Avogadro’s concept of equivalents for teaching cationexchange capacity1

Steve J. Thien2

ABSTRACT

A thorough understanding of cation exchangecapacity, which requires mastery of many chemicalconcepts in an introductory soils course, is crucial tofully comprehending the nature of soils. Studentperformance in this area had been especially low oncourse objectives requiring working knowledge of theconcept of chemical equivalents. Four specific dif-ficulties associated with understanding and usingequivalents are examined. Avogadro’s concept of anequivalent as 6.02 X 1023 charges is outlined as analternative pedagogical approach. The advantages ofsimplicity and readily apparent stoichiometry areoffered in some examples. The approach has sig-nificantly increased student performance on relatedcourse objectives.

Additional index words: Soil chemistry, Pedagogi-cal approach.

A MEDEO Avogadro, a 19th century Italianphysicist, is perhaps best remembered for his

hypothesis that laid the framework for our under-standing of molecular weights. In addition to dis-tinguishing between molecules and atoms, his workestablished that equal reacting units (moles) musthave the same number of molecules. The numberbears his name, is customarily denoted by the sym-bol N, and has a value of 6.02 X 1023 (8). Insteadof expanding on the concept of a mole as a finitenumber of reacting units, chemical educators in thepast have favored a combining ratio approach. Butnow, Hawthorne (6) has reported on a growingtendency in chemistry education to abandon thedefinition of a mole as a mass of material that hap-pens to react with 16 g oxygen (or 1 g hydrogen,or 12 g carbon), and, instead, to teach students thatthe mole is Avogadro’s number, N, of molecules, aclearly defined number of particles. An extensionof using this definition, shows an equivalent to sim-ply be Avogadro’s number, N, of charges. While inno way representing a new chemical theory, thechange signifies a different reasoning schemeneeded by its learners and users.

Implications of this pedagogical change willquickly spread to related chemical education inagronomic courses. While most agronomic teachers

will have probably learned their chemistry underthe aforementioned definition, their students willcome with another concept in mind. The role ofchemistry is so basic to agronomic education thatthe resultant information gap needs to be recon-ciled. The basic simplicity of using Avogadro’s con-cept, plus its ability to organize and clarify manypreviously difficult concepts to quantitative chemis-try make the new approach educationally attractive.

This paper uses cation exchange as the agronomicformat for illuminating the advantages of teachingagronomic chemistry based on pedagogical use ofAvogadro’s number.

DIFFICULTIES IN LEARNING EQUIVALENTCHEMISTRY

Cation exchange is a fundamental chemical con-cept used in understanding soil science. Its im-portance to comprehending the nature of soils isfrequently considered parallel to the impact ofphotosynthesis in studying crop science. Yet, thenecessity of defining and explaining this theory byusing difficult chemical terminology and conceptsfrequently presents a block to learning. Soils in-structors, especially those in introductory courseswhere students are initially exposed to cation ex-change, must recognize the challenge of such alearning situation. A thorough understanding ofthis concept is so fundamental and crucial to fullcomprehension of the properties of soils that every

teacher’s effort should reflect the requisite amountsof time, talent, resources, and pedagogical ap-proaches to insure maximum understanding.

The high degree of difficulty students exhibit incomprehending the nature of cation exchange hasbeen repeatedly acknowledged in informal discus-sions on agronomic teaching. My own discussionsabout this problem with students and their per-

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IContribution No. 1, College of Agriculture, KansasState Univ., Manhattan, KS 66506.

2Associate professor of agronomy, Kansas State Univ.,Manhattan.

36 JOURNAL OF AGRONOMIC EDUCATION

formance in meeting specific course objectives onthe various components of cation exchange havefocused on some specific difficulties. It seems thedefinition, cation-anion attraction, utility, andcause present little difficulty compared with master-ing a working use of the term "equivalent" to ex-press relationships. While there seems to be nosuitable alternative to using this terminology, theshift in basic chemical education to using a conceptof molecular chemistry originally explained byAvogadro offers some solutions to making the con-cept of chemical equivalents more easily learned.

Four areas of difficulty frequently mentioned bystudents lend insights to their problem. First,agronomy students are quick to acknowledge a dif-ficulty in understanding the conceptual definitionof equivalency. Through non-Avogadron generalchemistry texts, they have learned that the equiva-lent is a mass of material that combines with 1 g ofhydrogen (or I6 g oxygen, or 12 g carbon). Moststudents seem able to recite that relationship frommemory, or at least acknowledge an exposure to it.Why the difficulty then? The difficulty seems tobe not in understanding what is said, but in under-standing the basis for saying it. In other words, itcomes across more as an example than a definition.

Another problem is encountered because someagronomic texts suggest that an equivalent weightcan be arrived at by dividing an ion’s atomic weightby its valence. If a student fails to comprehend thetextbook definition above, examining this relation-ship strains the logic of the concept even more--asfollows. Both dimensions, atomic weight and val-ence, are essentially unitless relations representingratios of combining weights (or numbers of atoms)and ionic charges. Hence, the quotient should alsobe unitless, but it isn’t because an equivalent isgiven the units of grams. Assuring the student thatthe atomic weight can be assigned the units ofgrams, and now can be called the gram atomicweight, strains logic further. And it is no time totell a puzzled student that it "just works out thatway", or to freshen up on beginning chemistry. Amore nearly logical approach is needed to build aknowledge base from which to learn more aboutsoils. If you do not, you later have to point outthat the formula (atomic weight + valence) doesnot necessarily work for oxidation-reduction reac-tions.

A third difficulty in understanding and using theequivalent-as-a-mass concept is its lack of apparentstoichiometry. After all, why should 9 g aluminum,20 g calcium, and 23 g sodium all be chemicallyequivalent to 1 g hydrogen? That apparent whimsi-

cal relationship, which is also embedded in theprevious two examples, contributes considerableconfusion.

Students who may grasp the previous threepoints point out an additional difficulty. Theyponder how an equivalent of base, something beingcomprehended and defined in terms of so manygrams, reacts in the soil not with grams of anything,but with negatively charged sites that have no ap-parent weight parameter--more confusion aboutwhat an equivalent really represents.

Those who have mastered the concept of anequivalent find no real obstacles to comprehendingthe above examples. Frustration on the teacher’spart maybe forthcoming, however, when strugglingwith a student, or many students, who simply "donot see" the relationship. An "examplish" defini-tion, ghost units, lack of obvious stoichiometry,and disimilar reacting units offer little help in sucha situation.

From a pedagogical viewpoint, describing equiva-lents as a weight of material seems to be the basisof the difficulties described. Hawthorne’s (6) re-search shows that about one-third of the most re-cent chemistry texts give pedagogical approachesusing N, not just merely mentioning its value. Sixrecently published introductory soils texts (1,2, 3,4, 5, 7) give no treatment of equivalents as Avoga-dro’s number of reacting charges. One (4) mentionsthe value of N, so a conceptual gap already existsbetween educational approaches used in chemistryand agronomic texts.

MEETING THE DIFFICULTIES

Avogadro’s concept directly addresses the fourstudent-acknowledged problem areas with clarityand logic. A mole is conveniently explained as afinite number, Avogadro’s number, N of molecules(def. 1),

A mole equals 6.02 X 1023 molecules [1]

and an equivalent is likewise simply explained asAvogadro’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 aredox reaction, an equivalent refers to Avogadro’snumber, N, of electrons given off or taken up.Since cation exchange reactions represent nonredoxreactions, discussion here is confined to that partof the definition.

THIEN: AVOGADRO'S CONCEPT IN TEACHING 37

Those two definitions greatly simplify the con-cept of moles and equivalents and rule out ghostunits. As will be seen later, all units agree with thebasic rules of unit cancellation, lending the neededlogic and self-checking capability to problem setup.

Perhaps the greatest single advantage associatedwith Avogadro's concept of equivalents is thestoichiometry it contributes to problems. Becauseof the readily apparent stoichiometry, enormousareas of quantitative chemistry can be organizedand clarified for students by using the concept ofan equivalent as a standard number of reactingunits. For example, in an exchange reaction anequivalent of material contains 6.02 X 1023 positivecharges and replaces only that amount of other ma-terial from exchange sites that also possess 6.02 X1023 charges. This approach has the logic missingwhen one explains that 20 gCa2+ occupies the sameamount of the cation exchange capacity as 39 g K+.

In the same manner, a soil with 20 me/100 gexchange capactiy becomes a soil with (20/1000)(6.02 X 1023) charged sites per 100 g. It is easy tocomprehend how that much soil holds an amountof ions whose charges also sum to (20/1000) (6.02X 1023). If the ions were all monovalent, thenthere 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 occupythe same number of sites.

Critics of the pedagogical use of Avogadro'snumber say it is too large to be fully comprehendedand is bulky to use in equations. Whether largenumbers or combining weight ratios are more easilyunderstood and used depends on an individual'spre-conditioning. When chemistry texts do the pre-conditioning needed to work with exponentialterms, then it is advantageous to adapt this stoichio-metric scheme to teaching cation exchange.

CLASSROOM APPLICATION

A switch to Avogadro's concept of equivalentchemistry has resulted in dramatic improvement ofstudent performance on related course objectives.Before changing, less than 20% of a large introduc-tory soils class handled cation exchange problemson exams satisfactorily. One semester of collegechemistry is a prerequisite for this class, but a non-Avogadron approach was being taught. For thepast three semesters, a class handout (Fig. 1) hasbeen used in conjunction with a problem set,virtually identical in content under both systems.Satisfactory student performance on related testitems is now approaching 70 to 80%. Even though

Two basic computations involving equivalents are needed when workingwith cation exchange situations; either equivalents are converted to grams,or vice versa. After the conversions are made, further mathematicalmanipulations may be required to solve a problem, but they usually repre-sent simpler numerical logic. Examples of each conversion using the con-cept of an equivalent as Avogadro's number (N = 6.02 X 10") of chargesare given here. For easier calculation, leave all exponents of 10 as the 23rdpower.I. Converting equivalents to grams

Example A. How many grams of sodium are in 2.1 equivalents ofsodium?

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 equivalentsExample 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 102-')^(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 1021.

Fig. 1. Using equivalents in cation exchange applications.

taught another way in this chemistry course, virtual-ly all students will use the method outlined in Fig.1 on exams.

Switching to the Avogadron concept of equiva-lents has smoothed out a previously rough portionof an introductory soils course, both from an in-structor's and student's viewpoint.

SUMMARY

Complete comprehension of cation exchangetheory in soils is virtually impossible without aworking knowledge of chemical equivalents. Basicchemistry education trends toward teaching a moresimplified explanation of the equivalent. Thistraining method has merit because it adds order andclarification to a complicated concept. This paperis to alert readers to the change with the hope thatinterest in effective instruction will generate afollow-through evaluation of its usefulness in agro-nomic teaching situations.

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