In the Classroom
Amounts Tables as a Diagnostic Tool for Flawed Stoichiometric Reasoning John Olmsted III* Department of Chemistry and Biochemistry, California State University, Fullerton, Fullerton, CA 92834
A table of amounts, giving initial amounts, change in amounts during the reaction, and final amounts, has long been recognized as a useful tool for organizing concentration data in equilibrium calculations. Although Pauling’s classic general chemistry text does not use this device, even in later editions (1), the equally classic successor texts by Sienko and Plane (2) and Dickerson, Gray, and Haight (3) both employ abbreviated tables that summarize initial and equilibrium conditions. A full table listing initial concentrations, changes in concentration, and concentrations at equilibrium is now a standard feature of the pedagogy of general chemistry equilibrium calculations. An amounts table can be used to advantage in organizing the data and reasoning involved in limiting reagent problems because the stoichiometric reasoning underlying the construction of concentration tables is as valid for reactions that go essentially to completion as for those that reach equilibrium. This device has not been a common feature of the pedagogy of stoichiometric calculations, but at least one current general chemistry textbook develops amounts tables for stoichiometric calculations (4). Also, a recent preparatory chemistry textbook uses amounts tables extensively in presenting excess reagent and limiting reagent calculations (5). The pedagogical value of this approach has not yet been systematically assessed, but the pilot study reported here indicates that the errors students make when asked to complete amounts tables reveal the flaws in their stoichiometric reasoning. In this study, second semester general chemistry students were given a diagnostic test requiring them to construct an amounts table for a simple limiting reagent problem. This paper describes the diagnostic test, gives examples of the types of mistakes made by students, and discusses the weaknesses in reasoning indicated by those mistakes. Diagnostic Test The following question was used on the diagnostic test, and its correct solution is given: Question: Aluminum metal reacts with chlorine gas according to the following balanced equation:
2 Al(s) + 3 Cl2(g)
→ 2 AlCl3(s)
Set up an amounts table and use it to determine the amounts (in mol) of all substances present after reaction if 12.0 mol of Al reacted to completion with 15.0 mol of Cl2. Show the completed table as part of your work. Solution: ⫹ 3 Cl2 (g) → 2 AlCl3 (s)
Reaction:
2 Al (s)
Starting amounts: Change in amounts:
12.0 mol ᎑10.0 mol
15.0 mol ᎑15.0 mol
+10.0 mol
Final amounts:
2.0 mol
0.0 mol
10.0 mol
*Email:
[email protected].
52
0.0 mol
In order to probe for common errors in reasoning, the chosen reaction has simple but not 1:1 stoichiometry, requiring the appropriate use of stoichiometric ratios. Additionally, the starting amounts were chosen so that the reactant present in larger amount was limiting. To minimize the likelihood of computational errors, the starting amounts were expressed in moles and were integral multiples of the reaction coefficients. This diagnostic test was administered as a quiz during the second week of the second-semester general chemistry course. It was a quiz for credit, to ensure that students made their best efforts. Students were warned in advance that the quiz would require construction of an amounts table, and they were urged to review the relevant section of the textbook. Student performance on the quiz, which is summarized in Table 1, could be conveniently divided into three categories. Just under a third of the students worked the problem correctly. Another 15% made minor mistakes such as calculation errors but used correct stoichiometric reasoning, as evidenced by their correctly identifying the limiting reagent and having change amounts in the correct stoichiometric ratios. More than half the students showed a poor grasp of stoichiometric reasoning using amounts tables, despite having successfully completed first-semester general chemistry and having access to a text that presented amounts tables as a stoichiometric tool. The table also shows that students whose first semester instructor taught amounts tables as a stoichiometric tool scored substantially better than students whose first semester instructor did not emphasize this technique, even though the textbook introduced amounts tables as a stoichiometric tool. Table 1. Student Performance on Amounts Table Quiz Instructor a
Number of
I
II
Other 11
Total
Students
21
22
Correct solutions
3 (14%)
10 (45%) 4 (36%)
17 (31%)
Correct approach, minor errors
3 (14%)
5 (23%)
8 (15%)
0
54
aFirst-semester
instructor I did not teach amounts tables; first-semester instructor II did teach amounts tables; “Other” indicates students who took a different first-semester course.
Examples of Student Errors Examination of the manner in which students completed their amounts tables often showed where their stoichiometric reasoning was faulty. Examples of the amounts tables written by eight representative students appear below. About onethird of the students who did not reason correctly (exemplified by students A, B, and E) failed to take stoichiometric coefficients into account. Another common mistake (exemplified by students B, C, and D) was the incorrect application of an algorithm for determining the limiting reagent. Students E, F, and G, exemplifying those whose stoichiometric reasoning was seriously deficient, made errors that had in common only
Journal of Chemical Education • Vol. 76 No. 1 January 1999 • JChemEd.chem.wisc.edu
In the Classroom
their idiosyncracy. (Another common mistake that was not evident in our sample was reported by an instructor who administered the diagnostic quiz in the first-semester general chemistry course: many students insisted that masses were important and tried to compare masses of reactants rather than moles.) Solution of Student A: 2 Al(s) Start:
+
3 Cl2 (g)
→
2 AlCl3 (s)
15.0 mol ᎑12.0 mol
0.0 mol
Change:
12.0 mol ᎑12.0 mol
Finish:
0.0 mol
3.0 mol
12.0 mol
+12.0 mol
Here is the most common stoichiometric error. Student A assumed equal changes in all reagents, ignoring the stoichiometric coefficients. Solution of Student B: 2 Al(s) Before:
+
3 Cl2(g)
→
2 AlCl3 (s)
During:
6 mol ᎑ 5 mol
5 mol ᎑ 5mol
+ 5 mol
Completion:
1 mol
0 mol
5 mol
Solution of Student C: Reaction:
2 Al(s) 12.0 mol
Used:
+
3 Cl2 (g) 15.0 mol
→
2 AlCl3 (s) 0 mol
11.0 mol
15.0 mol
+10 mol
1.0 mol
0 mol
10 mol
Student C’s reasoning is not clear from the table, but this student showed the following additional work: 15.0 mol Cl2 = 5.0 mol Cl2 3
12.0 mol Al = 6.0 mol Al ; 2
Cl2 limiting reagent, 1.0 mol Al left over
Student C used the “mole/coefficient” algorithm—in this case correctly identifying the limiting reagent, but, like student B, made the mistake of treating the resulting quantity as though it were moles. Additionally, this student made one correct stoichiometric link (3:2 ratio of changes in Cl2 and AlCl3) but failed to make any stoichiometric connection between the two reactants. Solution of Student D: 2 Al Start:
+
3 Cl2 (L.R.)
12 mol Al 2 = 8 3 Student D used the correct stoichiometric ratio, but applied it to the starting amount of the wrong reagent. Furthermore, this student failed to make any stoichiometric connection among the changes for reactants and products. Solution of Student E: 2 Al(s) Starting mat.:
→ 2 AlCl3
Change:
12 mol ᎑ 8 mol
15.0 mol ᎑15 mol
0 mol +8 mol
Final:
4 mol
0 mol
8 mol
+
→
3 Cl2 (g)
2 AlCl3 (s)
Change:
12.0 mol ᎑ 5.0 mol
15.0 mol ᎑ 5.0 mol
+5.0 mol
Final amounts:
7.0 mol
10.0 mol
5.0 mol
0.0 mol
The following additional work indicates the source of the change entries:
0 mol
Student B divided the amount of each starting material by its stoichiometric coefficient, using an algorithm that is useful for determining the limiting reagent: the reactant with the smaller value for moles/coefficient is limiting. The amounts table that resulted is correct if the entries are understood not as moles but instead as “reaction units” (moles/coefficient) in all cases. Multiplying each entry in the “completion” row by the corresponding stoichiometric coefficient gives correct results (2 mol of Al left over, 10 mol of AlCl3 formed). However, the student clearly labeled the entries “mol”, indicating an application of the algorithm without understanding it.
Have:
How this student arrived at the change entries is indicated by the following additional work:
12.0 mol Al = 6.0 ; 2 mol
15.0 mol Cl2 = 5.0 3 mol
Student E correctly applied the “moles/coefficient” algorithm but did not know how to make use of the result. This student’s table reveals confusion about both stoichiometric ratios and the concept of a reaction going to completion. Solution of Student F: Al Before rxn.: After rxn.:
+
Cl 2
AlCl3
2 (12.0 mol)
3 (15.0 mol)
24.0
45.0
69.0/2
0
0
34.5 mol AlCl3
This table reveals confusion over several aspects of stoichiometry. Starting amounts are multiplied by stoichiometric coefficients, the results are added to obtain the amount of product, which is divided by two (the coefficient for AlCl3?) to give a final amount, and the stoichiometric coefficients are ignored in concluding that both starting materials are completely consumed. Solution of Student G: Reaction:
2 Al(s)
Start out:
2 mol
+
3 Cl2 (g)
(6)
(6)
Finish:
12 mol
15 mol
→
3 mol
2 AlCl3 (s) 2 mol of AlCl3 3 mol Al4Cl5
Again, the table reveals several points of confusion. The given starting amounts are ignored at first but appear intact at the finish, the notion of change is absent, and a new chemical formula has been introduced in order to consume both reactants. Solution of Student H: +
3 Cl2 (g)
→
Reaction:
2 Al(s)
i ∆
2x
3x
2x
2 AlCl3 (s)
2 (12 mol)
3 (15 mol)
2x
eq
(24 mol) x
(45 mol) x
2x
Student H, who was repeating second-semester general chemistry, shows a more sophisticated degree of confusion.
JChemEd.chem.wisc.edu • Vol. 76 No. 1 January 1999 • Journal of Chemical Education
53
In the Classroom
This student attempted to apply algebraic reasoning, but the entries in the table indicate the same degree of total confusion as shown by students F and G. This student has added the use of “x” in equilibrium tables to his or her arsenal of misapplied algorithms. Discussion The manner in which students complete amounts tables for a “simple” limiting reagent problem reveals the extent to which they have mastered stoichiometric reasoning. Many in our sample of second-semester general chemistry students made entries that indicated a shocking deficiency in this area, despite their having successfully completed the first semester course. Only about 50% of the students showed a satisfactory grasp of stoichiometric reasoning. The others fell into two groups containing approximately equal numbers. One group, exemplified by students A–D above, showed some ability to reason stoichiometrically but made errors that indicated an imperfect understanding. The second group, exemplified by students E–H, completed their amounts tables in ways that showed nearly complete ignorance of how to reason stoichiometrically. Interviews with some of these students determined that a few students realized that their stoichiometric abilities were weak; these students reported being able to pass by doing sufficiently well in other segments of the first-semester course to offset weaknesses in stoichiometric reasoning. Most students, however, reported having been able to solve standard stoichiometry problems correctly. These observations indicate that a significant fraction of students learned how to work problems without acquiring a deeper understanding of stoichiometric principles. This is consistent with other research into misconceptions in chemistry showing that the ability to balance chemical equations and solve routine stoichiometry problems often is not accompanied by a molecular understanding (6, 7 ). Completed amounts tables help to identify instructional strategies that can assist students whose stoichiometric reasoning skills are deficient. Students A–D appear to be on the threshold of a satisfactory understanding. For them, reminders of the stoichiometric ratios involved in chemical reactions may be sufficient to remove deficiencies. Students E–H, however, cannot achieve proficiency without intensive additional work. These students are likely to be at the Piagetian concrete operational stage, not yet capable of formal operational reasoning (8). If this is so, then they require a variety of instructional approaches, none of which may show much success (9, 10).
54
The responses of students in this sample indicate that the use of convenient algorithms, while undoubtedly advantageous for students whose grasp of stoichiometric reasoning is already established, confuses weaker students without enlightening them. This is indicated by the frequency of erroneous application of the “moles/coefficient” algorithm in our sample. This algorithm was emphasized by the instructors of both first-semester general chemistry sections from which this student population was drawn. The amounts tables make it clear that some students adopted this algorithm as a way of “solving” stoichiometry problems without any understanding of its proper application. The data in Table 1 show that students whose firstsemester instructor taught amounts tables as a tool for solving stoichiometry problems performed substantially better than those who were not so taught: 68% of the former group had a correct solution or correct approach, compared with 31% of the latter group. This difference might be attributed to exposure and familiarity or to student “mastery” of yet another algorithm, the amounts table. The difference in performance is nevertheless large enough to suggest that the use of amounts tables in teaching stoichiometry may improve student understanding of stoichiometric reasoning. Given the importance of stoichiometric reasoning as the foundation for quantitative treatment of equilibria, this possibility seems worthy of a more detailed study using a larger sample size and appropriate controls. Literature Cited 1. Pauling, L. General Chemistry, 3rd ed.; Freeman: San Francisco, 1970. 2. Sienko, M. J.; Plane, R. A. Chemistry: Principles and Properties; McGraw-Hill: New York, 1966. 3. Dickerson, R. E.; Gray, H. B.; Haight, G. P., Jr. Chemical Principles; Benjamin: Menlo Park, CA, 1970. 4. Olmsted, J., III; Williams, G. M. Chemistry, the Molecular Science, 2nd ed.; Wm. C. Brown: Dubuque, IA, 1997. 5. Sevenair, J. P.; Burkett, A. R. Introductory Chemistry; Wm. C. Brown: Dubuque, IA, 1997. 6. Yarroch, W. J. Res. Sci. Teach. 1985, 22, 449–459. 7. Nurrenbern, S.; Pickering, M. J. Chem. Educ. 1987, 64, 508– 510. 8. Good, R.; Mellon, E. K.; Kromhout, R. A. J. Chem. Educ. 1978, 55, 688–693. 9. Goodstein, M. P.; Howe, A. C. J. Chem. Educ. 1978, 55, 171– 173. 10. Herron, J. D. The Chemistry Classroom; American Chemical Society: Washington, DC, 1996.
Journal of Chemical Education • Vol. 76 No. 1 January 1999 • JChemEd.chem.wisc.edu
