Problem solving and chemical equilibrium: Successful versus unsuccessful performance

JOURNAL OF RESEARCH IN SCIENCE TEACHING VOL. 26, NO. 3, PP. 251-272 (1989)

PROBLEM SOLVING AND CHEMICAL EQUILIBRIUM: SUCCESSFUL VERSUS UNSUCCESSFUL PERFORMANCE

MOISES CAMACHO

InterAmerican University of Puerto Rico, Aguadilta, Puerto Rico 00603

RON GOOD

Louisiana State University, Baton Rouge, Louisiana 70803

Abstract

The purpose of this study was to describe the problem-solving behaviors of experts and novices engaged in solving seven chemical equilibrium problems. Thirteen novices (five high- school students, five undergraduate majors, and three nonmajors) and ten experts (six doctoral students and four faculty members) were videotaped as they individually solved standard chemical equilibrium problems. The nature of the problems was such that they required more than mere recall or algorithmic learning and yet simple enough to provide the novices a reasonable chance of solving them. Extensive analysis of the think-aloud protocols produced 27 behavioral tendencies that can be used to describe and differentiate between successful and unsuccessful problem solvers. Successful solvers’ perceptions of the problem were characterized by careful analysis and reasoning of the task, use of related principles and concepts to justify their answers, frequent checks of the consistency of answers and reasons, and better quality of procedural and strategic knowledge. Unsuccessful subjects had many knowledge gaps and misconceptions about the nature of chemical equilibrium. Even faculty experts were sometimes unable to correctly apply common chemical principles during the problem-solving process. Important theoretical concepts such as molar enthalpy, heat of reaction, free energy of formation, and free energy of reaction were rarely used by novices in explaining problems.

Introduction

In fecent years, cognitive psychologists, science educators, and others have focused their attention on problem solving in semantically rich science content domains such as genetics and mechanics. The early work of Newel1 and Simon (1972) provided the impetus for much of the more recent research on scientific problem solving. The work is generally based on information-processing theory, and the think-aloud interview, described in detail by Ericsson and Simon (1984), serves as the main data-collection technique. Experts and novices are asked to solve problems and their think-aloud behaviors are recorded while they attempt to solve the problems.

0 1989 by the National Association for Research in Science Teaching Published by John Wiley & Sons, Inc. CCC 0022-4308/89/03025 1-22$04.00

252 CAMACHO AND GOOD

According to Larkin, Heller, and Greeno (1980), the domain knowledge of experts is organized hierarchically and stored in the form of “large functional units” composed of coordinated principles that are easily retrievable, while novices’ knowledge is more fragmented and more difficult to access from long-term memory. These results are consistent with work done by Bhaskar and Simon (1977), who reported results from engineering thermodynamics. In a problem-sorting study in physics, Chi, Feltovich, and Glaser (1981) described experts as making classifications based on underlying principles of the domain (e.g., conservation of momentum, matter, energy), while novices sorted the same problems according to superficial information of objects given in the problems (e.g., inclined plane, pulleys).

Problem-solving differences in strategies knowledge have attracted the attention of several investigators. Larkin (1981) and Simon and Simon (1978) reported that experts not only demonstrated more strategic knowledge but tended to use better- quality heuristics than novices.

In the area of biology, Stewart (1983) and Smith and Good (1984) described many differences in the way experts and novices go about solving genetics problems. Although problem solving has been studied carefully in physics (particularly in classical mechanics) and in biology (mainly classical genetics) using the expert-novice model, comparatively little work has been reported in chemistry. The work that has been done by Gabel and Sherwood (1983) and Larkin and Rainard (1984) focused on stoichiometric problems. An area of chemistry that may be better suited for problem-solving research is chemical equilibrium, the focus of this study. Since chemical equilibrium can be seen as a synthesis of most general chemistry concepts and principles, the findings of this and related studies should add to our understanding of problem-solving theory in the sciences.

Purposes of this Study

In addition to the general purpose of extending problem-solving research to chemical equilibrium, specific purposes of this study were to

(1) Verify empirically whether chemical equilibrium is an adequate content domain

(2) Describe the behaviors of experts and novices during the process of solving

(3) Document scientific problem-solving behaviors reported by previous related

(4) Describe the performances observed in successful and unsuccessful subjects. (5) Observe how problem-solving performance is affected by conceptual knowledge

for problem-solving research.

chemical equilibrium problems.

studies.

of related chemistry concepts and principles and related mathematics skills.

Methodology

Problem Selection

The principal investigators and several chemistry professors selected seven chemical equilibrium problems (see the appendix) from a large pool of problems found in

PROBLEM SOLVING AND CHEMICAL EQUILIBRIUM 253

standard textbooks and exams. The problems had to be well structured, involve several concepts, and be appropriate in difficulty for the subjects in the study. A pilot study was used to test the viability of the problems and they were found to meet the criteria.

Subject Selection

Following sample-selection guidelines of Bogdan and Taylor (1975) and Smith (1983), 23 subjects were selected. Table I shows selected characteristics of the subjects. The sample of 13 novices consisted of five high-school students who had completed one semester of high-school chemistry with a grade of A or B; five undergraduates majoring in chemistry who had completed at least two semesters of college chemistry with a grade of. A or B; and three undergraduates majoring in biology who had completed two semesters of college chemistry with a grade of A or B. The sample of experts consisted of six doctoral students in chemistry and four faculty members in chemistry.

TABLE I Description of Subjects’ Chemical Background (HN = High School Novice,

BN = Undergraduate Biology Major Novice, CN = Undergraduate Chemistry Major Novice, DE = Doctoral Student Expert, FE = Faculty Expert

No. o f Semesters of Chemistrj

t i . School College Graduate Experience Subject Sex Age Courses Taken Vears of

HN1 M HN2 F HN3 F HN4 M HN5 M

BN 1 M BN2 M BN3 M

CN1 M CN2 F CN3 F CN4 F CN5 M

@E 1 M DE2 M CE3 N DE4 M GE 5 M DE6 F

FE 1 M FEZ M FE3 M FE4 M

17 !7

16 15

26 28 23

27 21 21 2 0 21

28 2 5 28 25 25 26

44 36 45 28

I -

l l

1 1 1 1 1

1 1 2

1 4 4 2 2

3 1 1 2 1 2

2 1

6

3 4 4

5 5 3 4 7

8 8 8 8 8 8

8 8 8 8

4 3 4 1 5 6 3 1 - 5 4 5

3 15 3 1 3 22 3 2

7


Data Collection

Each subject was videotaped as she or he thought aloud while solving each of the seven problems in chemical equilibrium (see the appendix). The videotape records (each about 90 minutes) were reviewed extensively following the guidelines of Ericsson and Simon (1984) and Smith (1983). From the reviews, a checklist of 27 problem- solving behaviors of experts and novices was prepared. Each of the authors of this article and a faculty member in the chemistry department (not a study subject) participated in the process, although it was the first author who did the major share of the analysis.

A quantitative evaluation of the degree of success of each subject (e.g., completely successful, mostly successful, partly successful, and unsuccessful) was based on criteria related to the number of problems (parts) actually solved and the adequacy of the reasons given to justify their answers.

Results

The results of this study are presented in terms of three major aspects of differential performance observed: (1) Quantitative degree of problem-solving success of each subject, (2) specific knowledge about chemical equilibrium, and (3) general problem- solving characteristics of subjects.

Degree of Problem Solving Success

The degree of success (e.g., completely successful, unsuccessful) was based on the number of correct answers and reasons provided by the subject to each problem. A score of one (1.00) was assigned if proper answers and reasons were provided.

When only part of the problem was solved, partial credit was assigned (e.g., 0.25, 0.50). Table II summarizes the degree of success of each high-school novice (HN), college biology-major novice (BN), college chemistry-major novice (DN), doctoral expert (DE), and faculty expert (FE).

The principal investigator evaluated the adequacy of the answers and reasons, which were then corroborated by the second author. As Table I1 shows, three subjects were completely successful (FE1, DE4, FE3) three were mostly successful (FEN, FJ32, DE6), and the remaining 17 were either partially successful (5) or unsuccessful (12).

Eleven of the 13 novices were unsuccessful, lending support to previous research that points to the importance of specific content knowledge (e.g., chemical equilibrium, genetics, physics) in problem solving. Two of the undergraduate novices (CN4, CN5) were sometimes as successful as four of the experts (DE1, DE2, DE3, DE5), showing that there is no clear dichotomy of experts and novices, as pointed out by Smith (1983).

Four of the six doctoral students in chemistry showed rather limited problem- solving success. Although they had about the same amount of formal coursework as their more successful colleagues, their teaching and related experiences were much less. This seems to suggest that not only is the total quantity of formal content knowledge available an important factor, but so is the nature and extent of the opportunities to apply that knowledge.

Table 111 is a summary of the relative difficulty posed by the problems to the subjects. It can be seen that the number of unsuccessful subjects (6) was the same in the first three problems (gas phase) but increased in Problems 4-7. This may indicate


TABLE I1 Degree of Problem-Solving Success P = partial credit, Y = Yes (Credit = l),

N = No (No Credit)

f Probi-ems Degree of- kJbJeLt5 1 3 4 5 6 7 Solved Success

_ _ _ _ - ______ HN I k P Y N N h F: 1.75 Unsuccessful IF? N N N N N N F i - - Unsuccessful f-'N3 N N N N N N N -- Unsuccessful 141.4 N h h F i N N N - - Unsuccessful HN5 N N l v K N N N - - Unsuccessful

BN1 P P F P F: Ei P 1.50 Uvsuccessful BN: E F I\' P N N N -0.50 Unsuccessful BN3 P P P N N N N 1.50 Unsuccessful

CN1 P N N N N N P 0.75 Unsuccessful L N i P P P P N 1.1 P 1 . 2 5 Unsuccessful CN3 P N P N N N N 1.00 Unsuccessful TN4 P P P P N P P 2.50 P a r t . Success CN5 P P P P P P P 3.50 P a r t . Success

DE 1 P P P P P P N 2.00 DE7 I' P P P N N P 1.50 DE 3 P P P P N N Y 3.00 DE4 Y P P Y Y Y Y 6.25 DE5 P P P P N N F 2.75 DLF P Y P Y P P Y 4.75

FE 1 Y Y Y Y Y Y Y 7.00 FEZ P P P Y Y Y Y 5.50 FE3 P Y Y Y Y P Y 6.00 FE4 P Y Y P P P P 4.50

P a r t . Success Unsuccessful P a r t . Success Comp. Success P a r t . Success M o s t . Success

Comp. Success Most. Success Comp. Success Most. Success

HN = High School Novice BN = College Biology Novice CN = College Chemistry Novice DE = Doctoral Expert FE = Faculty Expert

less contact with solution equilibria, or greater difficulty with conceptual knowledge involved in ionic equilibrium as compared to gas-phase equilibrium. Also, the number of subjects who were partially successful (13-15) was gmter on the gas-phase problems than in solution-equilibrium problems. One reason for these results may be that chemical species in solution equilibrium are more abstract or harder for a chemical representation and conceptualization (e.g., cations, anions, molecules) than whole molecules in the gaseous state.

By contrast, the number of completely successful subjects was similar through all problems, showing a similar degree of difficulty and stability of the chemical equilibrium knowledge that they exhibited. Only two subjects (FE1, DE4) were completely successful in solving Problem 1 due to the fact that application of a principle (ideal


TABLE 111 Problem Difficulty

Problem Number

Number o f Sub jec ts Who were

Unsuccessfu l

6 (26%)

6 (26%)

6 (26%)

R ( 3 5 % )

15 ( 6 5 % )

14 (61%)

9 (39%)

Number o f Sub jec ts Number o f Sub jec ts Who Were Mostly c r P a r t i a l l y Success fu l 5uccess f u l

Who Were Cori ipletely

___- -~ . __ 15 ( 6 5 % ) 2 ( 8 ” )

13 ( 5 i X ) 4 ( 1 7 ‘ )

13 (57%) 4 ( 1 7 Y I )

10 (43%) 5 ( 2 7 ’ i )

4 (17.n) 4 (177’;

6 (26%) 3 ( 13%)

8 (35%) 6 ( 2 6 9 )

gas law) was needed to make the prediction that the Kp would be bigger than the K, for that reaction.

Three subjects (13%) solved Problem 6, which required the isolation of a ratio of two concentrations from the equilibrium constant equation. In fact, several subjects could write the equilibrium equation but were unable to isolate the ratio. Since most equilibrium problems involved proportions (double ratios), it seems this was a potential source of difficulty for solving Problem 6. The lack of conversion skills (e.g., pH to [H’]) was a major source of difficulty for dealing with Problems 4-6. Thus, most unsuccessful and partially successful subjects said that they had the K, and pH but did not know what to do with these data.

Chemical Equilibrium Content Knowledge

Probably the major problem-solving differences observed were those related to the amount and quality of specific content knowledge expressed and applied by each subject. These relevant knowledge differences between successful and unsuccessful subjects are described in terms of a large number of chemical misconceptions and knowledge gaps observed at the levels of taxonomy, stoichiometry , kinetics, thermodynamics, acidimetry , and mathematical skills.

First, most novices showed a large number of knowledge gaps about the taxonomy of chemical equilibrium constants (e.g., K,, Kp, KO, Kb, Ksp), which are presented by standard chemistry textbooks with the same symbols and meaning as the seven problems used in this study. Clear evidence for this lack of equilibrium literacy is the very frequent question (mostly by novices) “What is the K,, Kp , K,, Kb, K,,?’ Also, the nomenclature knowledge gap was noticeable in naming and writing symbols for molecules (Problems 1-3) and ionic species in solution equilibria (Problems 4-7). These taxonomic difficulties were recurrent throughout the problems for most novices and were observed in two experts (DE1, DE2).


Second, in thermodynamics, most experts demonstrated proper specific knowledge of chemical equilibrium by applying several principles (e.g., gas laws, thermodynamics laws) to justify their answers and reasons. Most novices, however, showed many basic misconceptions and lack of access to important concepts, which can be summarized by the following examples:

(a) The fact that, in terms of energy content, all reactions are endothermic or exothermic was ignored or not accessed by most novices and a few experts in solving Problem 2. Since the reaction was exothermic (AH” = -22 Kcal), it was predictable that an increase in temperature inhibited or reversed the reaction by a simple thermodynamics argument and by applying LeChatelier’s principle. However, few experts completely succeeded in this conclusion (mainly E l , DW, FE4).

(b) Also only two experts (FEl, FE4) expressed clearly that only temperature changes can change the value of the equilibrium constant.

(c) Most novices ignored the meaning of molar enthalpy, heat of reaction, free energy of formation, free energy of reaction, and that for the elements the free energies of formation are zero by definition.

(d) Almost all novices confused the extent or completeness of a reaction (a thermodynamics concept) with the rate of a reaction in achieving equilibrium (a kinetics concept). This conceptual error was clearly observed in Problems 2 and 3, which asked the subjects to describe the position of equilibrium. In Problem 2, the general tendency was to say that an increase in temperature increases the kinetic energy of molecules, which react more rapidly to form more products, or that the position of equilibrium shifted to the right, which is generally correct. Likewise, most novices believed that more pressure or less volume would produce more molecular collisions causing greater rates of reaction and therefore more product, or completion of the reaction. Interestingly, this reflects a correct basic knowledge and is often true, but in this case (Problem 2) incomplete kinetics reasoning caused errors related to thermodynamics (i.e., the reverse reaction was not taken into account).

(e) Several subjects, who were successful in determining AGO for Problem 3, tended to justify the position of equilibrium as shifted to the right based on the negative free-energy change. However, when asked for another reason or method to describe the extent of equilibrium, few of them used the huge value of Kp (1.2 X Id2). This may reflect more contact with the notion of spontaneous reactions but isolated from the context and application of the equilibrium concept.

(f ) All novices and half of the experts showed misunderstandings of the nature of the equilibrium constant at a given temperature. In fact, few experts were completely successful in solving Problem 2, which requires the concept of constancy for differentiating between factors temporarily shifting the equilibrium state (e.g., volume, pressure, mass) and temperature, which is the only factor capable of changing the magnitude of the equilibrium constant. Only FE1 used the equilibrium equation (mass action law) to reason that to keep K, constant (K, = [NH3I2/[H2l3[N2]), addition of N2 would cause a proportional increase in NH3 and vice-versa.

(8) Most subjects were unsuccessful in predicting quantitatively the relation between Kp and K, (Problem 1) and between K, and K b (Problem 4). Only FEl and DE4 were completely successful in demonstrating this important behavior.

(h) Several subjects confused the negative free energy change (criterion of spontaneity) with the negative enthalpy change (exothermicity) of a reaction.


A sample of excerpts reflects several related misconceptions:

FE3: “They should be the same since concentrations are proportional to partial pressures.” (Problem 1 .)

The proportionality between concentration and partial pressure of a gas is a correct statement but these do not imply equality of K , and Kp. In fact, they are equal only when the change in number of moles of gases (sum of moles of product gases minus sum of moles of reactant gases) is zero. This confusion was expressed by several subjects.

DE3: “1 need the temperature . . . more temperature will increase production of NH3 due to more collisions . . . any of these factors will change the value of K,.” (Problem 2.)

As mentioned previously, a higher temperature will favor an endothermic reaction. On the other hand, only changes in temperature affect the magnitude of the equilibrium constant. These two similar ideas, which were verbalized by several subjects (e.g., DE3, FE4, CN3, HN5, CNl, DE2, DE6, CN5) reflect misunderstandings not only of chemical equilibrium but also of the principles behind it (e.g., LeChatelier, thermodynamics laws).

DE6: “Increase in pressure will increase production of NH3. I’m not sure about the effect of temperature.”

Several subjects seemed to give more importance to the pressure effect (which is correct in this case) than to the thermodynamics factor that should predominate for predicting the extent of a reaction, but very few mentioned why, except for saying more collisions.

Third, several kinetics misconceptions appeared to be intermingled with thermodynamics concepts.

The following results illustrate this confusion:

(a) Most novices and at least two experts (DE3, FE3) expressed lack of knowledge about the difference between rates of reaction and rate constants (Problem 1). In fact, only three experts (FE1, FE4, DW) applied the rate constants provided as a second method for calculating and/or checking the equilibrium constant value. The rest of the subjects said they did not remember methods other than the equilibrium equation.

(b) The perception of chemical equilibrium seemed to be a static instead of dynamic phenomenon. For instance, DE3 did not use the equilibrium concentrations of the four experiments because he observed they were “changing,” since they were different for each experiment. Note the comments of the following expert; DE3: “These values are changing. All I need are the rates . . . K, is the point when K I = K2 . . - since the reaction has not achieved equilibrium yet * *

anyway we need the reaction at equilibrium.” (c) DE3 did not use the rate constants (K1, Kz) because they were different in

magnitude, which he confused with the rates of forward and reverse reactions, whose equality defines the state of equilibrium. It seems that his static notion of equilibrium was a conceptual hindrance for using a simple method (con-


centrations) to determine the equilibrium constant, which could be corroborated by applying the rate constants definition.

(d) FE3 wrote the equilibrium constant equation and showed how it could be calculated by using the concentrations (Problem 1) and said: “I’m reasonably sure about this equilibrium equation method but not sure about rates.” Since no rates were provided, it appears he confused the rate constants with rates of reaction.

(e) All 13 novices and some experts indicated they did not remember other formulas or relations for calculating the equilibrium constant (Problem 1) .

Stoichiometric Knowledge Differences

As expected, this study showed that stoichiometric skills are a prerequisite for solving equilibrium problems that integrate any manifestation of stoichiometty (e.g., formulas, reactions, energy, solutions).

The following are examples of basic stoichiometrical knowledge gapslmisconceptions observed in most subjects:

(a) Only two subjects (El, DE4) succeeded in using the change in the number of moles ( A N = 2) in conjunction with the ideal gas law to predict the correct relation between Kp and K, (Problem 1).

(b) Most novices and some experts failed in determining the limiting reagent (Problem 2) in a reaction that already was balanced and the ratios by volume were 1:l as given in the problem and 1:3 (moles) in the equation.

(c) The majority of the subjects also were not successful in determining the limiting reagent (Problem 3) where the volumes ratio was 1:l and the moles ratio was 2:l. However, most of them wrote the reaction and balanced it.

(d) Most subjects had difficulty in calculating the amount of theoretical product (e.g., mass, volume) for problem 3 where at least two methods were possible (e.g., Avogadro’s principle, ideal gas law). Most novices tended to use the factor-label method (dimensional analysis), but abandoned it because the units did not cancel properly.

(e) Several subjects were not completely successful in calculating enthalpy change (Problem 2) and free energy change (Problem 3) because of the number of moles (2) involved in both cases, which seemed a matter of definition and proportional reasoning.

( f ) Most reactions (Problems 4-7) were not written properly in terms of number of moles, gram-atomic weights, and/or symbols.

(g) Most subjects misused or ignored the 2X (moles) produced by the dissociation reaction MX2 (1:2 stiochiomehy). This failure and the problem of not squaring properly tended to produce incorrect equilibrium equations. Thus, a square root was usually obtained instead of a cube root (X = z/K,,/4), and consequently a wrong prediction was made of the relative solubility.

(h) Several novices suggested that at STP gases have the same density (Problem 3).

The previous observations corroborate similar results of related studies (e.g., Nurrenbern, 1980; Gabel & Sherwood, 1983, Gabel et al . , 1984; Larkin & Rainard, 1984), which reported several stoichiometric limitations of high-school students.

These stoichiometry states of knowledge appeared to be confirmed and are illustrated by several excerpts:


NH1: “I can’t remember the formula for this. I believe the limiting reagent is N2 because smaller atomic number of 3, oh, I think this is the number of moles available.”

DE2: “I believe the limiting reagent is H2, but I’m not sure . . . neither of the two is limiting reagent since the number of moles is the same (5/22.4 1) . . . if I’m right five liters of SO3 are produced . . . SO2 could be the limiting reagent, but I’m not sure.”

BN3: “The limiting reagent is the one in less quantity so it is N2.” HN4: “I guess, only as many as five liters of SO3 are produced since 0 2 is the

limiting reagent.” BN2: “I would say the limiting reagent is N2 because less number of atoms, but

not sure . . . for the liters and grams of SO3 we have to know densities, oh, no, wait a minute! At STP you have the same number of molecules of SO2 and 02, so 5 liters are produced.”

CN3: “The limiting reagent may be SO2 . . . let’s play with some numbers . . . I think I’m not using the right formula.”

CN 1: ‘‘I really don’t remember about the limiting reagent . . . and molar enthalpy.” BNI: “I guess H2 is the limiting reagent . . . molar enthalpy, I don’t know what

that means . . . since it is STP they have the same density . . . I suppose SO2 is the limiting reagent because you have twice as much 02.”

These excerpts seem to reflect stoichiometric difficulty with reactions, energy, and the mole concept.

Chemical-Mathematical Skills

Lack of chemical-mathematical skills was a hindrance for problem-solving success. Many subjects demonstrated a lack of basic mathematical ability indicated by several behaviors:

(a) Difficulty in isolating a chemical parameter from the equilibrium equation (e.g., [H’], [H3C20-2] in Problems 4 and 5 , respectively).

(b) Inability to isolate a ratio of two concentrations. For instance, most subjects did not isolate [HCON]/[OCN-] in Problem 6.

(c) Difficulty in using logarithmic laws to convert pH into [H’] and vice versa; and for relating K, with pK,.

(d) Inability to deal properly with exponents, roots (Problem 7), and scientific notation.

(e) Inability to make proper approximations (e.g., Ca - x = Ca) to justify that the concentration of a weak acid minus the degree of dissociation is equal to concentration of the acid or base (Problem 4).

( f ) Difficulty in using the hydrolysis equation to reason quantitatively about the relative basicity of OCN- and CN- (Problem 4), and the buffer equation as a possible method for working weak acid-base equilibrium problems.

Several excerpts illustrate the importance and interrelatedness of mathematics skills and chemical equilibrium:

CN4: “K, = 0.2M/[Ht], oh, I just cannot remember how to do this . . . I’d say this solution is twice as acidic because of the K, . . . it seems the K,’s are the K,, but I’m not sure about that (Problem 4) . . . pH is the inverse log

PROBLEM SOLVING AND CHEMICAL EQUILIBRIUM 26 1

of that, but I’m not sure, I guess 3H’ per mole are present (Problem 5) . . . just change pH to [H’], but I’m not sure how to do this or to work with the K,. I guess the ratio HOCN/OCN equals K,” (Problem 6 ) . . . “If the Ksp is the same for (MX2, M X ) that would say to me that they are equally soluble . . . I don’t know what to do without having the weight” (Problem 7).

Clearly, CN4 had many chemical and mathematical limitations. Her representation of the task did not go beyond a superficial level, since she did not perceive the ionization of the salts, write the reactions or equilibrium equations, and could not perceive K, and Ksp as equilibrium constants.

HN1: “Well, I don’t know about the KO . . . OK, oh, one is twice as much concentrated, but I’m not sure (Problem 4). I think the stronger acid also is the stronger base . . . I don’t know how to get the pH value . . . What do you mean by K,,? Since they have the same Ksp the solubility is the same . . . I don’t remember the relation between these values” (the atomic weights of X in MX2, M X ) .

HNl’s present knowledge state reflects a lack of contact with the topic but he showed some promising expertlike behaviors throughout Problems 1 and 2. He was the only high-school student who was at least partially successful on some of the problems.

DE1: “Simply by knowing the KO values, the stronger acid produces the weaker

HN5: pH can be calculated by using litmus paper in the lab but here I don’t know

CN5: “I cannot tell which is more acidic. I think the bigger K , means more

base . . . but I don’t know another way right now” (Problem 4).

how to do it . . . equal Ksp implies the same solubility.”

dissociation” (Problem 4).

In Problem 6, CN5 wrote the correct equilibriutn equation but did not isolate the ratio from it. In Problem 7, he made a guess and predicted that “MX2 was more soluble than MX since the Ksp’s were the same.”

CNl: “I think the pH is concentration, but not sure . . . I don’t know how to do with the concentration ratio . . . Since the Ksp is the same, they have the same solubility.”

CN4: “I don’t know about the KO . . . the more concentrated is more acidic . . . I think the weak acid produces the weak base . . . I don’t remember the exact formula . . . pH is PK,? . . . I don’t know the formula . . . this is frustrating.”

DE3: “This is more acidic than this (due to the K,) by about 2.5 times . . . the stronger acid gives the weaker base. I don’t want to go into the business of the K , . . . What do you mean by concentration ratio? I’m not sure, I don’t remember the formula.”

DE2: The smaller the pK, value the stronger the base . . . the concentration of Hf is larger if the KO is larger. . . . We need to know how much it dissociates . . . I may be wrong . . . the K,, = X2/MX2 and KspS2/MX, so I get cubic root and square root . . . the more soluble is the one that gets more into solution . . . This may be useless.”


BN1: “What is K,? . . . We have to compare both pH and using K,, but I’m not sure . . . What is pH,? Empirically, you can measure pH and the number of moles . . . What is KSp? So you may use K,, to calculate concentrations but I forgot how to use it . . . I think they are equally soluble, or maybe MX2 is twice as soluble.”

It is evident from the above examples that these excerpts report common behaviors:

(1) The chemical and chemical-mathematical misconceptions were serious enough to prevent subjects from going beyond the memory or algorithmic level they reflected.

(2) It is educationally important that most of the subjects gave better ideas of how a problem could be approached experimentally (e.g., pH, litmus paper) than by applying chemical-mathematical knowledge at a theoretical level.

(3) Most revealed acidimetry (Problem 4-6), stoichiometry (Problem 7), and chemical- mathematical misconceptions (Problems 4-7) as main sources of lack of success. It might be that lab skills and/or lab activities are more concrete and easy to internalize than abstract concepts.

In conclusion, the number and diversity of similar misconceptions and difficulties observed corroborated related findings reported by others in genetics (Stewart, 1982, 1983; Smith, 1983), physics (e.g., McDermott, 1984; Lin, 1982; Clement, 1981), and chemistry (e.g., Larkin and Rainard, 1984; Gabel and Sherwood, 1983).

Differences in General Problem-Solving Characteristics

This study has confirmed several problem-solving characteristics of subjects reported by previous investigations in related scientific domains. First, the nature of problem- solving performance as a continuum, described by Smith (1983), can be observed by looking at Tables I1 and 111, which indicate gradual differences in problem-solving success instead of a clear dichotomy between experts and novices.

Second, the novices exhibiting expertlike behaviors (Smith, 1983) also were detected in at least two subjects (HN1, CN5), who demonstrated expertlike behaviors such as application of principles, use of checks and better heuristics, and perceiving the problem as a task of reasoning.

Third, the perception of the problems reflected quality differences in favor of successful subjects who perceived the problems as tasks deserving careful analysis and reasoning to reach a solution. By contrast unsuccessful subjects tended to refer to formulas they did not remember, showed a lack of contact with the topic, and saw the problem as one that could be solved in one or two short steps.

Fourth, strategic knowledge differences shown in Table IV indicate the types of heuristics observed through the solution paths. Successful subjects tended to use more and better heuristics and exhibited a greater degree of flexibility regarding the solution path. Unsuccessful subjects tended to be very static (inflexible) in the first 2-3 steps and used fewer and weaker heuristics.

Finally, two main types of representation of the problems were observed: an accurate chemical-quantitative representation and a superficial language-level repre-


TABLE IV Use of Heuristics by Subjects in this Study

~

Used by Used by Successful Unsuccessful

Heur is t ics Subjects Subjects

1.

2.

3 .

4 .

5.

6.

7 .

b.

9.

10.

11.

Moviny to the next p a r t of a problem when unahle t o solve a previous p a r t .

Use o f a simpler method when two o r more were ava i lab le .

Looking fo r s i m i l a r i t i e s in the problems.

Comparison o f answers with the problem requirement.

El iniinate var iab les which appeared problematic a n d work with one var iab le a t a time.

Check consistency o f assuniptions.

Check the answers zrd rea$nt?s fo r consistency.

l l s e o f knokledge development. s t1.a t egy . Use o f t r i a l and e r ro r .

Cse o f other rhmica l symbols ( the i r . o w n ) e a s i e r t o handle t h a r those s t a t ed i n problem.

i l c c ~f dpproxiciat ions which l i c i l i t a t e l ca lcu la t ion? o f r e s u l t s .

Rarely

Frequently

Frequently

Frequently

Frequently

F requen t 1 y

Frequently

Frequently

Karely

Frequcril ty

Frequently

Frequently

Rarely

Rarely

Rarely

Rarely

Parely

Rarely

Rarely

Frequently

Karely

Rarely

sentation. Successful subjects exhibited several behaviors that are evidence of effective chemical representation:

(a) They read the whole problem and then restated the problem in their own scientific language.

(b) They not only used properly the symbols pertinent to each problem but invented their own symbols when needed.

(c) They applied properly several principles, (e.g., gas laws, Avogadro’s principle, LeChatelier’s principle, thermodynamics laws).

(d) They used two methods to solve the problem and/or to check their answers and reasons.

(e) They used several assumptions (e.g., ideal gas behavior in Problems 1, 2, 3) and approximations adequately to simplify the solution of the problems.

( f ) Their questions reflected more checking of purpose and state of knowledge development than insecurity or lack of knowledge.


By contrast, most unsuccessful subjects did not exhibit the above critical behaviors. For most novices even the language of the problems was unrepresentable or incom- prehensible.

Evidence reflecting the absence of chemical representation includes the following: (a) a great number of questions indicating lack of basic knowledge, (b) poor translation of the English language into chemical symbolic language, (c) improper answers to the probings about their conceptual errors, (d) very frequent invocation of formulas, (e) many guessing behaviors, and ( f ) extensive use of “I don’t know” and “I’m not sure” behaviors.

Table V is a synthesis of the quantity and quality of behaviors exhibited by successful and unsuccessful subjects.

TABLE V Synthesis of Successful and Unsuccessful Performance Differences

Unsuccessfu l S u b j e c t s Tended: Success fu l S u b j e c t s Tended:

1. To r e a d p a r t o f a problem and s t o p t o ask f o r aspec ts o f t h e p rob 1 em,

ques t i ons t h a t r e f l e c t e d l a c k o f knowledge even a t t h e memory and a l g o r i t h m i c l e v e l s .

2. To ask seve ra l l o w - l e v e l

3. ’ c a r d on t h e cha lkboard b e f o r e

To w r i t e d a t a f rom t h e problem

making sense o f t h a t i n f o r m a t i o n .

To b e g i n t h e process by w r i t i n g t h e r e a c t i o n t h a t was g i v e n i n t h e problem.

4.

5. To o m i t o r i g n o r e w r i t i n g t h e e q u i l i b r i u m c o n s t a n t equa t ion .

6. To p e r c e i v e and t,alk about t h e problem as a t a s k o f r e c a l l f rom p a s t courses.

7. To say 1 r e a l l y d o n ’ t know how t o do t h i s , I j u s t remember t h e fo rmu la o r equa t ion .

8. To use e x a c t l y t h e same sywbols o r t o use them i m p r o p e r l y .

9 . To o m i t a l o t o f necessary s t e p s o r t o work w i t h o u t o rde r .

1.

2 .

3 .

4.

5.

6.

7.

R .

9.

To r e a d c o m p l e t e l y t h e whole problem f i r s t b e f o r e a s k i n g any q u e s t i o n s .

To ask h e t t e r q u e s t i o n s r e - f l e c t i n g more knowledge, unders tand ing , and f o r purposes o f c l a r i f i c a t i o n o r c o n f i rn ia t i on .

To r e r e a d t h e o b j e c t i v e s on t.he problem c a r d b e f o r e s t a r t i n s t h e s o l v i n g p rocess .

To b e g i n t h e s o l v i n g p r o w s . - by w r i t i n g t h e t - rac t i o r i , whether i t was g i v e n o r n o t , dnd b d l a n c i n g i t o r checkirrc; i f i t was, ba lanced.

To w r i t e down the c o r r e c t e q u i l i b r i u m c o n s t a n t e a r l j i n t h e process.

To p e r c e i v e and t h i t i k about t h e problem a > d t a s k o f r m s o n i n g dnd developnicrl t o f LI s o l u t i o n .

Eht t o rnent i u n dny fo rmu la o r equa t io r i u n t i : t l . f y had s a i d how t o s o l v e i t i n chemicdl t e r m .

l o L I ~ C ’ oi.her symhcils t o r e p r e s e n t ctieni:ral specit,:. i r t g u i l i b r i u n l rhe r i i t . w a s needed or appropr id1. t .

To dc t h e necessary L i e p s o f t h e process i t i e r d e r .


Unsuccessful Subjects Tended: Successful Subjects Tended:

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

To do unnecessary work o r repeat the same information given i n the problems.

To use or mention only the LeChatelier p r i nc ip le and t o c i t e i t improperly.

To provide wrong answers and j u s t i f y them wi th i l l o g i c a l reasons o r too much guessing.

To make no checks which could show wrong answers and reasons

To lose motivation o r show lack o f motivation and even t o be t i r e d soon.

To make no assumptions or approximat ions.

To show o r use many chemical misconceptions o f equi l ibr ium and re la ted concepts and principles.

To show only ro te memorization o f the meaning o f the Ke o r algorittnnic learning a t best.

To ignore the use o f the equil ibr ium constant equation as a means o f reasoning.

To show a lack o f basic mathematics knowledge such as logarittnnic laws. ra t ios. proportions, roots. exponents, and i s o l a t i o n o f parameters (e.g., concentration, Kc) from and equation.

To ignore the fac t that the problems were a l l equi l ibr ium problems.

To use t r i a l and e r ro r unnecessarily.

To use means-ends analysis

To use o r w r i t e t r i v i a l information. words. work.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

To do only the necessary work and use other informa- t i o n not given i n the problems.

To use several pr inc ip les and related concepts t o j u s t i f y t h e i r reasons.

To provide correct answers and reasons.

To make frequent, checks o f the work done t o discover any inconsistency among steps, answers, and reasons.

To be w e l l m t i v a t e d by the problems and the so lut ion process.

To make proper assumptions and approximations when needed.

To show adequate chemical conceptualizations o f the concepts and pr inc ip les i nvol ved . To show proper understanding o f the equi l ibr ium constant.

To use o r apply the e q u i l i - brium constant equation t o j u s t i f y t h e i r reasons when needed.

To show proper knowledge and use of these and other mathematical s k i l l s .

To categorize the ne*r problem as another equilibrium problem by saying i t i s s i m i l a r o r almost the same.

Not t o use t r i a l and e r ro r

To use a knowledge development strategy.

To use o r mention relevant information not stated i n the problems.

(Table continues on p . 266.)


TABLE V (Continued)

U n su cce s s f u 1 S 11 b j ec t 5 Te r,d ed : >U( L + , I > i U I \ [ , t ~ , t ’ ! . I,, ’ t.r.J+‘tl:

~ ~ ~ .. ~. ~ ~~ ~~ ...

24. To make care less c i s t akes 24. To nidke few cdrc lps i which were not ncted or m i i t Uhes . d’iscovered.

25. To be s i l e n t very often and 2 5 . To t h i r l . a l o u d f l u e f i t l y as d express a l o t o f insecur i ty w e ! -prepared teacher does. about t h e i r verba l iza t ion .

26. To use o r mention one nethod 2 6 . T o list? or expr-c.5 the know- t o solve the problem rw ledge o t i,,tirt> t h a n ovc none a t a l l . method or p r i i i r ’ p l e t o solve

t h c prubleill.

27. To make no coninients about the 2 7 . Tr Viake proper Lxwents problems and to s k i p informa- a b o u t or bejot id t h e p r o - t i on on the problems. tilrrns’ covtert.

Conclusions and Discussion

The type of research reported in this article should be seen as preliminary to confirmatory, hypothesis-testing research that uses random and other sampling techniques. Different samples will undoubtedly produce somewhat different results in expert-novice and related designs that are used to explore the problem-solving behaviors and knowledge structures of subjects. This should not inhibit such research, but it must be recognized that overgeneralization is a very real risk. We urge the reader to interpret our conclusions and discussion in this light.

The findings of this study have generally confirmed several important observations reported by previous problem-solving studies in science. Some of these observations are (1) Problem-solving expertise is a continuum rather than a simple dichotomy (Smith

& Good, 1984). (2) Frequent checks of the work done by successful subjects clearly determined part

of their success (Smith & Good, 1984). (3) Experts spend some time initially with each problem developing a representation

scheme before proceeding to formulas or equations (Simon & Simon, 1978). (4) Experts exhibited more coordinated and integrated knowledge (Larkin, Heller, &

Greeno, 1980; Simon & Simon, 1978). (5) Experts showed an insight for categorizing the problems at the beginning as equilibrium

problems (Chi et al., 1981). (6) Problem solving is negatively affected when rote memorization rather than meaning-

ful integration of knowledge is the clear choice of the subject (Stewart, 1982; 1983; Gabel & Sherwood, 1983; Smith & Good, 1984; McDermott, 1984).

(7) Experts demonstrated more powerful strategies such as knowledge-development and cross-checking as compared to novices who used more trial-and-error and algorithm recall (Simon & Paige, 1974; Smith & Good, 1984; Larkin, et al . , 1980).


The observations and results of this study have several implications for (1) a theory of problem solving, (2) utility of the think-aloud technique to observe problem-solving behaviors, (3) determining the adequacy of the chemistry domain in which to do problem-solving research, and (4) for developing chemical education instruction.

Development of Problem-Solving Theory

Research into problem-solving skills in scientific domains points toward the need for a holistic theory in which those problem-solving performance characteristics can be accounted for. Presumably the following observations about problem-solving characteristics are common across scientific domains:

(a) The differential application of a core of principles and concepts that serves as a conceptual umbrella for a formal domain has been a major observation of this study. Successful subjects consistently evoked and correctly used a large number of principles (e.g., gas laws, thermodynamics laws, Avogadro’s principle, LeChatelier’s principle) to guide the solution process and to justify their answers and reasons. This clear advantage of having an adequate conceptual framework (by successful over unsuccessful subjects) for guiding the solution of problems under a specific domain supported previous observations of Smith (1983), Chi etal. (1981), Bhaskarand Simon (1977), SimonandSimon(1978), andLarkin et al. (1980).

(b) Quality differences in the characterization of problems according to superficial features (by novices) or deeper shuctures (by experts) have been reported by several related studies (e,g., Chi ef al., 1981; Larkin, 1981; Smith, 1983). In this study, the classification or sorting superiority of the successful subjects was shown by behaviors such as these: (1) additional comments beyond those ideas explicit in the problems, (2) the use of more than one method to solve the problem, (3) the use of other chemical symbols not given by the problem, and (4) the types of questions asked. Sorting ability is very likely a consequence of the existence of an adequate theoretical framework of the domain.

(c) Related to the hierarchical conceptual organization appears to be the difference in procedural and strategic knowledge exhibited by successful and unsuccessful subjects (e.g., Smith, 1983; Simon and Simon, 1978; Larkin, 1981; Greeno, 198). For example, successful subjects demonstrated a knowledgedevelopment capacity, frequent checks for consistency of work done, and checks by using a second method, and better adaptability to the solution path (Tables IV and V). In contrast, unsuccessful subjects tended to use weaker strategies (trial and error, guessing) and appeared too inert to follow a productive solution path.

(d) The differential degree of motivation of subjects may be an important component to be considered in the growing development of problem-solving theory (Smith, 1983). In this study, motivation seems to be a success-determining factor. Most successful subjects clearly expressed interest in several ways during the activity of solving the problems: They made comments about the nature of the problems, tended to spend more time working for an acceptable solution, tried to detect errors and asked if they could go back to modify something apparently wrong, and they seemed to enjoy the challenge of solving the problems correctly. On the other hand, unsuccessful subjects, especially novices, exhibited a lack of motivation by the following behaviors: (1) trying to get rapid solutions in one or two steps, (2) asking frequently, “How many more problems are to be solved?,” (3) providing answers and reasons by using guessing, (4) verbalizing


that they felt very uncomfortable about these problems, (5) saying that they disliked these problems, and (6) tending to show facial and verbal lack of interest in the problem-solving activity.

Usefulness of Think-Aloud Technique

This study has described a large number of problem-solving behaviors that could not have been observed by direct paper-and-pencil measures (Table VI). These 27 chemical-equilibrium behavioral patterns documented similar performances in other scientific domains such as genetics (Smith, 1983; Stewart, 1983), chemical thermodynamics (Bhaskar & Simon, 1977), stoichiometry (Gabel and Sherwood, 1983; Nur- renbern, 1980), and physics (e.g., Chi et at., 1981; Larkin et a l . , 1980).

Increasing awareness of the affinity and universality of behaviors inside the diversity of scientific domains should provide the theoretical-empirical basis for a problem- solving behavioral composite with a wide range of applicability. It is important to note that the use of the think-aloud interview and its principal product of real problem- solving performances allowed the development of quantitative and qualitative criteria for determining the individual degree of success (Tables I1 and 111) , whose comparison with qualitative problem-solving differences (Tables IV and V) provide additional credibility for the think-aloud technique. An implication that follows not only from the chemical and chemical-mathematical misconceptions observed in this study, but from cognitive science research in general, is the increasing need for teachers, students, and researchers to develop their capacity of assessing the amount or quality of information they possess. For example, Good, Kromhout, and Bandler (1986) suggest that the development of interactive expert systems may help both teachers and students to evaluate their conceptual knowledge state in a given domain.

Adequacy of the Chemical Domain

The problems selected for this study have demonstrated empirically that chemical equilibrium is indeed a well-structured, semantically rich, and formal domain for observing problem-solving behaviors. However, little attention has been given to the field of chemistry as a fruitful source of problems that allow the observation of a wide spectrum of behaviors and skills. This study has provided empirical evidence that chemical equilibrium is a very suitable content domain for psychological research.

It is important to note that most problems provided by standard textbooks do not meet the criteria as adaptable to both experts’ and novices’ states of knowledge. For example, most problems in textbooks seem to share the following features:

(a) Some are too easy for experts or too complex for novices in terms of mathematical

(b) Most can be solved by one method only. (c) Most are solvable by direct application of a formula or equation, which probably

is a major factor in the fixation of the pervasive formula-leaming phenomenon described in this study and previous related research.

(d) Problems ask for answers or results (mostly numerical) without reasons or justifications.

demands.


(e) Emphasis is placed on quantitative aspects of leaming at the expense of qualitative

( f ) They usually are solvable in one or two steps. reasoning.

Chemistry Instruction

This study has helped to answer the question: What are the procedures used by successful problem solvers of chemical equilibrium problems? Experimental classroom research must be done to confirm hypotheses about how instruction can help students become better problem solvers. Our ideas (mainly experience-based conjectures) about teaching chemistry are similar to those of others who have done related problem- solving research in physics and biology.

(1) Select the problems that require analysis and reason. (2) Provide class time for individual and small-group problem-solving efforts. (3) Search for underlying student misconceptions and deal with them in class. (4) Encourage frequent checks by students to discover inconsistencies. (5 ) Check for basic mathematics knowledge. (6) Emphasize the organization of concepts and principles underlying chemical

equilibrium and how this organization is used to solve problems.

Apparently, the path to effective problem solving is neither short nor direct. Considerable content-specific knowledge is needed and the solver must see how the various pieces of knowledge come together in the solution process.

Suggestions for Further Research

Since there are many types of chemical equilibrium problems, a classification study similar to that done by Chi et al. (1981) is strongly recommended. This might provide valuable information about the organization of knowledge held by persons with various chemistry backgrounds.

A second area that needs further study is the successful novice. How do some novices achieve expertlike success in solving certain problems? Do they have different content knowledge structures, attitudes about problem solving, similar success in related content fields, etc.?

A third type of research that would be helpful is related to the think-aloud interview technique. We found that some faculty experts provided many more details about their thought processes when they explained things much as they would in a teacher-student role. Clearly this has potential problems for distracting the solver from the task at hand, but encouraging the subject to explain things as they might do for a student or younger person could provide a richer store of data about their knowledge of the science content. Especially for some novices who tend to remain silent unless prodded often, this technique might be helpful.

Experimental classroom research on various instructional techniques must eventually be done to test these and other conjectures about how students become better solvers. As with other important, complex questions, there are likely to be no simple answers. Expertise in solving science problems is achieved probably through a long, arduous


process requiring high motivation, just as it is in chess, medicine, mathematics, ice skating, computer programming, etc.

Appendix

Chemical Equilibrium Problems Used

(1) For the reaction, PC15(g) ++ PCl&) + Cl2(g) the forward and reverse rate constants are 20.00 and 3.64, respectively. In four experiments the equilibrium concentrations were

Experiment [Cl,] w151 W 1 3 1

1 0.055 0.0023 0.23 2 0.37 0.010 0.15 3 0.47 0.085 0.99 4 1 S O 1 .oo 3.66

(a) How would you calculate the value of K, or kq. (b) What is the relation between the K, and Kp (i.e., which is bigger? (c) What is the value of the K, for the reaction PCI3(g) + C12(g) cf PCls(g)?

(2) 5.0 liters of N&) are mixed with 5.0 liters of H2(g) in a reaction tank of 10 liters. The reaction is N&) + 3H2(g) t, 2NH3(g), AH = -22.0 kcal.

(a) Describe the effect of the following on the position of equilibrium and explain why? (1) Addition of N2(g) (2) Addition of NH3(g) (3) Increase of temperature (4) Decrease in volume

(b) Which of the above stresses will change the value of the equilibrium constant? (c) What conditions of temperature and pressure would you use to increase the

(d) Which reactant will limit the production of HN3(g)? Why? (e) What is the molar enthalpy of NH3(g)?

production of NH3(g)? Why?

(3) SO&) is a major air pollutant which is oxidized by 02(g) to form SO3(@.

(a) Write the equation. (b) If AGP (SO2) = -71.79 kcalimol, AGP (SO3) = -88.53 kcalimol, and the

(c) Describe the position of the equilibrium. (d) If 5 liters of SO2(& are mixed with 5 liters of Oz(g), how many liters and

(e) Which is the limiting reactant. Why?

K, = 1.2 X 10” (at 25”C), how would you calculate the AGO?

grams of SO3(g) would be produced at STP?

(4) How would you compare the acidity of the following solutions:

PROBLEM SOLVING AND CHEMICAL EQUILIBRIUM 27 1

and 0 . 1 0 ~ HOCN ( K ~ = 1.2 x 10-4)

0.20M HCN (KO = 4.0 X lo-")?

(a) How much more acidic is one solution than the other (i.e., how many times

(b) How would you compare the relative basicity of the anions OCN- and CN-? more)?

(5) How would you calculate the number of moles of sodium acetate Na'H3C202- which must be added to 1.0 liter of 0.50 M H3CCOOH (K, = 1.8 X lod5) to produce a solution with pH = 4.00?

(6) HOCN has an ionization constant of 1.2 X (pK, = 3.92).

(a) How would you calculate the concentration ratio of HOCN (K, = 1.2 X

(b) How would you calculate the above concentration ratio if the pH = 3.50? and OCN- needed to have a solution with [H'] = M?

(7) Assuming that two salts of the type MX;! and MX have the same Ksp (solubility product constant) with a value of 8.1 X

(a) How would you calculate the solubility (in moleslliter) of each salt? (b) Which salt is more soluble?

(Assume you have saturated solutions of the two salts.)

Acknowledgments

Special thanks to Ed Mellon, Professor of Chemistry at Florida State University, for his careful reading of the manuscript and helpful comments. Also, we appreciated the cooperative atmosphere within FSU's Chemistry Department during the completion of this study. Without the assistance of many chemistry faculty and graduate students, this study could not have been completed.

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Problem solving and chemical equilibrium: Successful versus unsuccessful performance