Some Notes on Avogadro's Number, 6.022 x 10²³
T.A. Furtsch,  Tennessee Technological University, Cookeville

Chemists use Avogadro's number every day.  It is a very valuable number for a chemist to know how to use, and use properly. Where did Avogadro's number come from?   Did Avogadro himself do all the calculations?  Was it just arbitrarily made up?   How can it be measured? Some possible answers follow.

Amadeo  Avogadro (1776-1856) was the author of Avogadro's Hypothesis in 1811, which, together with Gay-Lussac's Law of Combining Volumes, was used by Stanislao Cannizzaro to elegantly remove all doubt about the establishment of the atomic weight scale at the Karlsruhe Conference of 1860.

 

The name "Avogadro's Number" is just an honorary name attached to the calculated value of the number of atoms, molecules, etc. in a gram mole of any chemical substance. Of course if we used some other mass unit for the mole such as "pound mole", the "number" would be different than 6.022 x 10²³.

The first person to have calculated the number of molecules in any mass of substance was Josef Loschmidt, (1821-1895), an Austrian high school teacher, who in 1865, using the new Kinetic Molecular Theory (KMT) calculated the number of molecules in one cubic centimeter of gaseous substance under ordinary conditions of temperature of pressure, to be somewhere around 2.6 x 10¹⁹ molecules. This is usually known as  "Loschmidt's Constant."  (This value, n_o, is now listed at the NIST web site as 2.686 7775 x 10²⁵ m^-3)

When was the first time the term "Avogadro Number" was used?  The designation seems to originate in a 1909 paper entitled "Brownian Movement and Molecular Reality."  by Jean Baptiste Jean Perrin

(b. Lille, France, 30.9.1870-d. New York, 17.4.1942.)  This paper was translated into English from the French in Annals De Chimie et de Physique by Fredric Soddy and is available.  Perrin, was the 1926 Nobel Laureate in Physics for his work on the discontinuous structure of matter, and especially for his discovery of sedimentation equilibrium.  Perrin should be very well known to anyone who does calculations in molecular dynamics.  Most of these methods were developed by Perrin.   In his paper Perrin says "The invariable number N is a universal constant, which may be appropriately designated "Avogadro's Constant."

In the presentation of his Nobel prize in 1926 it was said of the work of Perrin: 

It may perhaps be said that in the work which we have just summarized Perrin has offered indirect evidence for the existence of molecules. Here, follows a direct evidence. Microscopic particles in a liquid are never at rest. They are in perpetual movement, even under conditions of perfect external equilibrium, constant temperature, etc. The only irrefutable explanation for this phenomenon ascribes the movements of the particles to shocks produced on them by the molecules of the liquid themselves. A mathematical theory of this phenomenon has been given by Einstein. The first experimental proof of this theory was given by a German physicist, Seddig. After him, the problem was taken up by two scientists simultaneously. One of them was Perrin; the other Svedberg. I have to speak of Perrin only. His measurements on the Brownian movement showed that Einstein's theory was in perfect agreement with reality. Through these measurements a new determination of Avogadro's number was obtained.

The molecular impacts produce not only a forward movement of the particles distributed in a liquid, but also a rotational movement. The theory of this rotation was developed by Einstein. Measurements in relation herewith were carried out by Perrin. In these measurements he has found another method for determining Avogadro's number. What then is the result of these researches ? How many molecules are there in two grams of hydrogen? The three methods have given the following answers to this question: 68.2 x 10²²; 68.8 x 10²²; 65 x 10²².

The work of Einstein and Perrin gave some of the first concrete evidence for the existence of molecules, entities many still did not recognize even into the early 1900's.   And Avogadro's Number has a value that must be measured experimentally.    Subsequent to the work of Loschmidt and Perrin many scientists carried out many experiments using a variety of techniques to arrive at the most accurate value for this the number of molecules in one mole of substance.  And by 1933 there was still no universal agreement as to what the number should be called.  In a paper entitled "Loschmidt's Number", published in 1933 (Science Progress, v. 27, 1933, pp. 634-649), S. E. Virgo, a physicist at The  University, Sheffield, England says:

This number is frequently referred to as "Avogadro's Number," the term "Loschmidt's Number" being then reserved for the number of molecules in a cubic centimetre of a gas under standard conditions. Unfortunately, these designations are often interchanged. Avogadro's important hypothesis on the identity of the numbers of molecules in equal volumes of different gases at the same pressure and temperature was formulated in 1811, and is appropriately associated with his name; but Avogadro made no quantitative estimate of either of the above-mentioned constants. The first actual estimate of the number of molecules in one cubic centimetre of a gas under standard conditions was made in 1865 by Loschmidt, and from this the number of molecules (atoms) in a gram molecule (atom) was later evaluated. From the quantitative view-point it thus seems preferable to speak of "Loschmidt's number per gram-molecule (atom)," and of "Loschmidt's number per cubic centimetre," as is almost invariably done in the German scientific literature. This terminology avoids ambiguity, and has been adopted here.

So, even by 1933, there was no clear agreement as to what the number should be called.   Virgo goes on to say that by that year more than eighty separate determinations had been made to discover the true value of the number "as it is a basic atomic constant its most probable value is of great importance in atomic physics."  The best modern values for what we now call "Avogadro's Number" are the result of the x-ray diffraction measurement of lattice distances in metals and salts. The earliest attempts at using this method are reviewed in Virgo's paper.   Calculations reflecting these methods are often found in modern general chemistry text books. For example,  from x-ray data the one can determine that titanium (Ti) has a body-centered cubic unit cell (i.e.there are two Ti atoms per unit cell) and an edge length of 330.6 pm.  One can also find that the density of Ti metal is 4.401 g/cm³.   The number of atoms of Ti in a mole of Ti (47.88 g), Avogadro's Number, can be calculated as follows: (General Chemistry, Whitten, Davis and Peck, Saunders College Publishing, 6ed, 2000, p. 523.)

 

 

Today's best experimental value of   6.022 141 99 x 10²³ mol^-1 atoms per mol (obtained from NIST web site) is the best average for measurements using the best methods available.  The experiments are often very difficult to carry out. That the number today has 9 significant figures is a testament to the quality of modern experimental methods.

Some Links related to this essay:

Avogadro's 1811 Essay in which he hypothesizes that equal volumes of gases contain equal numbers of molecules.(from Carmen Giunta's classical chemistry page)
"Loschmidt's Number", Science Progress, v. 27, 1933, pp. 634-649.
Avogadro's Hypothesis
A Biographical interview with Amadeo Avogadro