Avogadro’s
conceptof equivalentsfor teachingcation
1
exchangecapacity
2Steve J. Thien
ABSTRACT
A thoroughunderstandingof cation exchange
capacity, whichrequiresmasteryof manychemical
concepts
in an introductory
soilscourse,is crucialto
fully comprehending
the nature of soils. Student
performance
in this area hadbeenespeciallylowon
courseobjectivesrequiringworking
knowledge
of the
conceptof chemicalequivalents.Fourspecific difficulties associatedwith understanding
andusing
equivalentsare examined.
Avogadro’s
conceptof an
equivalentas 6.02 X 1023chargesis outlinedas an
alternative pedagogical
approach.
Theadvantages
of
simplicity andreadily apparentstoichiometryare
offered in someexamples.Theapproachhas significantly increasedstudentperformance
on related
course
objectives.
will have probably learned their chemistry under
the aforementioned definition,
their students will
come with another concept in mind. The role of
chemistry is so basic to agronomic education that
the resultant information gap needs to be reconciled. The basic simplicity of using Avogadro’s concept, plus its ability to organize and clarify many
previously difficult concepts to quantitative chemistry make the new approach educationally attractive.
This paper uses cation exchange as the agronomic
format for illuminating the advantages of teaching
agronomic chemistry based on pedagogical use of
Avogadro’s number.
Additionalindexwords:Soil chemistry,Pedagogical approach.
DIFFICULTIES IN LEARNINGEQUIVALENT
CHEMISTRY
Cation exchange is a fundamental chemical concept used in understanding soil science. Its importance to comprehending the nature of soils is
frequently considered parallel
to the impact of
photosynthesis in studying crop science. Yet, the
necessity of defining and explaining this theory by
using difficult
chemical terminology and concepts
frequently presents a block to learning. Soils instructors, especially those in introductory courses
where students are initially
exposed to cation exchange, must recognize the challenge of such a
learning situation.
A thorough understanding of
this concept is so fundamental and crucial to full
comprehension of the properties of soils that every
teacher’s effort should reflect the requisite amounts
of time, talent,
resources,
and pedagogical approaches to insure maximumunderstanding.
The high degree of difficulty students exhibit in
comprehending the nature of cation exchange has
been repeatedly acknowledged in informal discussions on agronomic teaching. My own discussions
about this problem with students and their per-
Avogadro, a 19th
A MEDEO
physicist, is perhaps best
century
Italian
remembered for his
hypothesis that laid the framework for our understanding of molecular weights. In addition to distinguishing between molecules and atoms, his work
established that equal reacting units (moles) must
have the same number of molecules.
The number
bears his name, is customarily denoted by the symbol N, and has a value of 6.02 X 1023 (8). Instead
of expanding on the concept of a mole as a finite
number of reacting units, chemical educators in the
past have favored a combining ratio approach. But
now, Hawthorne (6) has reported
on a growing
tendency in chemistry education to abandon the
definition of a mole as a mass of material that happens to react with 16 g oxygen (or 1 g hydrogen,
or 12 g carbon), and, instead, to teach students that
the mole is Avogadro’s number, N, of molecules, a
clearly defined number of particles.
An extension
of using this definition, shows an equivalent to simply be Avogadro’s number, N, of charges. While in
no way representing
a new chemical theory, the
change signifies
a different
reasoning scheme
needed by its learners and users.
Implications
of this pedagogical change will
quickly spread to related chemical education in
agronomic courses. While most agronomic teachers
IContribution No. 1, College of Agriculture, Kansas
State Univ., Manhattan, KS66506.
2Associate professor of agronomy,Kansas State Univ.,
Manhattan.
35
36
JOURNAL
OF AGRONOMIC
formance in meeting specific course objectives on
the various components of cation exchange have
focused on some specific difficulties.
It seems the
definition,
cation-anion attraction,
utility,
and
cause present little difficulty compared with mastering a working use of the term "equivalent" to express relationships.
While there seems to be no
suitable alternative to using this terminology, the
shift in basic chemical education to using a concept
of molecular chemistry originally
explained by
Avogadro offers some solutions to making the concept of chemical equivalents more easily learned.
Four areas of difficulty
frequently mentioned by
students lend insights to their problem. First,
agronomy students are quick to acknowledge a difficulty in understanding the conceptual definition
of equivalency.
Through non-Avogadron general
chemistry texts, they have learned that the equivalent is a mass of material that combines with 1 g of
hydrogen (or I6 g oxygen, or 12 g carbon). Most
students seem able to recite that relationship from
memory, or at least acknowledge an exposure to it.
Why the difficulty
then? The difficulty
seems to
be not in understanding what is said, but in understanding the basis for saying it. In other words, it
comes across more as an example than a definition.
Another problem is encountered
because some
agronomic texts suggest that an equivalent weight
can be arrived at by dividing an ion’s atomic weight
by its valence. If a student fails to comprehend the
textbook definition above, examining this relationship strains the logic of the concept even more--as
follows. Both dimensions, atomic weight and valence, are essentially unitless relations representing
ratios of combining weights (or numbers of atoms)
and ionic charges. Hence, the quotient should also
be unitless, but it isn’t because an equivalent is
given the units of grams. Assuring the student that
the atomic weight can be assigned the units of
grams, and now can be called the gram atomic
weight, strains logic further. And it is no time to
tell a puzzled student that it "just works out that
way", or to freshen up on beginning chemistry. A
more nearly logical approach is needed to build a
knowledge base from which to learn more about
soils. If you do not, you later have to point out
that the formula (atomic weight + valence) does
not necessarily work for oxidation-reduction
reactions.
A third difficulty in understanding and using the
equivalent-as-a-mass concept is its lack of apparent
stoichiometry. After all, why should 9 g aluminum,
20 g calcium, and 23 g sodium all be chemically
equivalent to 1 g hydrogen? That apparent whimsi-
EDUCATION
cal relationship,
which is also embedded in the
previous two examples, contributes
considerable
confusion.
Students
who may grasp the previous
three
points point out an additional
difficulty.
They
ponder how an equivalent of base, something being
comprehended and defined in terms of so many
grams, reacts in the soil not with grams of anything,
but with negatively charged sites that have no apparent weight parameter--more
confusion about
what an equivalent really represents.
Those who have mastered the concept of an
equivalent find no real obstacles to comprehending
the above examples. Frustration
on the teacher’s
part maybe forthcoming, however, when struggling
with a student, or many students, who simply "do
not see" the relationship.
An "examplish" definition, ghost units, lack of obvious stoichiometry,
and disimilar reacting units offer little help in such
a situation.
From a pedagogical viewpoint, describing equivalents as a weight of material seems to be the basis
of the difficulties
described. Hawthorne’s (6) research shows that about one-third of the most recent chemistry texts give pedagogical approaches
using N, not just merely mentioning its value. Six
recently published introductory soils texts (1,2, 3,
4, 5, 7) give no treatment of equivalents as Avogadro’s number of reacting charges. One (4) mentions
the value of N, so a conceptual gap already exists
between educational approaches used in chemistry
and agronomic texts.
MEETINGTHE DIFFICULTIES
Avogadro’s concept directly addresses the four
student-acknowledged
problem areas with clarity
and logic. A mole is conveniently explained as a
finite number, Avogadro’s number, N of molecules
(def. 1),
A mole equals 6.02 X 10 23 molecules
[1]
and an equivalent is likewise simply explained as
Avogadro’s number, N, of chemical charges (def. 2)
An equivalent
equals 6.02 X 1023 charges [2]
This definition is for a nonredox reaction. In a
redox reaction, an equivalent refers to Avogadro’s
number, N, of electrons
given off or taken up.
Since cation exchange reactions represent nonredox
reactions, discussion here is confined to that part
of the definition.
THIEN:
AVOGADRO'S CONCEPT IN TEACHING
Those two definitions greatly simplify the concept of moles and equivalents and rule out ghost
units. As will be seen later, all units agree with the
basic rules of unit cancellation, lending the needed
logic and self-checking capability to problem setup.
Perhaps the greatest single advantage associated
with Avogadro's concept of equivalents is the
stoichiometry it contributes to problems. Because
of the readily apparent stoichiometry, enormous
areas of quantitative chemistry can be organized
and clarified for students by using the concept of
an equivalent as a standard number of reacting
units. For example, in an exchange reaction an
equivalent of material contains 6.02 X 1023 positive
charges and replaces only that amount of other material from exchange sites that also possess 6.02 X
1023 charges. This approach has the logic missing
when one explains that 20 gCa 2 + occupies the same
amount of the cation exchange capacity as 39 g K+.
In the same manner, a soil with 20 me/100 g
exchange capactiy becomes a soil with (20/1000)
(6.02 X 1023) charged sites per 100 g. It is easy to
comprehend how that much soil holds an amount
of ions whose charges also sum to (20/1000) (6.02
X 1023). If the ions were all monovalent, then
there also would be (20/1000) (6.02 X 1023) ions.
If the ions were divalent, only (1/2) (20/1000)
(6.02 X 1023) ions would be necessary to occupy
the same number of sites.
Critics of the pedagogical use of Avogadro's
number say it is too large to be fully comprehended
and is bulky to use in equations. Whether large
numbers or combining weight ratios are more easily
understood and used depends on an individual's
pre-conditioning. When chemistry texts do the preconditioning needed to work with exponential
terms, then it is advantageous to adapt this stoichiometric scheme to teaching cation exchange.
CLASSROOM APPLICATION
A switch to Avogadro's concept of equivalent
chemistry has resulted in dramatic improvement of
student performance on related course objectives.
Before changing, less than 20% of a large introductory soils class handled cation exchange problems
on exams satisfactorily. One semester of college
chemistry is a prerequisite for this class, but a nonAvogadron approach was being taught. For the
past three semesters, a class handout (Fig. 1) has
been used in conjunction with a problem set,
virtually identical in content under both systems.
Satisfactory student performance on related test
items is now approaching 70 to 80%. Even though
37
Two basic computations involving equivalents are needed when working
with cation exchange situations; either equivalents are converted to grams,
or vice versa. After the conversions are made, further mathematical
manipulations may be required to solve a problem, but they usually represent simpler numerical logic. Examples of each conversion using the concept of an equivalent as Avogadro's number (N = 6.02 X 10") of charges
are given here. For easier calculation, leave all exponents of 10 as the 23rd
power.
I. Converting equivalents to grams
Example A. How many grams of sodium are in 2.1 equivalents of
sodium?
Step 1. Determine the number of charges in 2.1 equivalents.
(#equivalents)(charges/equivalentl* = #charges
(2.11(6.02 X 10") = 12.6 X 10" charges
Step 2. Convert number of charges to number of ions.
(# charges) ^ (— charges/ion)! = ~ ions
(12.6 X 10") -Ml) = 12.6 X 10" ions
Step 3. Convert number of ions to weight of ions.
(^ ions)(grams/ion}§ = grams
(12.6 X 10")(3.8 X 10~"g/Na'ions) = 48.3glMa'
II. Converting grams to equivalents
Example B. How many equivalents of calcium are in 2 grams of
calcium?
Step 1. Convert grams of ions to number of ions.
(gram) ^ (gram/ion) § = #ions
(2) + (6.6 X 10""l = 0.3 X 10" ions
Step 2. Convert number of ions to number of charges.
(#ions)(#charges/ionlt = #charges
(0.3 X 10" ions)(2) = 0.6 X 10" charges
Step 3. Convert number of charges to equivalents.
(•# charges) "^ (# charges/equivalent)* = #equivalents
(0.6 X 10 2 -')^(6.02 X 10" 1 = 0.1 equivalents Ca2+
* by definition from Avogadro's number,
t obtained from the Periodic Table, i.e., the valence.
§ obtained by dividing the weight of one mole of the ion (obtained from
the Periodic Table) by the number of ions in one mole; i.e., 6.02 X 10 21 .
Fig. 1. Using equivalents in cation exchange applications.
taught another way in this chemistry course, virtually all students will use the method outlined in Fig.
1 on exams.
Switching to the Avogadron concept of equivalents has smoothed out a previously rough portion
of an introductory soils course, both from an instructor's and student's viewpoint.
SUMMARY
Complete comprehension of cation exchange
theory in soils is virtually impossible without a
working knowledge of chemical equivalents. Basic
chemistry education trends toward teaching a more
simplified explanation of the equivalent. This
training method has merit because it adds order and
clarification to a complicated concept. This paper
is to alert readers to the change with the hope that
interest in effective instruction will generate a
follow-through evaluation of its usefulness in agronomic teaching situations.
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JOURNAL OF AGRONOMIC EDUCATION