APPROACH TO THE CONCEPTS OF “AMOUNT OF SUBSTANCE”,
“MOLE” AND “AVOGADRO’S CONSTANT” THROUGH THE USE OF
ANALOGIES
C. Aguirre-Perez
University of Castilla-La Mancha Cuenca/Spain [email protected]
Abstract
Within the field of chemistry education, many articles dealing with the problem of learning difficulties of
students in the concept of mole can be found in the science education literature, and therefore, all
other difficulties related to it. In our case, the core problem is that the magnitude of Avogadro's number
is far away from what students can conceive or imagine. Furthermore, it is well known that students
tend mechanically to operate and handle the exponential numbers without being aware of the real
scale of values they manipulate. So we could understand to what extent is hard for them to assimilate
this concept. Here, we try to develop a series of comparative analogies with other fundamental
magnitudes, well known magnitudes for teacher training students (Length, Mass and Time), in a way
that they can relate that number, Avogadro’s number, closer to quadrillion (1024) with amounts of more
known elementary entities. These amounts are very surprising used in these orders of magnitude. We
have found that, with the analogies and similarities we have posed, students come to understand
much better these magnitudes are much better and are able to resolve issues and problems related to
the concepts of mole and Avogadro number in different contexts.
1
INTRODUCTION
In recent years, and in the field of chemistry education, many articles dealing with the problem of
learning difficulties of students in the concept of mole can be found in the scientific teaching-learning
literature and, therefore, all other concepts related to it such Avogadro’s number., atomic, molecular
and molar masses , and so on. From the standpoint of educational research, the problem could be
confine within the field of interrelations between dimensions and categories of the Chemistry proposed
by Jensen (1):
DIMENSION
COMPOSITION/ STRUCTURE
ENERGY
TIME
CATEGORY
1.- MOLAR
Relative composition: simple and
composed substances; mixtures
and solutions. Allomorphes
Calorimetric entropy and
formation heats, free Energy land
equilibrium constants
Chemical kinetics.
Experimental laws. Arrhenius’
parameters and activation
entropies and heats
2.- MOLECULAR
Molecular and structural
formules. Rationalization of
allomorphes as a variation of
absolute composition absolute
(polymers) or in structure isomers
Molecular
interpretation
of
entropy,
atomization
and
formation heats, average bond
energy. Molecular mechanics
Mechanisms
of
molecular
reactions. Molecular vision of
activation
entropies
and
activated complex
3.- ELECTRIC
Electronic formulas (Lewis and
electronic configuration).
Variation in the nuclear
composition (isotopes) an
electronic (ions)
Calculations of energy based
in the electronic structure.
Spectral interpretation,
calculations of atomization
heats
Mechanisms of ionic and
photochemical reactions.
Isotopic effect. Calculations of
activation energies. Indices of
Electronic reactivity
Chart 1 (Taken from Alzate, Mª V. (2)
Ie, how to make students to advance in terms of understanding the structure of matter from the
observable level (molar) to the smallest level (electric) 1 Æ 2 Æ 3 so that the path can then reverse 3 Proceedings of EDULEARN09 Conference.
6th-8th July 2009, Barcelona, Spain.
ISBN:978-84-612-9802-0
004028
Æ 2Æ1
from quantum mechanics, understanding that certain words carry a certain level of
ambiguity in its comprehension that must be gradually corrected using it in different contexts,
problems, exercises, questions, etc?
STATE OF THE QUESTION
This inevitably leads to the need to take into account the three levels of representation proposed by
Johnstone (3) (macroscopic, microscopic and symbolic) which are deeply related to the categories of
Jensen
macro
micro
symbolic
Fig. 1
Jhonstone Triangle of the representation levels in Chemistry
What De Jong and van Driel (4) call interface macro-micro-symbolic
INTERFACE MACRO-MICRO-SYMBOLIC
Macroscopic Domain
(Substances, phenomena, etc.)
Symbolic Domain
(formulas, equations, etc.)
Microscopic (submicroscopic)
(Atoms, molecules, etc.)
Chart 2
These authors put particular emphasis on the distinction between teachers. as experts who make
without any difficulty both the direct and the reverse path between the micro and macro worlds linked
by the symbolic domain and the related concepts with the mole, and the students considered as
novices that can easily be overwhelmed by the cognitive demand required by the simultaneous
operation of the interface between three domains
As a synthesis of both categorizations a summary table that integrated the symbolic level, which is not
considered by Jensen, could be proposed and, at the same time, the atomic and molecular levels
covered by Johnstone, De Jong and van Driel as microscopic level (actually submicroscopic) would be
diversified:
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CATEGORIES
DOMAINS
( Jensen)
(Johnstone, De Jong)
MOLAR / MASSE / VOLUME
MACRO
SYMBOLIC
ATOMIC / MOLECULAR
MICRO
ELECTRONIC / NUCLEAR
Chart 3
Where we have changed the categories of Jensen "molar" and "electric" for “atomic / molecular” and
“electronic / nuclear" which seem more appropriate given the different chemical entities to which we
can refer in these two levels: molecules , atoms, ions, electrons, protons and nuclei. We have also
placed the symbolic level as a link between the two domains (or the three categories).
Among the Spanish authors who had paid more attention to the difficulties of the teaching / learning
the mole concept is bound to mention the group of Professor Carlos Furió (5) (6) . This group
approaches the problem from a historical and epistemological point of view. In general, in their works
they assume that the teachers’ knowledge of science history (in this case chemistry) and the various
twists that led to the development of concepts such as atom, element, substance etc. and the various
models / theories that at each historical moment have underpinned these concepts, concepts that
have evolved over time, will be a sufficient starting point for addressing the problem. It is also rather
stressed in them the need to introduce in the teaching of chemistry directly and as soon as possible
the concept of "amount of substance" as a fundamental magnitude whose only unit in the three
systems of units is the "mole" following the guidelines of the International Union of Pure and Applied
Chemistry (IUPAC) of 1965 which says:
A mole is the amount of substance of a system that contains as many "elemental entities"
(e.g., atoms, molecules, ions, electrons) as there are atoms in 12 g of carbon-12 (Guggenheim
1986, p. 3).
This definition Clearly establishes that the magnitude "amount of substance" is different from mass
although related to it by Avogadro's constant. In this expression the word "substance" can be replaced
each time by the name of the substance, for example, "amount of hydrogen chloride, HCl, or the
amount of sucrose C12H22O11. It is also related to the magnitude "volume" in the case of gases in
which there is a simple mathematical relationship in the event that they are in the so-called standard
conditions (0 º C and 1 atm pressure). As a summary of their approaches we reproduce the following
chart which summarizes the conceptual and operational relationships of the mole with other quantities
(mass, volume and number of entities). In It the need to distinguish clearly the amount of substance
quantities (n), mass (m), volume (V) and number of elementary entities (N) and to specify the
functional definition of the quantity of a substance as a magnitude used to count microscopic
elementary entities is stressed. Note that the three operational definitions of mole in the above cases
are, in principle, non-dimensional values:
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Mass (m)
Volume V (N.C.)
n = m/M
n =V/Vm
Amount of
substance
(n)
n = N/NA
Number of
elemental entities
n = P V/R T
Volume V
(Any conditions)
(N)
Fig. 2: taken from “ENSEÑANZA DE LAS CIENCIAS” 1999, 17 (3) p. 364
Other authors such as Gabel (7) justify the difficulties encountered in this research in what teachers
(experts), during their explanations and considerations, inadvertently move from one to another level
(macro, micro and symbolic) with the result of the students’ failure in the integration between these
levels. Based on that premise Angelini et al (8) propose a teaching strategy based on the use of a
model of particles (atoms, molecules, ions) increased with the postulates of the kinetic theory of
gases, which are subsequently used to interpret the behaviour of macroscopic systems (solutions,
chemical changes, balance). Subsequently it must be supplemented with a standard diagnostic
evaluation of the results with students
Pozo (9) points out in very clear and synthetic way in a tabular chart the difficulties of learning the
mole and its related concepts and in which there is very clearly reflected that the core problem lies in
the order of magnitude or Avogadro’s number, 1023, which is far beyond of what the students can
conceive or imagine. Moreover, we can add, this is a number of entities (molecules, atoms, ions,
electrons) so immensely small that their magnitude (of the order of nanometers, 10-9 m, Angstroms,10­
10
m or even below) also escapes of the understanding of most of our students. So we can conclude
that there is a conceptual physical-mathematical obstacle that stands between the teaching and
learning this latter conceived from a constructivist point of view as the understanding and assimilation
of concepts in a meaningful way that allows its implementation in different operational circumstances
and contexts. It must be taken into account that we are simultaneously handling concepts and entities
which are mathematically expressed by exponential notations that differ in more than thirty orders of
magnitude.
DIFICULTIES WITH THE CONCEPT
OF MOLE
THE AVOGADRO’S NUMBER IN
CALCULATIONS
RELATED CONCEPTS
1.- Complex definition. Students can not
understand the definition and use it in
an algorithmic way to establish a
1.- A so huge number that is far
beyond of what an student can imagine
1.- Phonetic similarity between a large
number of concepts (mol, molecule,
mole, molarity, etc..) Which, however,
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relation between moles and masses.
are totally different.
2.- It is badly used in many textbooks.
2.- Difficulty in distinguishing and
coordinating the relations of the moles
with the coefficients of the balanced
chemical equations
3.- The mole is a bridge between the
macroscopic and the microscopic
worlds: but students, in the most
occasions, are not capable to
distinguish
where are they (for
instance, they can not distinguish
between the number of atoms an the
number of atom moles.
3.- Application of the molar volume of
gases to all kinds of substances
(including liquids and solids)
4.- Use of false laws of conservation of
moles.
Adapted from: Pozo Municio, J.I.. Aprender y enseñar ciencia.
Morata. España, 1998. p. 185
Chart 4
Poskozim and others (10) report, after analyzing 155 North american textbooks, that only 41 of them ­
about one in four- contain analogies relating to the size in order to illustrate the magnitude of the
Avogadro’s number.These analogies were classified into 5 categories, as represented in chart 5:
Analogíes of NA en textbooks
Analogy based on
Premise: NA is so huge
that:
Examples of typical analogies
I. Small or tiny objects
(marbles, peas, grains of
sand, tennis balls, etc.
The volume occupied by a
number of such small
objects
would
be
incredibly tiny or large
An NA of marbles spread out on the
surface of Earth would result in a
layer of marbles over 75 km thick
Total references
14
A NA of sand grains spread across
all the surface of the USA would
lead to a sand layer about 8 cm
thick
II. the counting
It would take an incredibly
long time to count such
number of objects even
involving
the
whole
population of the Earth in
the task and without taking
rest
The world's population would need
almost five billion years to
collectively count one NA of objects
one per second
12
III. people
The total population of the
Earth now is incredibly
small in comparison
Se necesitarían 100 billones de
planetas
con
poblaciones
equivalentes a la terrestre (6 mil
millones) para acomodar un NA de
personas.
6
It would require 100 thousand
billion planets with populations
equal to the earth (6 billion) to
accommodate a NA of people
IV. water
It can be compared to the
number of milliliters of
There are twice the NA of milliliters
of water on this planet.
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6
water existing on Earth
If 18 grams of water (1 mol and NA
molecules) are spread throughout
the Earth's surface, would have
approximately
100,000
water
molecules
in
each
square
centimetre.
V. money
It would be impossible to
spend such number of
dollars (or euros)
It would take more than a thousand
billion years to spend a NA of
dollars (or euros) at a rate of one
billion per day.
3
Chart adapted from Poskozim, P.S. et al. (1986). Analogies for Avogadro’s Number. Journal of Chemical Education. Vol. 63.
Number 2 pp. 125-126,
Chart 6
It is, therefore, why in this article we suggest a number of models and analogies that allow students to
gradually move closer to the concepts of mole-Avogadro’s number with particular emphasis on the
order of magnitude in a visual, analogical and operational way.
Firstly, considering the amount of substance as a fundamental magnitude according to the following
table for the International System:
MAGNITUDE
Unity Name
Symbol
Length
L
metre
m
Mass
M
kilogramme
kg
Time
T
second
s
mole
mol
Amount of substance n
Chart 6
In which the three most important fundamental magnitudes (L, M, T) are represented along with the
"amount of substance", since, at least, one of them is present in all the physical derived magnitudes;
so, we shall make an analogical comparison of the amount of substance through the number of
particles, NA, involved in a mole with one of each these three fundamental magnitudes.
1.1
Length
As it is known within each fundamental unit multiple and sub-multiples can be established.
Multiples
Prefix
Symbol
Equivalence
exa
E
1018
peta
P
1015
tera
T
1012
giga
G
109
mega
M
106
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kilo
k
103
hecto
h
102
deca
da
10
Submultiples
deci
d
10-1
centi
c
10-2
mili
m
10-3
micro
µ
10-6
nano
n
10-9
pico
p
10-12
femto
f
10-15
atto
a
10-18
Chart 7
In the case of length we shall firstly use the resort known as "cosmic zoom" or "Powers of 10"
consisting on a film (transferred to slides or PowerPoint presentation) in which from a distance of one
meter can go from man to the infinity of the universe and then reverse the process to penetrate the
myriad microscopic, ie, a cosmic journey in power of 10 jumps from the world of atoms, passing
through the Earth's surface to reach the largest observable structures of the universe in 40 jumps or
steps. Given that the estimated size of the universe gives values of 156,000 million light-years wide
(1,48x1027 m or 1,48.1024 km) and if the number of Avogadro, 6.023 x1023,or 0.602x1024, the quotient
1,48x1024 / 0,602x1024 ≈ 2.5 times NA, so the width of the universe in kilometres is about 2.5 times the
number of Avogadro and is about one billion (109) times the exametre (1018 m).
Continuing with this analogy we can now compare the atomic and subatomic sizes wit astronomical
and geographical distances.
ASTRONOMICAL AND GEOGRAPHICAL DISTANCES
In the following chart we can see that the size of the atom radius is about ten thousand times the size
of the nucleus, one hundred thousand times the radius of the proton / neutron and one hundred million
times higher than the electron
Absolute and relative sizes of the atom and the subatomic particles
Scale in metres (m)
entity
Relative Scale
10-10 m
atom
100 000 000
10-14
nucleus
10 000
10-15
Proton/neutron
1000
10-18
Electron / quark
1
004034
Chart 8
Given that we can establish a geographical analogy as follows:
Considering the hydrogen atom (consisting of a proton and an electron) the nucleus (proton
radio) measuring 5 cm in radius (the equivalent of a ping pong ball), How far would be the
electron?
The answer is D = 5x 105 cm = 500 000 cm ≡ 5 km
Obviously it is important to convey to students the impossibility of representing at scale the usual
visual images in texts or that they can be displayed either on the blackboard or on a screen, the actual
size of the nuclei with respect to electrons of the cortex.
The same happens if we make the comparison with astronomical distances:
Imagine the Rutherford planetary model, If the Earth were the nucleus of the hydrogen atom
(one proton) and was located where the sun is now, How far would be the electron? Radius of the Earth: 6 400 km
D = 6 400 x 100 000 = 6.4 x 108 = 640 x 106 km (640 million kilometres) Ie, the electron would found close to the orbit of Jupiter and its size would be 1000 times smaller (radius = 6.4 km as an asteroid not too large)
On the other hand if the proton were of the size of the Sun the electron would be beyond the orbit of
Pluto.
PLANETS
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Average distance to the
Sun
(millions of Km.)
57,9
108
150
228
778
1.430
2.880
4.500
5.900
Equatorial Diameter
(Km.)
4.878
12.102
12.756
6.786
142.796
120.660
52.400
49.560
2.240
Density (Earth = 1)
0,98
0,95
1,00
0,72
0,24
0,13
0,21
0,28
0,36
Mass (Earth = 1)
0,055
0,815
1,00
0,107
317,7
95,15
14,53
17,15
0,002
Gravity (Earth = 1)
0,38
0,90
1,00
0,38
2,34
1,16
0,79
1,1
0,4
Uranus Neptune
Chart 9
The above example can be a valid explanation of the experimental facts discovered by Rutherford
when bombarding a gold foil with alpha particles: The majority of the particles pass through the foil
without deviating which is explained by the fact that most of the atom is empty space.
MASS
Solve the following problem:
1.- The graphite is pure carbon. Many atoms are set when writing on paper. Let's
approximately evaluate its number with the following data: 2 mm of pencil lead have been
spent doing pencil strokes whose total length is 1600 cm. If the density of graphite is 2.25
g/cm3 and the diameter of the pencil lead measures 0.5 mm, find the number of atoms fixed in
1 cm of stroke.
Sol: 2.75 x 1016 atoms / cm.
004035
Pluto
2 .- Dalton, as you know, is the creator of modern atomic theory. The pointed out, as also the
Greeks Leucipus and Democritus, that a sample of substance (considered as a collection of
atoms) could be subdivided till the atomic limit in which the sample could not be subdivided
further.
Consider a sample of one mole of hydrogen atoms. With a good sharp knife (which must
sharpen repeatedly during the operation), we cut the sample in half, and continue cutting each
time half of the preceding sample to reach the limit of a single atom. The question is, How
many cuts should I do to make the postulate of Dalton?
Sol:: 79
1.2
Time
We propose the following analogy:
Suppose that from the first day of the appearance of the Earth, 4800 million years ago, you
started to make hydrogen atoms in an analytical balance at a rate of x atoms per second.
Yesterday evening at half past three you had placed in the balance exactly 1.008 g of
hydrogen. What is the value of x?
Sol: 4 x 106 four millions atoms per second
2
TEACHING METHODOLOGY
During the last two years in the optional subject Basic Chemistry for students of the specialities:
Primary Education (PE), Infant (IE), Special Education (EE) and Hearing and Language (AL) of the
Teacher Training School of Cuenca (UCLM) we have developed a series of strategies to approach the
concept of mole and Avogadro’s number inside the 3rd topic ((Chemical quantities). We have tried to
adapt with it the tasks and the learning of these concepts to the methodology of ECTS within the
European Higher Education Area and within the pilot program implemented by the University of
Castilla-La Mancha:
CONTENTS OF THE SUBJECT "BASIC CHEMISTRY” (4 ECTS)
Face to face
lesson
Contents
Topic
1 hour
Presentation of the subject. Work methodology. Assesment criteria.
4h
Introduction to Chemistry. Matter and its properties. Matter. quantity of matter.
States of matter. Physical properties. Chemical properties. Intensive and
extensive properties. Pure substances. Elements and compounds, chemical
symbols. Homogeneous and heterogeneous mixtures. Solutions. Separation of
mixtures. Decantation. Filtration. Distillation. Chromatography. Physical
changes. Chemical changes. Conservation of the mass. Conservation of the
energy. Conversion of different forms of energy.
1
4h
Atomic Structure: First approach: atomic Theory of Dalton. Siza of atoms.
Subatomic particles. Electrons. Protons. Neutrons. Atomic and massic
numbers. Unities of atomic mass. Isotopes and atomic weight. Atomic
structure. Nucleus. Quamtum model of the atom. Quantum numbers. Levels,
sublevels
and atomic orbitals. Pauli’s exclusion principle. Electronic
configurations.
2
3h
Chemical quantities: Measure of the quantity of matter. Quantity of substance.
Mole. Moles and particles. Avogadro’s number. Moles of a gas volume.
Percentages. Empirical and molecular formulas.
3
3h
Periodic classification. Electronic configuration and properties: Development
of the periodic table. Electronic configuration and periodicity. Blocks of
elements. Tendencies in the periodic table. Atomic Volume. Ionization Energy.
Electronegativity. Metals and non metals. Properties of the elements and their
position in the periodic table.
4
004036
3h
Chemical bonds: Molecules and ions. Electrons of valence. Stable electronic
configurations. Ionic bond. Ionic compounds. Covalent bond. Molecular
compounds. Lewis structures. Bond energies. Polarity of the covalent bond.
Metallic bond. Intermolecular forces. Interaction among molecules. Molecular
structure and physical properties.
5
4h
States of
Matter. Gases: kinetic molecular theory. Gases. Molecular
interpretation of temperature. Pressure. Hypothesis of Avogadro. Diffusion.
Behaviour of gases. Laws of gases. Liquids. Solids. Changes of state.
6
4h
Solutions. Colligative properties: measure of the concentration. Molarity.,
Molality. Boiling point elevation. Freezing point depression. Osmotic pressure:
Vapour pressure lowering. Raoult’s law
7
4h
Chemical Reactions: Chemical equations. Balance of reactions. Types of
chemical equations. Combination reactions. Decomposition reactions.
Displacement reactions. Combustion reactions. Interpretation of the chemical
equations. Stechiometry. Stechiometrical calculations. Limitant reactant.
Efficiency. Energy of the chemical reactions. Speed of the chemical reactions.
Types of chemical reactions: acid-base; redox and precipitation.
8
Chart 10
2.1
Developed tasks
Initial motivation
With the idea of developing the curiosity of students and motivate them towards the understanding of
the large numbers we have made the following didactic proposals
1. Observe the video "Powers of 10". Objective: to visually familiarize with exponential notation that
they already know in a symbolic and mathematical way.
2. Task: develop the calculations for the short story "The chess board and the grains of wheat."
Compare with the annual production of wheat in Spain. Objective: After the first reactions of
astonishment about the enormity of the resulting value, emphasize that the value of NA is 32,650
times greater, ie, that it would take a thousand years to reach an equivalent amount of grains of
wheat.
1. Los conocimientos didácticos del profesor deben abarcar el componente o dimensión
epistémica (significados institucionales, sus adaptaciones y cronogénesis), dimensión
cognitiva (significados personales, conflictos cognitivos descritos en la literatura), dimensión
instruccional (patrones de interacción, tipos de configuraciones didácticas, su articulación,
optimización de los recursos tecnológicos y temporales) y dimensión afectiva
2. En las evaluaciones realizadas se constató una mejora del 30% en promedio en eficacia en
la resolución de cuestiones y problemas que involucran grandes números en Física y Química
en relación con los dos cursos anteriores
3. CONCLUSIONS
1. We believe that the activities of teaching and learning such as those we have proposed here
reinforce in our students the joy of discovery the applied knowledge or the relationship between theory
and practice.
2. We believe it is important to relate different disciplines together to to try to escape the restricted
limits of a single subject. So we have handled concepts related to physics, chemistry, biology,
ecology, economics, astrophysics, etc..
3. This helps students to understand that the scientific concepts, the systems of measurement, the
units and scales are related to the world around us in a much more important way than it might appear
at first sight.
4. It improves the learning and management of the mathematical calculation especially with regard to
the powers, the exponential and the logarithmic calculations.
5. The knowledge of teacher training should include the component or epistemic dimension (meaning
004037
institutional adaptations and), cognitive dimension (personal meanings, cognitive conflicts described in
the literature), instructional dimensions (interaction patterns, types of educational settings, their
articulation , optimization of technology resources and time) and affective dimension
6. In the the evaluations a 30% of improvement in average efficiency was found when students
resolved issues and problems involving large numbers in Physics and Chemistry in relation with the
two previous courses.
3
REFERENCES
The citation number of a bibliographical reference in text must be enclosed in square brackets [1] .A
list of the references should be given at the end of the paper.
References [Arial, 12-point, bold, centred and capitalize the first letter]
[1] Jensen, W.B. (1998). Logic, History, and the Chemistry Textbook I, does Chemistry have a Logical
Structure? Journal of Chemical Education, 75,7, 817-828
[2] Alzate Cano, Mª V. (1997). Campo conceptual Composición/Estructura en Química. Tendencias
cognitivas, etapas y ayudas cognitivas. Tesis Doctoral. Universidad de Burgos 1997.
[3] Johnstone, A.H. (1993). The developmento of chemistry teaching: A changing respone to
changing demand. Journal of Chemical Education. 1993, vol. 70, no9, pp. 701-705.
[4] De Jon, O. & van Driel, J. “Prospective teachers’ concerns about teaching chemistry topics at a
macro-micro-symbolic interface” Paper presented at a 1999 NARST annual meeting. Boston. USA.
[5] Furió, C y otros (1993)
[6] Furió, C. ; Azcona, R. y Guisasola, J. (1999). Dificultades conceptuales y epistemológicas del
profesorado en la enseñanza de los conceptos de sustancia y de mol. Enseñanza de las Ciencias, 17
(3). p. 359-376.
[7] Gabel D. J.(1993), Use of the particle nature of matter in developing conceptual understanding, J.
Chem. Educ., 70 [3], 193-194.
[8] Angelini, M.C. et al (2001). Estrategia didáctica para vincular distintos niveles de
conceptualización. Estudio de un caso (parte 1). Educación Química.12 (3) p. 149-157.
[9] Pozo Municio, J.I.. Aprender y enseñar ciencia. Morata. España, 1998. p. 185
[10] Poskozim, P.S. et al. (1986). Analogies for Avogadro’s Number. Journal of Chemical Education.
Vol. 63. Number 2 pp. 125-126,
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