Constructing a Graphic Organizer in the Classroom ... - ACS Publications

Research: Science and Education

Constructing a Graphic Organizer in the Classroom: W Introductory Students’ Perception of Achievement Using a Decision Map To Solve Aqueous Acid–Base Equilibria Problems Stephen DeMeo Department of Curriculum and Teaching, Hunter College of the City University of New York, New York, NY 10021; *[email protected]

Research has shown that graphic organizers can facilitate student learning in a number of ways. For example, Davidowitz and Rollnick have found that students who created schematic flow diagrams found them helpful to “see the bigger picture” and to link concepts to lab activities (1). Recently, Snead and Snead reported that lower-ability science students increased their level of achievement using concept maps (2). Hawk’s study of almost 400 middle school life science students indicated that graphic organizers as a teaching strategy improved student scores (3). Graphic organizers are not new to chemistry teaching and learning. Many teachers provide visual routes such as “mole maps” to students in order to facilitate solving mass, particle, and molarity exercises (4–5). Other common maps reported in this Journal include flow diagrams for qualitative analysis (6) and a decision tree to aid in the calculation of pH and the solubility of salts (7). One type of graphic organizer that I have used in my introductory chemistry lecture is what I term a decision map. I use the word “decision” because the map begins with a question for the user to answer. A decision map is directional since after answering the initial question, the user proceeds through a number of steps and ends with a solution. Between the question and solution, the map contains prerequisite knowledge that must be known in order to advance through the steps. The utility of a decision map lies with its ability to be used when solving many different types of problems in a subject area. The decision map that will be presented here involves aqueous acid–base equilibria. This subject, which takes up a significant part of the curriculum in a first-year chemistry course, is arguably one of the most difficult areas of chemistry for introductory chemistry students to understand (8). This is mainly because related problems require students to choose the dominant equilibrium reaction from negligible competing reactions, and to distinguish the major chemical species from the minor ones that exist in solution. I will discuss how I implemented a workshop in which students constructed an acid–base decision map. The students who participated in the study were enrolled in a second-half of a yearlong introductory chemistry course for science and premedical majors. Because chemistry textbooks currently available to introductory students do not include map making activities or presentations, the goal of the workshop was for students to create a graphic organizer to help them understand the content presented in the text as well as to solve end-of-the-chapter problems. The problems I focused on were those for finding the pH of a solution. I will present results from questionnaires that elicited student perceptions of their learning before and after they participated in the acid–base workshop. 540

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Because these were first-year students, I used a noncalculus approach and certain widely used simplifications and assumptions in order not to overburden the student with further complexities. Some of these assumptions include the well-known 5% rule and the simplification that salts dissociate fully in water. For those interested in a more rigorous approach, many articles in this Journal have been written that raise specific issues with how acid–base chemistry is addressed in chemistry textbooks (9–11). The First Questionnaire Before the workshop, I lectured on acids, bases, salts, and buffers. These topics closely followed the information contained in two chapters of Zumdahl’s Chemistry: “Acid– Base Chemistry” and “Aqueous Equilibria” (12). My series of lectures included board notes, examples, readings, and homework problems from the textbook. My lectures preceded the workshop because I didn’t know the extent of students’ problems with the content. Would they feel that they needed a map-making activity or did they gain enough insight to adequately solve problems from the lecture alone? Whichever the case, I believe students must have a need for a workshop for one to be successful. Therefore after these chapters were discussed in class, a short, anonymous questionnaire was handed out to determine whether a supplemental workshop was necessary to help students understand the material and solve problems. I gauged their perceived ability by asking them the following four questions: 1. How would you describe the content in the two units, acid–base and aqueous equilibria? 2. How would you describe your ability to solve acid– base and aqueous equilibria problems (that you did for homework?) 3. How do you feel about the organization of these two units? 4. Any additional comments can be written below.

In the first three questions students had to choose a single best answer. The choices represented a range of responses, for example, in the first question students could choose whether the content was extremely difficult, moderately difficult, not very difficult, or easy to understand. The last question elicited open-ended comments. The first time this voluntary questionnaire was used 61 students responded. The results indicated that most students thought the content was moderately difficult to understand (53%), that students were fairly comfortable with their ability to solve problems, and that the organization of the material by the teacher was adequate for their own understanding of

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the chapter. While this was encouraging, it was also apparent that 23% of the students thought that they could only solve half of the acid–base problems. Two other groups of similar introductory students were also given this questionnaire in subsequent classes that I taught. While the percentage of students who thought that the material was moderately difficult to understand was about the same in all three groups, more students in the subsequent groups thought the material was “extremely difficult to understand” (31% and 18% respectively). When it came to the second question concerning problem-solving ability, 28% of students in one of the subsequent groups said they could only solve a few problems. On the last question, the majority of students in all three groups were satisfied by how I was organizing the information for them in class. About half the students in all the groups wrote additional comments. These centered on three main issues: increasing in-class problems solving, a concern about the questions on the exam (test questions seemed more difficult than in-class problems), and identifying the different types of problems they encountered. From these three issues, it could be said that students are well aware that this and other introductory chemistry courses are dominated by word problems. The last issue that many students wrote about concerned identifying problem types. This is not a trivial concern when one considers the content of acid–base equilibria and the

various chemical species that must be considered when solving pH problems. Categorizing problems by type is one important way to begin the problem solving process. If a problem is unable to be identified, then it is less likely that a student will choose a correct methodology to solve the problem. Also, if students perceive that there are too many types of problems, then they are more likely to be confused during the identification process. From talking to students and reading their comments it was clear that the material in these two chapters was becoming increasingly complex. Therefore, based on student feedback, I decided to conduct a week-long workshop that would be open to anyone who felt the need to revisit this content area.

Figure 1. Diagram of part 1 of the acid–base decision map.


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The Workshop I began the workshop by stating its purpose: to construct a generic tool, a graphic organizer that could be used to solve a wide range of problems from the easy to the very complex. In the class before the workshop, students were given a homework assignment. Half the class was asked to write out the prerequisite knowledge needed to solve pH problems, such as finding the pH of a 0.1 M acetic acid solution. The other half was asked to create the beginnings of a map starting with the question, “For each compound, do I have an acid, base, or a salt (include water as your major species)?”.


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After the acid–base decision map was made and handed out in class, the map was “tested” with about a dozen examples of varying difficulty. Below I will solve one of the more complex problems my students encounter, a determination of the pH of a buffer solution.

Sample Problem What is the pH of 75 mL of a buffer solution containing 0.10 M HC2H3O2 and 0.20 M NaC2H3O2 after 9.5 mL of 0.10 M HCl have been added to it? Reasoning Using Part 1 of the Map 1. HC2H3O2 is a weak acid since it has an “H” and is not on my list of strong acids that was memorized. The major species is HC2H3O2. 2. NaC2H3O2 is a salt (sodium is a metal and C2H3O2᎑ is a polyatomic anion). We will assume it dissociates completely in water. The major species are Na⫹ and C2H3O2᎑. 3. HCl is a strong acid. It dissociates completely in water to give H⫹ and Cl᎑. 4. My major species are: H⫹, Cl᎑, C2H3O2᎑, HC2H3O2, Na⫹, and H2O.

Reasoning Using Part 2 of the Map


On the day of the workshop, the class was asked to divide itself into groups of four: two students who worked on prerequisite knowledge joined two students who made maps. After about 30 minutes the class reconvened from their individual groups and one or two students were asked to put their work on the board. This work was discussed with the entire class and I made points of clarification, corrections, and additions to the content written on the board. I began to organize this information in the form of boxes and arrows, and eventually a map-like structure began to take shape. After the first question was answered the small groups were reformed and asked to focus on two other questions: “Do any of the major species react with each other (disregard water for now)?”, and “Do any of the major species react with water?”. As they did the first time, groups had about 30 minutes to work and then had to go to the board to present their answers where they would be discussed and clarified. At this point many of the classroom’s blackboards were being filled up, so a student was designated as a class scribe to copy and keep track of any changes. I used this student’s notes to produce a computer generated, three-page map that was handed out on the next day of class. Figures 1–3 represent diagrams of the three parts of the acid–base decision map in its entirety. The acid–base map involves the concept of major species that Zumdahl introduces in his textbook.1

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5. The strong acid, H⫹, and the weak base, C2H3O2᎑, react completely as represented in the equation: H⫹ ⫹ C2H3O2᎑ → HC2H3O2 6. Now we have to do some stoichiometry and calculate the moles of the major species before reaction. Moles of H⫹

⫽ 0.10 M ⫻ 0.0095 L ⫽ 0.00095 mol

Moles of C2H3O2᎑

⫽ 0.20 M ⫻ 0.075 L

Moles of HC2H3O2 ⫽ 0.10 M ⫻ 0.075 L

⫽ 0.015 mol ⫽ 0.0075 mol

Moles of H2O (Left out: it won’t be in the equilibrium expression) 7. As you can see in 6, above, the H⫹ is the limiting reactant, so let us calculate the moles of the major species after reaction. Moles of H⫹

⫽ 0 mol ᎑

⫽ 0.015 mol ⫺ 0.00095 mol ⫽ 0.014 mol

Moles of C2H3O2

Moles of HC2H3O2 ⫽ 0.0075 mol ⫹ 0.00095 mol ⫽ 0.00845 mol Moles of H2O (Left out: it won’t be in the equilibrium expression) 8. Our major species now are: Cl᎑, C2H3O2᎑, HC2H3O2, Na⫹, and H2O. 9. Let us find the concentration of our major species. The total volume of the solution is 75 mL ⫹ 9.5 mL ⫽ 84.5 mL ⫽ 0.0845 L. Molarity of H⫹

⫽0M

Molarity of C2H3O2᎑ ⫽ 0.014 mol/ 0.0845 L Moles of HC2H3O2

⫽ 0.00845 mol/0.0845 L

⫽ 0.1657 M ⫽ 0.100 M

Moles of H2O (Left out: it won’t be in the equilibrium expression)


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Table 1. Distribution of Student Responses to Questionnaire 2 Questions (Numbers)

Responses in Each Category (%) A Not Helpful

B Hampered Me

C Somewhat Helpful

D Helpful

E Extremely Helpful

1 (N = 59)

12

2

27

34

25

2 (N = 59)

7

3

32

34

24

3 (N = 58)

7

3

17

45

28

A Use Every Time

B Use Sometimes

C Use When Told To

D Will Never Use

10

12

4 (N = 58)

36

42

Reasoning Using Part 3 of the Map

The Exam and the Second Questionnaire ᎑)

10. Our major species (HC2H3O2 and C2H3O2 both react with water. Na⫹ and Cl᎑ are spectator ions and don’t react. HC2H3O2 ⫹ H2O

C2H3O2᎑ ⫹ H3O⫹

C 2 H3 O 2 ᎑ ⫹ H 2 O

HC2H3O2 ⫹ OH᎑

11. You have to find the Ka for the first reaction and the Kb for the second reaction. The larger K value will indicate which reaction is dominant. Basically, you are saying that the difference in Ks is large enough that the presence of one of the reactions won’t affect the calculated pH value based on the rules of significant values. The Ka of the first reaction is 1.8 ⫻ 10᎑5 and the Kb of the second reaction is 5.6 ⫻ 10᎑10. The first reaction will dominate and so an equilibrium table for this reaction will be set up. 12. The equilibrium table is:

The workshop lasted for one week (2 lectures and 1 recitation for an approximate total of 3.5 hours). The acid–base exam was given approximately six days after the workshop and comprised five word problems covering two chapters. After students took the exam, but before it was returned, a second anonymous and voluntary questionnaire was given to students to determine whether the map helped them solve problems, understand concepts, and organize the material. This questionnaire also tried to determine what extent the map would be used in the future by these chemistry students. Two open-ended questions were asked concerning how the map could be improved and whether the students had any additional comments to make. Like the first questionnaire, the multiple-choice responses in the first four questions of the second questionnaire represented a range of effectiveness. These are the six questions on the second questionnaire: 1. How do you feel about the decision map helping you solve problems?

Initial Concentration

Change in Concentration in Terms of X

Equilibrium Concentration

2. How do you feel about the decision map helping you understand concepts?

[HC2H3O2]

⫽

0.10 M

⫺X

0.10 ⫺ X

3. How do you feel about the decision map helping you organize the material?

[H3O⫹]

⫽

0.00 M

⫹X

X

[C2H3O2᎑]

⫽

0.2657 M

⫹X

0.1657 ⫹ X

13. The equilibrium expression is: Ka ⫽ 1.8 ⫻ 10᎑5 ⫽ (X ) (0.1657 ⫹ X ) Ⲑ (0.10 ⫺ X ). 14. Dropping the X in the numerator and denominator we get: Ka ⫽ 1.8 ⫻ 10᎑5 ⫽ (X ) (0.1657) Ⲑ (0.10) 15. Solving for X, we get: X ⫽ 1.0863 ⫻ 10 ᎑5. This approximate value is OK to use (1.0863 ⫻ 10᎑5 Ⲑ 0.10 ⫻ 100 ⫽ 0.01%, which is less than our 5% cutoff limit). 16. To get pH, we look for H3O⫹ or OH᎑ in the equilibrium table and find that [H3O⫹] is equal to X. Therefore, pH ⫽ ⫺log(1.0863 ⫻ 10᎑5) ⫽ 4.96.

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4. How do you feel about using the decision map? 5. What don’t you like about this map and how would you improve it? 6. Any additional comments can be written on the back side of this page.

Table 1 shows the number of student responses to the four multiple-choice questions that comprised the second questionnaire. In the first two questions, the majority of students believed that the decision map helped them solve problems and conceptually understand acid–base knowledge. The results of question 3 indicated that an overwhelming majority believed that the map helped them organize the material. A secondary majority believed that the map was extremely helpful. Question 4 centered on how students feel about using the map in the future. A majority of the students surveyed believed that they would use the map sometimes when given


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of them to solve problems; they integrated it to the point where they retained the necessary concepts and steps needed to solve problems. The most repeated comment concerning how the map could be improved centered on the size of the map. Students desired an additional small, one-page map to supplement the three-page version. Replicating Results

Figure 4. An acid–base decision map condensing parts 1, 2, and 3 into a single map.

an acid–base problem. A large secondary group thought they would constantly refer to the map when faced with acid–base problems. Twelve percent of the students said that they would never use the map again. Questions five and six of the second questionnaire were open-ended, requiring students to write out their responses. Many of these statements reaffirmed responses to the multiple-choice questions, as indicated in these example responses: • Many thanks! I thought it helped make things clearer. I wasn’t confused after we had the workshop and went home to do problems. It clarified the concept of continually rewriting the M.S. [major species], when to react with water and when with each other. • I was skeptical of it at first but very impressed by how clearly the map worked to organize acid–base info. • There really was nothing about the map that I didn’t like. I found the map very very helpful and I think you should give them out to all Chem 2 students. There were a lot of materials to digest in those 2 chapters (it was a mass of confusion) and the map was helpful in sorting out what the major points were.

It is also apparent that conducting a workshop was important for students, as opposed to just handing out the map and referring to it as problems arise. These students also wrote that after awhile they did not need the physical map in front

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Over the next two years, the acid–base workshop with its accompanying questionnaires was repeated two additional times. The students in these latter groups were enrolled in the same second-semester general chemistry course, used the same textbook, were given the same homework assignments, and took very similar exams as the first group. I was the instructor to all three groups of students. In order to determine whether students’ perceptions about aqueous acid-base equilibria would be consistent over time, I compared the percent of student responses in all three groups. Overall, the results between the groups were fairly uniform. While some inconsistencies existed, the largest percentage in each of the four multiple-choice questions was aligned throughout all three groups. This indicates that the decision map generally was perceived by students to be helpful to solve problems, understand concepts, and organize material. About 80% of the students in all three groups either said they would use the map “sometimes” or “all the time” when solving acid–base problems in the future. The tabulated data collected from these two additional groups are included in the Supplemental Material.W In open-ended questions 5 and 6, positive comments strongly outnumbered negative ones. As happened with the first group, students complained about the length of the threepage map. As one student candidly wrote, “It is overwhelming in its breadth; condense it.” To meet this need, I made and handed out a one-page map to students after the longer three-page decision map was finished and used in class for a period of time (see Figure 4). Since the one-page map was just a shorter version of the longer map, it wasn’t “workshopped” or constructed piece by piece with the students. Relating Students’ Perceptions of Achievement to Exam Grades To find out whether a relationship existed between how well students performed on the exam and their thoughts about the map, students were organized into two groups: those that received a score of 80 or above on the exam and those that received a score of 69 and below. (This was done mainly because my introductory chemistry courses have been characterized by a bimodal distribution of grades.) Of 50 students who responded to the original second questionnaire, 24 received a score of 80 or better on the exam, while 19 earned a score of 69 or lower. Only 7 students were in the C range (a score of 70–79). I organized students into these two different groups in order to determine which group believed that the map helped them solve acid–base problems. It would be disconcerting to find that the lowest scoring group used the map extensively without being able to solve the problems on the exam.


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Table 2. Distribution of a Second Sample of Student Responses to Questionnaire 2 a Questions (Numbers)

Responses in Each Category (%) A Not Helpful

B Hampered Me

C Somewhat Helpful

D Helpful

1 (N = 33)

0

3

30

45

21

2 (N = 33)

3

3

24

61

9

3 (N = 32)

6

0

19

50

25

A Use Every Time

B Use Sometimes

C Use When Told To

D Will Never Use

E Know It So Well

6

3

9

4 (N = 32) aThis

38

44

E Extremely Helpful

questionnaire was distributed in a colleague’s class.

The comparison between higher- and lower-achieving groups indicate that students who did better on the exam felt that the map helped them to a greater extent in terms of problem solving and understanding concepts and, to a lesser degree, in helping them organize information. Both groups were consistent on question 4. A significant proportion—75% of the higher achievers and 74% of the lower achievers—said that they will use the decision map either “every time” or “sometimes” when given an acid–base problem. The inconsistencies in the first three questions raise a further query: Why did the lower achievers perceive that the map did not help them in terms of problem solving and understanding concepts, but said that they would use the map to much the same extent as higher achievers? To answer this is to go beyond perceptions and try to determine what prevented lower achievers from using the map in a meaningful way. Was it an issue of conceptual or methodological knowledge, or both? Was it predominately an issue of practice or mathematical ability? The purpose of relating exam scores with students’ perceptions of the decision map is not to claim for any causal relationship between the two; there are many reasons why students achieve and do not achieve on tests. My intention was to argue against the possibility that the workshop retarded or hindered student achievement. A Stand-Alone Workshop Another chemistry professor in my department conducted the acid–base workshop with a similar group of introductory chemistry students. While I had conducted a series of lectures before the workshop, this colleague chose to curtail the lectures and instead use the workshop as the main instructional method to teach her students about aqueous acid-base equilibria. Students were not given the first questionnaire which I used to gauge student need for a workshop; consequently at the end of the workshop only the second questionnaire was handed out. The results of my colleague’s workshop, collected in Table 2, were slightly more positive than those collected from my initial workshop. In the first three questions, more students

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in my colleague’s class indicated that the map was “helpful” to solve problems, understand concepts, and organize knowledge in comparison to my group of students. The results show greater consistency on responses to question 4. A large percentage of students in both of our classes said that they would use the decision map either “sometimes” or “every time” when given an acid–base problem. It must be noted that an additional multiple-choice response was added to question 4, which involved students using the map in future problem situations. After reading a few comments that suggested that the map would not be used to solve any future problems because it was so well known to students, I included an additional choice to question 4: “I will never use the decision map again because I know the map so well.” It was interesting to find that 9% of the students surveyed chose this as their best response. Student responses from both instructors’ classes also showed consistency in the open-ended questions. Students in my colleague’s class had many positive things to say about the map and the workshop. Negative comments were fewer in number and did not involve the length of the decision map. This was because my colleague distributed both versions of the map to her students at the same time (the threepage map and the one-page condensed version as seen earlier in Figure 4). The similarity between the results generated from my colleague’s class and my initial workshop suggest that a standalone workshop can also be successful. Prefacing the workshop with detailed lectures is not necessarily the only route that can lead to favorable student responses. The benefit of conducting a stand-alone workshop is of course the fact that less class time can be spent on the topic; and if time is not an issue, then time saved can be spent on additional and more advanced problems. Conclusion The acid–base workshop and the decision map involve two key aspects, the first being student participation. From my experience I believe it is crucial that students become involved in the map-making process while in class. An in-class


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experience is important because it is there that students can voice their beliefs, ask questions, have their misconceptions challenged, and where the instructor can facilitate student understanding. I believe the positive results of the acid–base workshop indicate that active group learning can help students; it can lead to the generation of a graphic organizer that students believe is useful to solve problems, understand concepts, and organize knowledge. While student participation might seem obvious for success, many similar chemistry courses are still heavily lecturebased. In such a course learners can be less cognitively active since they are preoccupied with listening and copying notes. It is often implicitly assumed that students should make sense of the lecture experience outside of class when they supplement their learning through study groups, reading, and doing homework problems. Unfortunately the teacher is not present during these times when questions arise and points of clarification are needed. As a result of this study, I have altered the way I teach this subject matter. I no longer preface the workshop with a series of lectures on acid–base chemistry and aqueous equilibria. The workshop has become my central pedagogical means to facilitate student learning in this subject area. The second key issue involves the decision map. For the decision map to be useful, I believe it must integrate conceptual and methodological content knowledge—the what we know with the how we know—in a form that the user must not blindly follow. The map must involve making decisions; the user must think through each section of the map and relate each to what is being asked in the problem. As an example of a graphic organizer, the decision map is more than a typical flow diagram with its characteristic yes-no responses and absence of conceptual linkages. Decision maps are different from concept maps in that they incorporate a method, a step-by-step process to solve a problem. The four decision maps presented here should be viewed as malleable templates rather than rigid diagrams. They serve as examples of how teachers can become involved with their students to make and structure knowledge. These maps are not prescriptions for student success; I fully support those teachers who during the construction of one or more of these maps add, modify, and remove information in order to help their students learn.2 It is clear that these two topics encompass an abundance of conceptual as well as methodological knowledge. Students who see more differences between problems than similarities will face a much more challenging time as they try to understand acid–base equilibria. The acid–base map shows promise in allowing students to integrate the knowledge surrounding different problem types, and thereby simplifying their problem-solving process into a general framework. But in order to do this the map cannot be seen as a magic bullet that could be used to slay problems. Rather, it is something that the user must understand and know how to use; it requires a good deal of practice to make it one’s own. Continued research is necessary to make a link between student perceptions and actual achievement. More in-depth analysis is needed to explore which specific concepts students feel the map helps them to understand. It would also be

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valuable to know exactly how the map facilitates problem solving and knowledge organization. Perhaps qualitative interviews could bring more light to bear on these issues. While it is always pleasing for students to appreciate the lessons that you conduct, it is still disconcerting that perhaps the map is only helping the top half of the class. A significant portion of students remains as lower achievers who are in need of help. One possible solution is to introduce less complex graphic organizers earlier on in the year. Some areas where I have used decision maps in the first semester of general chemistry include nomenclature, types of reactions, and stoichiometry. While teaching the same students in the second-half of the year (where the acid–base map is used) is problematic at larger institutions where students have several sections to choose from, smaller schools or courses that are team-taught could try this to lessen the cognitive demand on students. Continued reflection, experimentation, and evaluation are necessary to determine how the workshop and the decision map can be most effective for all students. W

Supplemental Material

Additional data tables, as well as full-page versions of all four decision maps discussed in this paper, are available in this issue of JCE Online. Acknowledgments I would like to thank Pamela Mills for helping me develop these maps, as well as William Sweeney for his interest in this project. Notes 1. Because the content of the map is very well known and closely follows the Zumdahl text, I will not discuss the specific chemistry here. 2. One example would be to modify the equilibrium change table to show the chemical equations written at the top with the concentration values directly below the chemical formulas.

Literature Cited 1. Davidowitz, B.; Rollnick, M. Australian J. of Educ. in Chem 2001, 57, 18–24. 2. Snead, D.; Snead, W. L. J. Res. in Childhood Educ 2004, 18, 306–20. 3. Hawk, P. P. Science Education 1986, 70, 81–87. 4. Kent, J. C.; Curtright, R. D.; Brooks, D. W. The Sci. Teach. 2000, 36–39. 5. Krieger, C. R. J. Chem. Educ. 1997, 74, 306–309. 6. Oliver–Hoyo, M.; Allen, D.; Soloman, S.; Brook, B.; Ciraolo, J.; Daly, S.; Jackson, L. J. Chem. Educ. 2001, 78, 1475–1478. 7. Cavaleiro, A. M. V. S. V. J. Chem. Educ. 1996, 73, 423–425. 8. Burness, J. H. J. Chem. Educ. 1990, 67, 224–225. 9. Hawkes, S. J. Chem. Educ. 1996, 73, 421–423. 10. Carlton, T. S. J. Chem. Educ. 1997, 74, 939–941. 11. Pardue, H. L.; Odeh, I. N.; Tesfai, T. M. J. Chem. Educ. 2004, 81, 1367–1375. 12. Zumdahl, S. Chemistry, 4th ed.; Houghton Mifflin Co.: Boston, MA, 1997.


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