In the Classroom
Applying the Reaction Table Method for Chemical Reaction Problems (Stoichiometry and Equilibrium)
W
Steven F. Watkins Department of Chemistry, Louisiana State University, Baton Rouge, LA 70803;
[email protected] The concept of balanced chemical reactions is presented early in introductory chemistry courses and texts, after which students are exposed to numerous examples and exercises that utilize the molar reagent ratios to perform quantitative calculations. The difficulties students at all levels have with these problems are well known to chemistry teachers (1), and methods of systematization have been suggested (2, 3). A systematic approach to chemical reaction calculations, which is an extension and elaboration of the equilibrium table method first observed by this author in the text by Masterton and Slowinski (4) nearly thirty years ago, is presented. Most texts utilize an “ICE” (initial, change, equilibrium) table when discussing equilibrium (5), but this author has found none that apply the technique systematically to stoichiometry problems. When students are first introduced to chemical reactions, they are told (or are left to infer) that at least one reagent is entirely consumed in the reaction. Later, the concept of “completion” is supplemented by the concepts of reversibility and equilibrium in closed systems. Throughout the course of study, reactions are presented and characterized by generic and specific names such as “weight–weight”, “combustion”, “decomposition”, “replacement”, “limiting reagent”, “theoretical yield”, “precipitation”, “titration”, “redox”, “partial pressure”, “gas phase equilibria”, “acid–base equilibria”, “solubility equilibria”, and others. In addition, chemical reaction calculations associated with each of these reaction types also require different quantitation measures, such as moles, mass, solution concentration, solution volume, gas volume, partial pressure, or activity. In most texts as each reaction type is introduced, analyzed, and discussed, calculational procedures are developed that often appear to the beginning student to be completely ad hoc creations. Few authors attempt to correlate analytical strategies for the “different” reactions, so the students are left with the daunting task of learning a growing number of seemingly unique procedures. However, almost all chemical reaction calculations reflect one of two basic variations on a single theme: •
Given the initial conditions before the reaction begins, determine the final conditions after the reaction is complete (or equilibrium is established).
•
Given the final conditions after the reaction is complete (or equilibrium is established), determine the initial conditions before the reaction begins.
The Reaction Table Method is a generalization of the ICE table that takes advantage of this common theme. It presents a single calculational paradigm within which systematic and consistent problem solving strategies can be developed for both stoichiometry and equilibrium problems. The method is designed to help chemistry students recognize common concepts, organize calculations, analyze problems, draw logical inferences, and resolve problems with confidence and accuracy. 658
Rationale for the Reaction Table The Reaction Table is a “spreadsheet”, the cells of which contain all of the explicit, implicit, and derived numerical data, as well as all algebraic symbols, expressions, and formulas that describe the chemical reaction problem. It is important to point out that the contents of the Reaction Table cells are not unique to this method—they are equivalent to those calculations usually presented piecemeal in traditional problem analyses. The utility of the Reaction Table spreadsheet format derives from four powerful features: 1. All numerical data and algebraic expressions are organized in the same consistent visual pattern for every chemical reaction problem. Thus, once the student learns to set up the basic Reaction Table correctly, it is very easy to find and use the numeric data, to generate the algebraic expressions and relationships required for problem resolution, and to extend and modify the table as the problem warrants. 2. The cells of the Reaction Table that contain the “answers” to the problem can be identified before calculations begin, so students know the goal they are seeking. Students can also learn to identify and distinguish between necessary and unnecessary information; learning to disregard the latter greatly improves problem solving efficiency. 3. Algebraic expressions are derived in a consistent and regular manner—there is no ad hoc formulation for each problem type. 4. There are often built-in checks for consistency and accuracy.
Reaction Table Layout The Reaction Table is composed of rows and columns. Each reactant and product in the balanced chemical reaction is represented by a column. The rows can be divided into three distinct sets, distinguished from one another by the kinds of data contained in the cells: •
Initial data: These rows describe the amounts of all reagents (reactants and products) before reaction begins; normally, at least some of these data are provided, either explicitly or implicitly, in the problem statement.
•
Reaction coordinate: This single row describes the reaction process and the change that takes place in the amount of each reagent; the data in this row are always formulated as algebraic expressions in exactly the same way, regardless of the problem or reaction “type”.
•
Final data: These rows describe the amounts of all reactants and products after the reaction has reached completion or after equilibrium has been established.
Journal of Chemical Education • Vol. 80 No. 6 June 2003 • JChemEd.chem.wisc.edu
In the Classroom
Solving Chemical Reaction Problems Every chemical reaction calculation involves three basic steps:
Table 1. Core Reaction Table for aA + bB → cC + dD A
1. Writing the balanced reaction
Change in moles
2. Drawing and filling the table cells with numerical data or algebraic expressions
Final moles
3. Finding one or more algebraic relationships required to solve for “unknowns”, then evaluating all algebraic expressions to derive numerical values in the “answer” cells
Step 1: Writing the Balanced Reaction It is imperative that students actually write down on paper the balanced chemical reaction equation. This focuses the students’ attention on the reagent formulas and molar ratios, both of which must be correct for problem resolution. Surprisingly, students often resist writing the reaction equation, so the instructor is advised to place strong and repeated emphasis on this step. Step 2: Drawing the Table The Reaction Table organizes all of the data required for solution of the problem. The fundamental core of every Reaction Table is the three “mole rows”, in which all reagent and product amounts are specified in units of moles. The Reaction Table (see Table 1) containing these three rows is most easily written right under the balanced reaction: aA + bB → cC + dD The cells in the first row contain the numbers of moles of each reactant and product before reaction begins. The second row shows the changes that occur in the numbers of moles as the reaction proceeds. These cells always contain the same algebraic expressions, derived in the following way: 1. Write an “x” in every cell (since the molar change is generally unknown). 2. Multiply each “x” by the stoichiometric coefficient of the reagent at the top of the column. 3. For reactions that proceed from left to right, apply a minus sign to each reactant change (because reactant amounts decrease during the reaction) and a plus sign for each product change. For reactions that proceed from right to left (as in reversible equilibria), reverse these signs. Note that correct sign placement ensures that the value of “x” will always be positive.
The cells in the last row contain the numbers of moles of each reactant and product after the reaction has reached completion (or equilibrium has been established). Since the final amount of each reagent is just the sum of the initial amount and the change (“first row plus second row equals third row”), completion of the table is automatic. The symbolically-completed Reaction Table is shown in Table 2. Initially, students are required to enter numerical values or algebraic symbols into each of the cells. In order to do so, they must learn to parse a problem statement and to distinguish useful information and data from noise (educative information irrelevant to problem resolution). The students
B
C
D
Initial moles
Table 2. Symbolically Complete Core Reaction Table for aA + bB → cC + dD A
B
C
D
niB ᎑bx
niC
niD
Change in moles
niA ᎑ax
+cx
+dx
Final molesa
nfA
nfB
nfC
nfD
Initial moles
Where nfA = niA − ax, nfB = niB − bx, nfC = niC + cx, and nfD = niD + dx. a
must also be instructed that there are two kinds of data in any problem statement: “explicit” and “implicit”. Explicit data (specific numerical values for initial or final amounts of reagents) are easily found and entered into the appropriate table cell where they are available for later reference and computation. Implicit values must be inferred from key words or phrases in (or absent from) the problem statement. Beginning students have the most difficulty with this key aspect of problem analysis. For example, the initial amount of a product is usually not mentioned, but students easily learn to infer that, since there is no product before reaction begins, the appropriate amount to be entered into that table cell is zero. Sometimes, the initial amount of a reactant is an “unknown” sought in the problem, in which case an algebraic symbol (other than x!) should be entered into the cell for that reagent. It is important that either a number (perhaps 0) or a symbolic algebraic expression must appear in each cell, especially when the method is first being learned. Later, students can learn to identify “silent reagents” for which neither the initial nor final amounts are needed for problem resolution. Learning which reagents to ignore in a problem is an important part of problem-solving strategy (see example problem 1 in the Supplemental MaterialW ). The process of writing the balanced chemical reaction and setting up the Reaction Table in the manner described provides a consistent, organized visual display of all numerical values and algebraic expressions needed to solve a chemical reaction problem. The relative ease with which the first two steps proceed also offers a valuable psychological advantage: starting every problem the same way reduces the intimidation of “all those words” and can diminish or eliminate the “deer in the headlights” syndrome so familiar to chemistry teachers. Students learn to parse complex problem statements because they know what they seek—explicit and implicit numbers or algebraic symbols to put into the table cells. They also learn to draw logical inferences from key words such as “completion”, “rigid container”, and “neutralize” and to identify and ignore noise in the problem statement.
Step 3: Solving For Unknowns The last and often most difficult step is to identify an algebraic expression that can be solved for “x”. Thus, all
JChemEd.chem.wisc.edu • Vol. 80 No. 6 June 2003 • Journal of Chemical Education
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In the Classroom
chemical reaction problems have an identical resolution strategy: equate an algebraic expression in the table to a numerical value, and solve for x. The power of the Reaction Table method derives from the fact that the algebraic expressions contained in the Table cells are not unique to each problem “type”, but evolve in a consistent and natural manner as the result of an unvarying set-up procedure. For example, a simple strategy for “limiting reagent problems” is to assume that each reactant in turn is the limiting reagent, solve for each value of x, and then choose the smallest value. It can be pointed out to the students that choosing a larger value of x leads to negative moles, a physical impossibility. Reaction Table Modified for Mass Although the three “mole rows” are common to every chemical reaction problem, many problems do not specify reagent amounts in mole units. It is thus useful to expand the Reaction Table to include the actual data so students do not have to do extra calculations on scratch paper. For example, in a “weight–weight” (stoichiometry) problem, two rows of initial data and one row of final data can be added to the three central “mole rows” as shown in Table 3. Students must be warned not to enter the product of formula mass and stoichiometric coefficient (e.g., fwA × a) in the first row; stoichiometric coefficients appear only in the “Change in moles” row. The second row records the initial mass of each reagent, some of which may be implicitly zero. After students are comfortable with the method, cells for silent reagents may be left blank. If one or more initial masses are “unknowns” in the problem, students must be warned to use algebraic symbols other than x. The “initial moles” are calculated from data in the first two rows: ni = mi兾fw. The “Change in moles “and “Final moles” rows are completed as above, and the “Final mass” of each reagent is calculated as mf = nf × fw. This formula is generally manifest as an algebraic expression, solution of which depends on the problem type. The three “mole rows” remain essentially unaltered except that the initial mole values are now derived quantities. Note also that there is an automatic validation system, because conservation of mass requires that the total initial mass of all reactants and products be equal to the total final mass, ∑mi = ∑mf Of course, validation can occur only if silent reagent amounts are also calculated (see example problem 2 in the Supplemental MaterialW ). Reaction Table Modified for Solutions The Reaction Table can be modified for chemical reactions in solution (e.g., aqueous titrations) as shown in Table 4. The initial concentration of each reagent is entered into the first row, with initial volumes in the second row. Again, the initial numbers of moles are derived from the first two rows: ni = Mi × Vi. The next two rows are completed as above. The final volume for each reagent is normally the common sum of all initial volumes: Vf = ∑Vi; inclusion of this row ensures that dilution effects are automatically taken into account. Final molarities are then calculated as Mf = nf兾Vf. 660
Table 3. Mass Reaction Table for aA + bB → cC + dD A
B
C
D
Formula weight
fwA
fwB
fwC
fwD
Initial mass
miA
miB
miC
miD
∑mi
Initial moles
niA ᎑ax
niB ᎑bx
niC
niD
∑ni
+cx
+dx
Change in moles
Total
Final moles
nfA
nfB
nfC
nfD
∑nf
Final mass
mfA
mfB
mfC
mfD
∑mf
Table 4. Solution Reaction Table for aA + bB → cC + dD A
B
C
D
Initial molarity
MiA
MiB
MiC
MiD
Initial volume
ViA
ViB
ViC
ViD
Initial moles
niB ᎑bx
niC
niD
Change in moles
niA ᎑ax
+cx
+dx nfD
Final moles
nfA
nfB
nfC
Final volume
Vf
Vf
Vf
Vf
Final molarity
MfA
MfB
MfC
MfD
Table 5. Precipitation Reaction Table for aA(aq) +bB(aq) → cC(s) + dD(aq) A
B
C(s)
D
Initial molarity
MiA
MiB
(fwC)
MiD
Initial volume
ViA
ViB
(miC)
ViD
Initial moles
niB ᎑bx
niC
niD
Change in moles
niA ᎑ax
+cx
+dx nfD
Final moles
nfA
nfB
nfC
Final volume
Vf
Vf
(Vf)
Vf
Final molarity
MfA
MfBB
(mfC)
MfD
Table 6. Gas Reaction Table for aA(g) + bB(g) → cC(g) + dD(g) A
B
C
D
Total
piB ᎑bx
piC
piD
∑pi
Change in pressure
piA ᎑ax
+cx
+dx
Final pressure
pfA
pfB
pfC
pfD
Initial pressure
∑pf
Table 7. Constant Volume Equilibrium Table for a A + bB cC + dD A
B
C
D
MiB ᎑bx
MiC
MiD
Change in molarity
MiA ᎑ax
+cx
+dx
Final molarity
MfA
MfB
MfC
MfD
Initial molarity
Journal of Chemical Education • Vol. 80 No. 6 June 2003 • JChemEd.chem.wisc.edu
In the Classroom
Note that the Table can also be used to illustrate reaction principles. For example, in an endpoint titration, both reactants (e.g., an acid and a base) are completely consumed, nfA = nfB = 0 From this equality, the endpoint titration equation is easily (and correctly) derived, M iA ViA a
=
M iB ViB b
Heterogeneous reaction calculations combine column data and expressions from “mass” and “solution” tables as appropriate. For example, the Reaction Table for a precipitation reaction is shown in Table 5. The third (solid precipitate) column contains mass data rather than solution data, and the final mass of the precipitate is mfC = nfC × fwC (see example problem 3 in the Supplemental MaterialW ). Reaction Table Modified for Gases Chemical reaction problems involving ideal gases sometimes cite explicit pressure–temperature–volume data, from which moles of gas may be derived in an expanded Reaction Table. However, Avogadro’s law problems involve unspecified but constant pressure–temperature or volume–temperature data, so explicit molar quantities cannot be derived. For example, if all reactants and products are ideal gases at unspecified but constant volume and temperature, the partial pressure of each gas is p = (RT兾V )n = kn, where k is a constant of unknown value for all reagent gases. As a result of the direct proportionality of p and n, partial pressure data can replace mole data in the Reaction Table as shown in Table 6 where total pressures (sums of initial and final partial pressures) may be explicit data in the problem. An Aside: Chemical Kinetics The second “mole row” in all Reaction Tables represents the total absolute change of moles, and the symbolic expression in each cell represents the absolute change in the number of moles of that reagent. For example, ∆nA = ᎑ax is the decrease in the number of moles of reagent A during the reaction, and x = ᎑∆nA兾a is the “relative absolute change” in the number of moles of reagent A. The relative absolute change of every reactant and product is the same positive value
x = −
∆ nA ∆ nB ∆ nC ∆ nD = − = + = + a b c d
If the total change in moles (∆n) of each reagent is replaced by the differential rate of change (dn兾dt), a familiar relationship from Chemical Kinetics emerges x = −
1 d nA a dt
= −
1 d nB 1 d nC 1 d nD = + = + b dt c dt d dt
where x is now equated to the rate law for the reaction.
The Reaction Table and Equilibrium A form of the Reaction Table has long been used in textbooks for calculations involving equilibrium reactions (4, 5). Consider, for example, solution equilibria such as acid–base, redox, or solubility in which the data consist of solution volumes and molarities. Assuming the reaction proceeds initially to the right, the reaction table would be identical to that used for a “completion” problem (Table 4). If the reaction occurs in a constant volume solution then ViA = ViB = ... = Vf and the table need display only concentration measures (although the underlying changes are still in the numbers of moles) as shown in Table 7 where MfA = MiA − ax, et cetera. For gas-phase equilibrium in a closed rigid container, Table 6 would be the appropriate form. In any case, solution strategy differs from a “completion” problem in that final molarities or partial pressures are algebraic expressions that are used in the mass-action expression (Kc or Kp). All textbooks discuss solution of the resulting polynomial. If students are introduced to the Reaction Table Method early in the course, and develop analytical procedures first for simple “mole–mole” problems, and then for increasingly complex homogeneous and heterogeneous reactions, they are not intimidated by equilibrium. The instructor can present equilibrium problems as an extension of the Reaction Table Method and spend his or her time on those aspects unique to equilibrium, such as polynomial solution and the use of approximations, and the calculation of initial reaction quotient for proper sign placement in the “Change” row. Conclusion The Reaction Table Method provides a consistent framework for the analysis and resolution of all chemical reaction problems in any setting, including completion and equilibrium reactions in solids, solutions, gases, and mixed phases. Because students learn one standard procedure for the set up of every problem, they are less intimidated by complex word problems and are better able to integrate new information and concepts as each reaction type is presented. An Internet site with many worked examples is available (6). WSupplemental
Material
Sample problems and analyses are available in this issue of JCE Online. Literature Cited 1. 2. 3. 4.
BouJaoude, S.; Barakat, H. Sch. Sci. Rev. 2000, 81 (296), 91–98. Doff, H. J. Chem. Educ. 1962, 39, 298–299. Kauffman, G. B. J. Chem. Educ. 1976, 53, 509. Masterton, W. L.; Slowinski, E. J. Chemical Principles, 3rd ed.; Saunders: Philadelphia, 1973; p 355. 5. Brown, T. L; LeMay, H. E., Jr.; Bursten, B. E. Chemistry: The Central Science, 8th ed.; Prentice Hall: Upper Saddle River, NJ, 2000; p 572. 6. Watkins, S. F. The Reaction Table. http://wb.chem.lsu.edu/ htdocs/people/sfwatkins/MERLOT/rt/00rt.html (accessed Dec 2002).
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