From b and the volume of the total gas, an estimate of the number of molecules in the gas can be obtained. In the early twentieth century Perrin measured b for mercury vapor, and combining this with results from viscosity measurements, he calculated Avogadro's number to be 6.2×10²³,^[5]which is a very good value.

Estimates from liquid solutions

Einstein wrote in his 1905 Ph.D. thesis about the size of molecules and the closely related problem of the magnitude of N_A. He derived equations for diffusion coefficients and viscosities in which Avogadro's number appears. From experimental values of the diffusion coefficients and viscosities of sugar solutions in water Einstein gave the estimate N_A = 2.1×10²³. In a later paper derived from his doctorate work^[6] he gave a better estimate from improved experimental data: N_A = 4.15×10²³, close to Maxwell's value of 1873. Later (1911) it was discovered that Einstein made an algebraic error in his thesis^[7] and in the paper based on it. When this was corrected the very same experimental data gave N_A = 6.6×10²³.

The phenomenon of Brownian motion was first described by Robert Brown in 1828 as the 'tremulous motion' of pollen grains (small solid particles of diameter on the order of a micrometer) suspended in water. Einstein's famous 1905 paper on the theory of Brownian motion^[8] gives a method for determining N_A, but not yet a value.

The first to give a value to N_A from Brownian motion was Perrin in 1908. He considered the distribution of Brownian particles in a vertical column in the Earths's gravitational field, and he used a similar mathematical approach to that which leads to the distribution of gas molecules in a vertical column of the atmosphere, see the article on the barometric formula. This formula contains the mass of the particle, the gravitational accelerationg, and the Boltzmann constantk. For suspended Brownian particles one has to make a correction for the buoyancy of the particles in the liquid (Archimedes principle) by using expressions that contain the densities of the particles and the liquid. Measurument of the numbers of particles at two different heights allows the determination of Boltzmann’s constant, k, and Avogadro’s constant through k = N_A/R. The molar gas constantR was already known in 1908 with high precision. Perrin in his first experiments prepared a monodisperse colloid of a gum called gamboge. The particle masses were determined by direct weighing of a specified number, and their radii (hence their volumes and densities) by using the Stokes-Einstein law for diffusion. Perrin’s first value for Avogadro’s number was N_A = 7.05×10²³. In 1909 Perrin ^[9] coined the name Avogadro's constant when he wrote: Ce nombre invariable N est une constante universelle qu'il semble juste d'appeler constante d'Avogadro [This invariant number N is a universal constant, which may, with justification, be called constant of Avogadro].

Estimates from the electron charge

Robert Andrews Millikan^[10] and his student Harvey Fletcher^[11] gave in 1910 and 1911 the first reasonably accurate values for the chargee of the electron. In 1917 Millikan^[12] gave the improved value e = 1.591×10⁻¹⁹ C. The current accepted value is the value of the elementary charge 1.6022×10⁻¹⁹ C.

The charge carried by a mole of singly charged ions in an electrochemical cell, which isknown as Faraday's constant, F, was already known for quite some time when the electron charge was determined. It was 9.6489×10⁴ C/mol. As F = e ⋅N_A, the 1917 value of the electron charge gave Avogadro's constant as N_A = 6.064×10²³.

Estimates from black-body radiation

In 1900 Planck gave birth to quantum theory by showing that the distribution of black body radiation as a function of temperature could be explained by assuming that oscillators in thebody of frequency n could only take up or release energy in integer packets of hn,where the proportionality constant h is now known as Planck's constant.

Planck pointed out that a comparison of his theoretical distribution with the experimental curve allowed the determination of h and Boltzmann's constantk. From the ratio of k and the molar gas constantR Avogadro’s constant could be determined. Planck's estimate was N_A= 6.175×10²³.

How To Explain Avogadro's Number To Kids

Estimates from counting alpha particles

Avogadro's Full Number

In 1908, Ernest Rutherford and Hans Geiger^[13] concluded that their scintillation technique for detecting α particles (He nuclei) recorded 100% of the particles which are emitted during the radioactive decay of radium. They found that a gram of radium emitted 6.2×10¹⁰ particles per second.Counting atoms clearly provides a method for determining Avogadro’s constant. Counting gives the number of a particles produced per second and one only has to measure the volume of helium gas produced per unit of time to know the number of atoms per volume, i.e., Avogadro's constant.

Avogadro's Number Practice

In 1911 Rutherford and his friend Boltwood, who spent a year's leave at Rutherford's laboratory in Manchester, measured the amount of helium produced by two radium samples after 83 days and after 132 days, respectively.^[14] The first experiment gave 6.58 mm³ of helium gas at 0°C and 760 mm pressure, while the second gave gave 10.38 mm³ of gas.From this the helium production was found to be 2.09x10⁻² mm³/day, and2.03×10⁻² mm³/day, which are satisfactorily consistent results. Boltwood and Rutherford did not state the value of Avogadro’s constant, which can be deduced from their experiments and the rate of production of a particles. But, knowing the amount of radium in the sample, and the amount of helium emitted per gram of radium, one can easily deduce that N_A = 6.1×10²³.

Estimates from crystal lattice spacings

Although X-rays have been used since 1912 to determine the lattice spacing of crystals, it was not until 1930 that X-ray diffraction was used to determine Avogadro’s constant. Before the 1930s, X-ray wavelengths were not known with enough accuracy, but today lattice spacings of certain crystals, especially the silicon crystal, form the most reliable source of Avogadro's constant.

The number density ρ of a crystal is defined as N_A/V_m, where V_m is the molar volume (the volume of one mole). It is reasonable to assume that the number density of the crystal is the same as the number density n/V_u of the unit cell. The volume V_u of the unit cell can be obtained from the lattice spacing of the crystal, provided the geometry of the unit cell is known. The silicon crystal is cubic and has n = 8 atoms per face-centered unit cell of edge length a = 543 pm.Avogadro's number N_A follows from

where it is used that V_u = a³ in the case of a cubic unit cell. This equation is the same as equation (143) of Ref.^[15]

Since the early 1990s an extensive international effort has been under way to reduce the relative standard uncertainty of the Avogadro constant measured by X-ray diffraction, so that serious consideration can be given to replacing the current SI unit of mass—the international prototype of the kilogram—by a definition based on a natural constant, such as the lattice spacing a of a silicon crystal.

In an outline of the problems, we notice that the exact molar volume V_m of silicon must be known to obtain an exact mole or an exact fraction of a mole. This requires knowledge of the molar mass and hence of the isotopic composition and the amount of impurities of the silicon sample. The three naturally occurring isotopes of Si are ²⁸Si, ²⁹Si, and ³⁰Si, and the amount-of-substance percentages of natural silicon are approximately 92%, 5%, and 3%, respectively. The first problem—the determination of the molar mass—is at present the limiting factor in the accuracy of Avogadro's constant. The second problem is the determination of the molar volume, or, equivalently, of the crystal density ρ. Determination of the lattice spacing a is a third source of error, and, to that end, the wavelength of the X-rays has to be known with great accuracy; this can be obtained from combined optical and X-ray interferometry.

References

1. ↑CODATA value retrieved December 27, 2020 from: | Avogadro constant at NIST
2. ↑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 [Essay on a manner of determining the relative masses of the elementary molecules of bodies, and the proportions according to which they enter into these compounds], Journal de Physique, de Chimie et d'Histoire naturelle, vol. 73, pp. 58-76 (1811).
3. ↑A. Pais, Subtle is the Lord ..., Oxford University Press (1982), chapter 5
4. ↑ J. Loschmidt, Zur Grösse der Luftmolecüle, Sitzungsberichte der kaiserlichen Akademie der Wissenschaften zu Wien: Math-Naturwiss., Klasse 2, vol. 52, pp. 395–413 (1866). Pre-publication abstract of the author, Anzeiger Kais. Akad. Wiss. Wien, Math-Naturwiss. Klasse 2, 162 (1865).
5. ↑J. B. Perrin, Les Atomes, Paris, Alcan, (1913). Reprint by Gallimard, 1970. Translation: Atoms, by D. L. Hammick, Van Nostrand, New York (1916).
6. ↑A. Einstein, Eine neue Bestimmung der Moleküldimensionen [A new determination of molecule dimensions], Ann. d. Physik, 19, 289, (1906)
7. ↑A. Einstein, Erratum to the 1906 paper, Ann. d. Physik 34, pp. 591–592 (1911)
8. ↑A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen [On the motion, required by the molecular kinetic theory of heat, of particles suspended in liquids at rest], Ann. d. Physik, 17, 549, (1905)
9. ↑Jean Perrin, Mouvement brownien et réalité moléculaire, Ann. Chim. Phys. vol. 18, pp. 1–114 (1909).Online
10. ↑R.A. Millikan, A new modification of the cloud method of determining the elementary electrical charge and the most probable value of that charge, Phil. Mag., vol. 19, pp. 209-228 (1910)
11. ↑H.Fletcher, A Verification of the Theory of Brownian Movements and a Direct Determination of the Value of Ne For Gaseous Ionization, Phys. Rev., vol. 33, pp. 81-110 (1911)
12. ↑R. A. Millikan, The Electron: Its Isolation and Measurements and the Determination of Some of Its Properties, University of Chicago Press, Chicago (1917).
13. ↑E. Rutherford and H. Geiger, An Electrical Method of Counting the Number of α-Particles from Radio-Active Substances, Proc. Roy. Soc., vol. A81, pp. 141-161. (1908)
14. ↑B.B. Boltwood and E. Rutherford, Production of helium by radium, Phil.Mag., vol. 22, pp. 586-604 (1911)
15. ↑P.J. Mohr and B. N. Taylor, CODATA recommended values of the fundamental physical constants: 2002, Reviews Modern Physics, vol. 77, p. 1 (2005)

Further reference:

• J. Wisniak, Amedeo Avogadro The Man, the Hypothesis, and the Number, Chem. Educator, vol. 5, pp. 263–-268 (2000) (This paper errs in giving the value for Avogadro's constant obtained by Loschmidt. It gives it a factor ten too high).