The first and a recent
experimental determination of Avogadro's number.

Luis M�rquez-Jaime^{1}, Guillaume Gondre^{1} and Sergio
M�rquez-Gami�o^{2}^{*}.

^{1} Department
of Mechanics, The Royal Institute of Technology (Kungliga Tekniska H�gskolan,
KTH), SE-100 44 Stockholm, Sweden. lsmarquez@gmail.com

^{2}Departmento de Ciencias
Aplicadas al Trabajo, Universidad de Guanajuato, A. Postal 1-607, Le�n, Gto.,
37000 M�xico. smgamino@fisica.ugto.mx

*Corresponding author.

When a friend as Jes�s is gone, your feelings
of loss are overwhelming! Such that yourself can be lost, until you realize he
is still influencing your own decision-taking processes. He is here as a model
to follow! Sergio.

ABSTRACT

It has been
almost two centuries since Avogrado presented his hypothesis, then, more than
fifty years later Austrian Loschmidt developed a method for measuring the
diameter of air molecules which was a big step for calculating a first
approximation of the amount of molecules or atoms per unit volume. Computing Avogadro�s
with Loschmidt�s diameter results give ten times the real value. It was until
the beginning of the 19th century when Jean Perrin measured an accurate value
of Avogadro�s constant, which is the amount of elementary entities in one mole
of substance. After him, many scientists have measured this constant using more
accurate methods, one of them is the X ray diffraction method, which can reach
accuracy of 99.99% using ^{28}Si.

Key words:
Avogadro�s constant, Avogadro�s number, Loschmidt, Perrin, X-ray diffraction.

FIRST
EXPERIMENTAL DETERMINATION OF AVOGADRO�S NUMBER

It is not
simple to decide who first determine Avogadro's number experimentally, in the
following paragraphs we talk about some of the methods used for measuring this
physical constant history. Afterwards, we compare it with a recent method that
gives more accurate values for the constant. Avogadro�s constant (formerly
called Avogadro�s number) is known as a physicochemical constant and is
equivalent to the number of �elemental entities�, usually atoms or molecules
contained in one mole of substance [1]. The value recommended since 2006 by the
U.S. National Institute of Standards and Technology (NIST) is: *N _{A}
*= 6.022 141 79(30) x 10

Avogadro's
constant is named after the italian scientist Amedeo Avogadro (Figure 1), who
in 1811 published �Essai d�une mani�re de determiner les masses relatives des
mol�cules �l�mentaires des corps et les proportions selon lesquelles ells
entrent dans ces combinaisons�. In this paper Avogadro proposed that the volume
of a gas at a given pressure and temperature is proportional to the number of
molecules or atoms regardless to the nature of the gas� [3].

Figure 1. Amedeo Avogadro

In 1865, the
Austrian physicist Johann Loschmidt published �Zur Gr�sse der Luftmolek�le�
which is translated into English as �On the Size of the Air Molecules�. In this
work, based on the Kinetic Gas Theory and previous results of Clausius, Maxwell
and Meyer, Loschmidt derived an approximation of the size of the diameter of
air molecules in normal conditions (0,970 nm, which is 2.7 times the real
value). He also cites in his work �This result is only worthy for being an
approximation, but it is not 10 times bigger or too small�, then defined a
condensation coefficient with the mean free path of a molecule at temperature
of 273 K [4]. Knowing the diameter for molecules, and assuming that in the
liquid state molecules touch each other, Loschmidt derived an approximation for
the number of particles (atoms or molecules) of an ideal gas in a given volume
(the number density) at standard temperature and pressure, *N _{L} *=
2,092 2 � 10

Brownian
motion led to a more accurate determination at the beginning of the 20th
century by Perrin [11] (1909, *N _{A} *= 6,7 x 10

RECENT
DETERMINATION OF AVOGADRO�S NUMBER

A modern way
to get the Avogadro�s number is the X-ray diffraction experiments using the
silicon ^{28}Si, which determines the size of an edge for a cubic
crystal system [15]. The unit cell volume is then deduced, and used to
calculate a value for Avogadro's constant. First, one must get an adequate
crystal, pure in composition and regular in structure, with no significant
internal imperfections. In principle, an atomic structure could be determined
from applying X-ray scattering to non-crystalline samples. However, crystals
offer a much stronger signal due to their periodicity since they are composed
of many unit cells repeated indefinitely in three independent directions.
Hence, using a crystal concentrates the reflection from the unit cell. The
crystal is then placed in an area where monochromatic X-rays (single wavelength)
are emitted. They are produced by a radioactive beam. The crystal is mounted
for measurements so that it may be held in the X-ray emission and rotated. The
principle is to place the crystal in a tiny loop, which will be rotated in
respect of three different angles of rotation. Since both the crystal and the
beam are very small, the required accuracy is about 25 micrometers. When the
mounted crystal is irradiated, it scatters the X-rays into a pattern of spots.
The relative intensities of these spots provide the information to determine
the arrangement of molecules within the crystal in atomic detail. The reflected
or deviated X-rays produce spots recorded on a screen placed behind. The
intensities of these reflections are recorded with photographic film. Because a
crystal structure consists of an orderly arrangement of atoms, the reflections
occur from what appears to be planes of atoms [16]. A rotation step-by-step
through 180� must be done in order to collect several data all around the
crystal. Indeed, one image of spots is insufficient to reconstruct the whole
crystal. It represents only a small slice. A full data set may consist of
hundreds of separate images taken at different orientations of the crystal.
Finally, these data are combined computationally to produce and refine a model
of the arrangement of atoms within the crystal. One must determine which
variation corresponds to which spot. The final refined model of the atomic
arrangement is called a crystal structure [17].

Since the
crystal structure is determined, one can get the value of the parameter, a,
length of one side. As the unit cell is cubic, the volume is the cube of the
length. Hence, the unit cell volume is now known. The Avogadro�s constant is
determined using the ratio of the molar volume, *V _{m}*, to the
unit cell volume,

The factor
of eight arises because there are eight silicon atoms in each unit cell.
Silicon occurs with three stable isotopes: the major stable isotope of silicon,
28 14Si, has fourteen protons and fourteen neutrons. It appears to over 92% of
the element in nature. The atomic weight *A _{r} *is based on the
stoichiometric proportions of chemical reactions and compounds [18]. Given that

In 2005,
Fujii et al. measured Avogadro�s constant with a X ray diffraction experimental
setup, obtaining *N _{A} *= 6.0221353(18) � 1023 mol‐1 [19].

CONCLUSION

In summary,
Avogadro�s constant was not measured by Avogadro, but it was measured based on
his hypothesis. Loschmidt was the first to measure a similar concept called
Loschmidt number that includes the number of molecules or atoms per unit volume
for given a pressure and temperature. The accuracy was not very good (10 times
the accepted value), due to his assumptions i.e. that the molecules are
spherical and that all touch each other in a liquid phase in the same way, that
does not happen exactly that way, but was sufficient to get an idea on the
statistical errors and to prove Avogadro�s hypothesis. Maxwell also obtained a
value for the Loschmidt�s number that was more precise. The Avogadro�s number
is the number of elemental entities per mole. Perrin was the first to give a
real accurate estimation of the Avogadro�s constant (differing ~10% of the 2006
NIST recommended value), and also to use the concept of mole in the definition,
He also proposed that the constant would be named after Amedeo Avogadro. Till
today more methods have been used for measuring the constant with better
precision, one of the newest methods use enriched silicon‐28 due to its known
properties. In the X-rays diffraction method, the result of the experiment is
quite more accurate than the approximation using Loschmidt assumptions. It can
give 99.99% accuracy. This is explained by the precision of the used
instruments, which enable placing accurately the crystal to the beam and enable
measuring the inference from reflections. Also, the molar mass and the atomic
weight of the 28Si are known with precision. The 28Si is stable and enables to
repeat the experiment in order to validate it. On another hand, one can say
that it is uncommon to get all the instrumentations to manage this experiment,
especially to get the adequate crystal or the X-ray beam which requires
specific recommendations. Moreover the experimenter must have good experience
and knowledge to analyze the results from the spots. In addition, the
reflection of the X-ray will not always give clear inference. Hence, the X-ray
diffraction enables to get accurate atomic structure but is not easy to
implement.

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