Avogadro's Number
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Some Notes on Avogadro's Number, 6.022 x 10
23
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 10
23
.
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 10
19
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 10
25
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 10
22
; 68.8 x 10
22
; 65 x 10
22
.
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/cm
3
. The number of moles 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 10
23
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 8 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