William G. Davies
I
and John W. Moore Eastern Michigan University Ypsilanti. MI 48197
I
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in Introductory Chemistry
Thcrc is no question that problems involving pruportionalitv. e i ~ e ~ i ithoce l h havine to du with stoichiometric relationships, are major hurdle for many students in introductory chemistry c ~ u r s e s ? .One ~ approach to this difficulty is to work around it by avoiding problems that involve proportions,' but this is not appropriate for students who wish to pursue scientific careersandis probably a disservice to many others as weL2 We believe that the terminology and conventions associated with the International Svstem of Units (SI units)3.4 can provide considerable aid to both teachers and students who face the difficulties of stoichiometric calculations. We have already described other advantages of SI units in this Journal5 and will not repeat them here.
a
Conventions Assoclated with SI Units Making a measurement involves comparing some unknown nhvsical auantitv. " , sav " the mass of a sample of salt. with a known unit. The unknown quantity is measured by counting how many units are required to equal it, and the result of the measurement is expressed as the product of that number times the unit. As a result of international agreement, seven hase S I units have been defined and adopted:the mete; to measure length; the kilogram to measure mass; the second to measure time; the ampere to measure electric current; the kelvin to measure thermodvnamic temperature: the candela to measure luminous intensity; and the mole to measure amount of suhstance. Prefixes such as milli- or mega- are available to adjust the S I hase units to sizes that are aunronriate for a widevariety of calculations. Physical quantities other than the seven mentioned above are expressed as the product of a number times a compound unit, that is, a unit that itself is a product or quotient of SI base units. For example an energy might be expressed as 10.4 J = 10.4 kg m2 sr2.In some cases a pure (unitless) number may be encountered-when counting atoms, for example-hut in most calculations physical quantities are involved. A manual of symbols and terminology for physical quantities and units is available from IUPAC.3 I t suggests symbols whose universal adoption would make algebraic formulas that summarize physical and chemical relationships as unambiguous as possible. S I units have been designed to he used together with a convention known as the quantity calculus4 when calculations with physical quantities are done. The rules of the quantity calculus are simple: (1) Units as well as numhers obey the conventional rules of arithmetic and aleehra: (2) A svmbol in an algel~raicformuln should r~presentu quantity regardless of units: nnd (3) The eauals sicn should be used onlv to con. nect two cpantities that have the same dimensions.kule (1) is almost universally adopted among chemistry instructors and needs no further explanation. Rule (2) says that algebraic formulas should he the same whether, say, a density is expressed in g/cm3, kg/m3, or any other unit that has dimensions of mass over length cubed. This rules out a number of algebraic formulas that can still he found in chemistry textbooks. For example,
..
.. .
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Jnv no. moles HCI = 1000 where A is molarity and V is the numher of milliliters of solution, is not in accord with rule (2) because if V is not in milliliters (or cm3) the formula will give the wrong answer.
Actually formula 1 is a remnant of an earlier convention in which onlv numbers (and not units) were used in calculations. Rule (3) is simply the hasis for dimensional analysis; unit cancellation provides a check on the correctness of a calculation only if all algebraic equations are required to have the same dimensionality on both sides. A major obstacle to more consistent use of the quantity calculus in chemistry is the fact that several important terms which are currently in almost universal use do not lend themselves to a system where a symbol represents a number times a unit. The most common such term is the phrase "number of moles." In the case of other quantities phrases like "number of grams" or "numher of centimeters" are seldom heard. We speak instead of "mass" or "length," terms that do not imply a specific unit. Terms that include "number of. . ." fit more naturally into the methodology of eqn. (1) above (where pure numbers must he substituted) than into the quantity calculus. Such terms are also ambiguous, since it is uncertain whether a phrase like "number of centimeters" means a numher like 2 or a quantity like 2 cm. The phrase "number of moles" is used widely because until recently there was no alternative. Lack of a word that bears the same relationship to "numher of moles" that "mass" bears to "number of erams" has made the mole seem somehow unique and moridifficult to handle than other units. Fortunately the SI definition d mole and the II'PAC manual have suppiied the name amount of substance and the symbol n for the quantity that the mole measures. Of course we must remember that the term amount nou has a specific scimtific definition. Toavoid confusion it is hest not to use amount in its grneral, nmchemiral sense. After a few mmths'expcrience it beromes natural to substitute a phrase like "quantitv of heat" for the notentiall\. nmfusine "amount of heat." Such care in phraseology is mire than reiaid by the precision which the term amount of substance brings to our ability to describe stoichiometric calculations. Another i m ~ o r t a nconventional t term that needs to he revised is moleckar weight. The strict definition of molecular weight refers to relative molecular mass and is dimensionless. In the case of 02, for example, the molecular weight would be 32.00. Sometimes, however, the molecular weight is also regarded as the actual mass of a molecule (32.00 amu for Oz), and sometimes as the mass per unit amount of substance (32.00 glmol). The third meaning is most convenient for stoichiometric calculations, and IUPAC recommends that the term molar mass and the symbol M be used whrn mass per unit amount of iuhstance is meant. If wemeasure the amount n x and the mass mx of substance X, the molar mass M x is given by:
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Bent, H. A,, J. Coll. Sci.Teaching, 6 (1).48 (1976). 2Arons,A. B., J. Coll. Sci. Teaching 6 (4), 248 (1977). (a) McGlashan, M. L. and Paul, M. A,, "Manual of Symbols and Terminology far Physicoehemical Quantities and Units," IUPAC Additional Publication, Butterworths, London, 1975; (b) MeGIashan, M. L., "Physieochernical Quantities and Units," Royal Institute of Chemistry, Monographs far Teachers, No. 15, London, 1971; Paul, M. A,, Chemistry, 45 (9), 14 (October 1972). Guggenheim, E. A,, J. CHEM. EDUC.,35,606 (1958). Davies, W. G., Moore, J. W., and Collins, R. W., J. CHEM. EDUC., 53,681 (1976). Volume 57, Number 4, April 1980 / 303
M x = - mx nx Note that even though molar mass is commonly expressed in nnits of grams per mole, it is bad practice to define molar mass in terms of specific units, that is, as the "number of grams per mole." Such a definition suggests falsely that if we express a molar mass in different nnits, we need a different name and a different symbol for it. For instance, when calculating the root mean square velocity ii of molecules of an ideal gas whose molar mass is 32.00 glmol at a temperature of 800 K, the formula
when we speak of the hydrogen ion concentration, for example). Concentration is defined as the amount of solute per unit volume of solution, that is, When solved for n x , eqn. (2) fulfills the same function as eqn. ( I ) , but is independent of the choice of concentration o r v d ume units. Those who adopt the IUPAC definition of concentration do face the difficulty that most current textbooks use the word concentration to include mole fraction, molality, and sometimes mass fraction. However, it is a simple matter to use the term solution composition in this wider sense, reserving the name concentration for the restricted definition of eqn. (2). The term molar concentration was not adopted by IUPAC because of potential confusion with the reserved use of the adjective molar to mean "per unit amount of substance," as in molar volume or molar mass, where the amount of substance unit is in the denominator. The question of what to do about pH also arises, because quite a different scale would be calculated if the logarithm of the number of moles per cubic meter (using SI base units) were taken. The answer is that since the logarithm of a unit is not defined, the concentration must be divided by appropriate units before taking a logarithm. Those units are chosen to be molIdm3, and for unit activity coefficient pH can be descrihed by
can he used regardless of the nnits of M. There is no need to invent anew symbol such as KMM (kilogram molar mass) to perform the calculation. Use of unity factors such as 1000g k g to convert units to the desired form is an important aspect of the quantity calculus. A third conventional term that acts as an impediment is molarity, which most textbooks define as "the number of moles of solute per liter of solution," that is, in terms of specific units. When different units are used, such as the SI base units mol/m3 (which, by the way, are useful in the electroplating pH = -loglc~+/maldm-? (3) industry), there issomeuncertainty asu, whether thequantity although the actual definition of pH is an operational one.3b s h d d he called moltuity. (This pwhlem is also c k ~ e l yrelatad Equation (3) maintains the conventional pH scale. to the convention of eqn. (I),which requires pure numbers.) In order to describe the composition of a solution without Roadmapping Stolchlometry Problems presupposing particular units, IUPAC has suggested that we An important pedagogical advantage of adopting SI units abandon the term molarity and use concentration instead (as Conversion Relationships Commonly Used In Chemistry Related Quantities
Volume
Amount of substance
Amount of substance
Amount of W consumed or produced
+.
Proportionality Factor
Volume of solution
Electric charge
-
Roadmap
mas
density, p
V o m
M I S
molar mas. M
n
number of particles
Avagadro constant. L
n-N
Amount of Y consumed or DrCduCed
Stoichioretric ratio,
M
u
m
L
S(WI YJ
n
S~W,
nw
v
Amount of Qas
Amount of X consumed or produced
Definition
RT/P
u
."A
Heat energy absorbed from surroundings
Molar enthalpy change. AH,
nx-q,
Amount of Solute X
Concentrationof solute, cx
it-nx
Amount of electrons
Faraday constant, F
o o n .
304 1 Journal of Chemical Education
SX
n,
and the IUPAC recommendations is that they provide a simple symbol for each of the quantities regularly used in chemical calculatious. These symbols can he used very effectivelv to "roadmao" the loeic of stoichiometric calculations. Iwrnuie mua such calculatiuns involve a series of stepi in each uiwhich one uumtitv (/I is cnlculated from another auantitv Q2 to which the firit quantity is proportional. ~ a c step, h therefore, involves a relationship of the form: quantity 1 = proportionality factor X quantity 2; i.e., Q1= P X Qz or quantity 2 =
1
proportionality factor x quantity 1;i.e., Q2=
(i)x Q1
Such "conversions" from one quantity to another are easily roadmapped by the diagrams
the ratio ? L A ~ N o ~ / ? Z B ~ C I is ~ dimensionless. For this to he true the units mol AgNOa and rnol BaClz must cancel. In other words there should be no distinction between the units rnol AgNO3 and mol BaCIz. In general, even though the SI definition of the mole states that one must specify clearly what elementary entities are being considered, the IUPAC recommendations do not encourage making a distinction between the units moles of A and moles of B. We feel that this lack of distinction is an impediment to the beginning student, because it prevents the cross-check permitted by unit cancellation. Therefore we have introduced a new term which we call the stoichiometric ratio. This quantity reflects the fact that proportionality factors relating the amount of one substance to an amount of another suhstance can be obtained from subscripts in correct chemical formulas and from coefficients in balanced chemical equations. In step 2 of the solution of the example, for instance, the relevant stoichiometric ratio is
Q~ZQI~~~QIZQQ~ This technique is best illustrated by applying it to an example prohlem. Suppose a sample of BaC12 whose mass is 0.385 g is titrated with a solution containing 0.0983rnol AgN03 per liter. The equation for the reaction is
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BaCMaq) + 2AgNOdaq) 2AgCl(s)+ Ba(N0s)daq) What volume of AgNOa(aq) is required to achieve the equivalence point? A typical solution to this prohlem might appear as: 1mol BaClz 2 mol AgN08 1liter V = 0.385 g X 208.2 -e 1 rnol BaCb 0.0983 rnol AeN07 - V = 0.037filiter from which it is obvious to a student that the units on the right-hand side divide out to give liters. Much less obvious is why the prohlem solver replaced the units in the exact sequence:
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-
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g mol BaClz mol AgN03 liter In our experience a great many students will ask "I see that the units cancel, hut how did you know to use the molar mass of BaCIz?" That the units cancel is a necessary, but not a sufficient, condition for a correct solution. What is needed is a concrete guide to the logic of solving the prohlem, and such a guide can be provided in the form of a roadmap. The first step in solving the example problem involves conversion from a mass to an amount of suhstance for a sample of BaCI2.It can be roadmapped as:
The corresponding calculation is obtained by setting the desired quantity, in this case ns,az. equal to the quantity on the left of the roadmap times the proportionality factor. In this case the proportionality factor is the reciprocal of the molar mass. ne,clz
= rnB.a,
X
1 --= 0.385 g
M B ~ I ~
m"l BaC12= 0.001 85 mol BaC12 208.2 g The swund step required to sulve the example makes use of a t~alancedchemicul equation. The coeificient 2 on the left side of the equation shows that an amount of AgNOn twice as great as the amount of BnCl? nvailahlc must he consumed in the reaction to achieve the endpoint. Consequently we can calculate the amount of AgNO, as X
nAgN03 = 2 X nBaC1~
(5)
nnpN03=2
(6)
and ~ B ~ C I Z
1mol BaC12 in which we do make a distinction between rnol AeNO* " " and rnol BaC12. This stoichiometric ratio enables us to relate the amount of AgN03 consumed nA,No, to the amount of BaClz consumed nB.a, by the formula - 2 m0l AgNO8 nAsNOs = BaClz) - 1 ma1 BaClp nBeCls The corresponding roadmap is
(
Note that (including reciprocals) eleven other stoichiometric ratios may be obtained from the coefficients of the equation in the example. If we restrict our attention to an individual compound, say BaClz, even more stoichiometric ratios are available. For example, 2 mol C1 CI 2 mol C1 1mol Ba = l mol BaCI, Stoichiometric ratios have three pedagogical advantages. First, it is easy to tell students how to find them, and after a little drill work most students can use this skill without thinking about it. Second, because a distinction is made hetween moles of A and moles of B, unit cancellation can be used to check the correctness of a calculation. This has already been illustrated in the solution of the example. Third, using the term stoichiometric ratio permits a concise statement of a point that is fundamental to all stoichiometric calculations: Whenever we have information about how much of one chemical substance is present and we want to know how much of some other suhstance will he involved in a chemical compound or reaction, we must use a stoichiometric ratio to relate the amounts of the two substances. This statement gives students an explicit and easily rememhered guideline to follow whenever they are confronted with stoichiometric prohlems. Application of this rule to the complete solution of the example will illustrate its value in helping a student to devise a logical procedure. The reasoning might go like this: The problem asks for the volume of AgNOs(aq) which is required to react with a given mass of BaC12. Since two different substances are involved we will need a stoichiometric ratio (obtained from the halanced equation) to relate their amounts
(m)
The concentration of AgNO, relates the amount of AgNOa and the volume of solution, and the molar mass of BaC12 relates the mass of BaC12 and the amount of BaC12. Hence the complete roadmap is
Since any coefficient in achemical equation is apure number, Volume 57, Number 4, April 1980 / 305
and the calculation follows 1 mol BaC12 2 mol AgN03 V=0.385pX 208.2 g 1 mol BaC12 X
'liter = 0.0376 liter 0.0983 mol AgN03
Note that the roadmap does not have to specify whether to multiply or divide by each proportionality factor. Inspection of units makes it obvious which operation is appropriate in each case. With the roadmao there is a rationale for each logical step toward sol~nionof the prohlem. I t is always clear which auantitv is related to which and what oro~wrtionalit\~ factor ii calledfor. Indeed the terminology w;ha;e describei can be used to write the examole calculations in the form of an equation: ~~
V = rne.cl, X --X
1
AgN03 1 S -X -
(
)
~
Meacg BaClz ca.lroa We do not sueeest that such a formula be taueht " to students as a method for solving problems. Nevertheless i t is worth knowing that aformula can he obtained, if only to realize that stoichiometric calculations are not really any different from
~."
306 / Journal of Chemical Education
calculations (such as those involving density) that are usually taught with the aid of a formula. Such a realization might help some students over the barrier that stoichiometry often provides. Conclusion The roadmap method just described can obviously be used to solve anv, vroblem involvine a series of ~rooortionalitv re. . .~ lationships. We have listed the more important chrmical relationrhi~sof this kind in thc table. wine II.'PAC names and symbols for the quantities and p r o p k o n k i t y factor involved. A single-ste~roadmao is also orovided for each relationshio. We d i not have space to illustrate the use of all these r o a k maps, hut the one example provided should give a general indication of the procedure to be followed. We believe that by emphasizing the proportionality relationships among the quantities involved in stoichiometric calculations and by providing an outline form in which the logic of a problem solution can be expressed we are able materially to increase problem-solving skills of introductory chemistry students.
