Introduction
The International System of Units (the SI unit system)is based on the seven base units:
meter, kilogram, second, ampere, kelvin, mole, and candela [1]. These units are
defined in terms of a standard chosen by the General Conference on Weights and
Measures (CGPM) with permission of the International Union of Pure and Applied
Chemistry (IUPAC). The definitions of these units are under constant review and
are changed from time to time.
One of these base units is the mole, which is the unit of
the amount of a substance. The mole was introduced into the SI unit system by
the 14th CGPM in 1971 [1]. It is defined as the amount of substance in a system
that contains as many elementary entities as there are atoms in 12.00000 g of
carbon-12 [1]. The number of atoms in exactly 12.00000 g of carbon-12 is called
Avogadro's number, NA [2, 3]. If the mass of a single
carbon-12 atom in grams is determined, Avogadro's number can easily be
calculated by dividing the 12.00000 grams per mole 12C, from the
definition of mole, with the mass of a single carbon-12 atom in grams. There
are various methods for the determination of Avogadro’s number in the
literature that are not based on carbon-12 [4–12].
The author asked 37 chemistry teachers, 67 prospective
chemistry teachers, 142 freshman science education students, and 142 high
school students, as well as several chemistry professors, about the definitions
of mole and Avogadro's number. More than half of them defined these concepts as
either "Avogadro's number of particles" or "6.022 ´ 1023."
It is quite obvious that both these answers are incorrect. They were also asked
about the origin of Avogadro’s number. The results showed that most of them did
not know the relationship between Avogadro’s number and carbon-12. They thought
that Avogadro’s number is a constant number assigned as 6.022 ´ 1023
in order to define the mole. They were also asked why one mole of any substance
has a mass in grams that is numerically identical to the mass of a single unit
(atom or molecule) in unified atomic mass units (symbol u). None of them could
answer this question correctly. In other studies related to the subject, it is
reported that chemistry teachers [14] and students [12, 14–17] from various
countries, and even some chemistry textbook writers [18], also have the same
difficulties. The reason for this may be that they have not studied this
concept using an activity based on carbon-12.
In this manuscript we describe an activity in
which 52 first-year students in the Department of Science Education, divided
into 7 groups, determined the mass of a single carbon-12 atom in grams using
mass spectroscopy data and obtained Avogadro’s number by dividing the mass of a
single carbon-12 atom in grams. Then, using the definition of a mole, they
calculated Avogadro's number by dividing 12.00000 g mol–1 (carbon-12
mole weight) with this mass. Using the data from the seven groups, they
calculated an average value for Avogadro’s number. In addition, they calculated
the conversion factor that relates the (unified) atomic mass unit to grams by
using the definition of the amu and the mass of a single atom of carbon-12 in
grams.
Procedure for Evaluating Avogadro’s Number
J. J. Thomson determined the ratios of the unit charge to
the masses, e/me and e/mp,
for the electron and the proton [19]. The first measurement of the charge on
the electron, e, was made by R. A. Millikan [19, 20]. The masses of the
electron and the proton in grams, me and mp,
were calculated from the values of e/me
and e/mp and
e. The currently accepted values for their rest masses are 9.109 381 88(72) ´ 10–28 g and 1.672 621 58(13) ´ 10–24 g, respectively [21].
The mass spectrum of methane contains the peaks of 1H+
and 12C+ species in addition to those of the other
species [22]. The positive ion mass spectrum of methane is shown in
Figure 1. The mass spectrum of methane [22].
This spectrum was enlarged to page size for measurement.
Figure 1. The m/z =
1 and m/z = 12 peaks (red and blue peaks) indicate the presence
of 1H+ and 12C+ species.
In the mass spectra, although the abscissa axis includes
the mass number, this axis is generally related to the mass. There is certainly
a mass in grams or kilograms or atomic mass units corresponding to each mass
number; therefore, on the condition that x1 and x2
are the abscissa lengths of the peaks of the 1H+ and 12C+
species, one can write mp µ x1 and 
µ x2,
where mp and 
are the masses of the
1H+ (proton) and 12C+ ions. Then,
the mass of 12C+ ion in grams can be calculated from
(1)
The mass of a 12C atom is
(2)
Thus, according to the definition of a mole,
Avogadro’s number is
(3)
In this context,
knowing that the standards chosen for the definitions of the mole and
the unit of atomic mass are based on thesame chemical element is very important. The amu is a standard
mass unit that is equal to one-twelfth the mass of a neutral atom of the
carbon-12 isotope. According to this definition, the atomic mass of carbon-12
is 12.00000 u. Because the mass of a single carbon-12 atom in atomic mass units
(12.00000 u) must be equal to its mass in grams, the factor that relates the
amu to the gram can be calculated from
(g atom–1 12C) = 12.00000 (u
atom–1 12C) (4)
Performance of the Activity
The activity was used in a class of fifty-two first-year
students taking a general chemistry course in the Department of Science
Education. The students were divided into seven groups. Each group was given a
copy of the spectrum shown Figure 1 and asked to measure the abscissa lengths
of the peaks of the 1H+ and 12C+ species
with an ordinary ruler graduated in millimeters. Each group calculated the
masses of the 12C+ and 12C species and
Avogadro’s number using eqs 1–3. They obtained a final value of Avogadro’s
number by calculating the average of all seven values. In addition, they
substituted their average value of the mass of a single carbon-12 atom in grams
into eq 4 to obtain the conversion factor that relates the amu to gram.
The positive ion mass spectrum of methane in Figure 1 is a
reproduced version of the spectrum published elsewhere [22]. For this
reproduction, the spectrum was redrawn in its
original form by using a common word processor, and
then, it was enlarged to fit on a page.
Natural gas contains methane in quite a large proportion
and it can be used as the methane source. The mass spectrum of natural gas is
quite similar to that of methane [23 ]. The mass spectrum of methanol also
contains the peaks of 1H+ and 12C+
ions [24]; therefore, it can be used for the same purpose.
In the performance of the activity, students
worked as a cooperative group [25].
Results and Discussion
The students calculated an average value of 6.0 ´ 1023
atom 12C per mol 12C for Avogadro's number. The results
are summarized in Table 1. They also derived the relationship between the amu
and gram and found that the amu multiplied by Avogadro's number is equal to one
gram (NA u = 1 g).
The abscissa lengths of the peaks in the spectrum were
measured with an ordinary ruler graduated in millimeters. The reliability of
the result is highly dependent upon the accuracy of the measurement of the
abscissa lengths; however, the purpose of this study was not to determine
Avogadro's number with many significant figures, but to make students aware
that the definitions of mole and Avogadro's number are based on carbon-12.
Because the definition of a mole is based on a neutral
atom of the carbon-12 isotope [1], although it does not change the
Table 1. Results for the Determination of
Avogadro’s Number
|
Group
|
x1
(mm)
|
x2
(mm)
|
 ´
10+23 (g)
|
NA (1023 mol–1)
|
|
1
|
4.1
|
48.8
|
2.0
|
6.0
|
|
2
|
4.1
|
48.8
|
2.0
|
6.0
|
|
3
|
4.1
|
49.0
|
2.0
|
6.0
|
|
4
|
4.1
|
48.6
|
2.0
|
6.0
|
|
5
|
4.1
|
48.8
|
2.0
|
6.0
|
|
6
|
4.1
|
48.8
|
2.0
|
6.0
|
|
7
|
4.1
|
48.8
|
2.0
|
6.0
|
|
Average
|
|
|
2.0
|
6.0
|
result, formally, the mass of an
electron should be added to the mass of the 12C+ ion.
The standards chosen for the definitions of the mole and
the unit of atomic mass are based on thesame chemical element. Since 1971, the number of atoms in 12.00000
g of carbon-12 has arbitrarily been assigned as the standard for the mole [1].
Since 1961, one-twelfth of the mass of a single carbon-12 atom has arbitrarily
been assigned as the standard for the unit of atomic mass [1]. If the standards
are changed, the values of Avogadro's number and atomic masses will also
change. For example, according to the base of J. J. Berzelius' atomic mass
scale, an atomic mass unit is one-hundredth of the mass of a single oxygen atom
[3]. According to this scale, the mole and Avogadro’s number may be defined as
“the amount of substance of a system that contains as many elementary entities
as there are atoms in 100 g of oxygen” and “the number of atoms in exactly 100
gram of oxygen,” respectively. Then, the value of Avogadro’s number would be
3.77 ´
1024 atoms O per mole O.
In summary, the size of Avogadro's number is dependent on
the standard chosen for the definition of the mole. In fact, we cannot
accurately know the value of Avogadro’s number because it is such a large
number that measuring devices are not sufficient for this purpose. In terms of the
basis of 12.00000 g carbon-12, Avogadro’s number is a number with 24 digits,
but its current value has only nine significant digits [6.022 141 99(47) ´ 1023]
[21]. In the future, scientists may determine more than these nine digits.
The following points should be taken into consideration in
the presentation of the mole and Avogadro’s number in textbooks and classrooms:
(a) The
relationship between the mole and the SI unit system should be presented.
(b) The
relationship between the standards chosen for the definitions of the mole and
the atomic mass unit should be explained.
(c) An activity relating to the determination of
Avogadro’s number, which is in accordance with its definition, should be
carried out.
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The
Chemical Educator, Vol. 9, No. 1,
Published on Web 12/12/2003, 10.1333/s00897030740a, © 2004 The Chemical
Educator