Two Suggestions for Improving Chemical Equilibrium Instruction

Transcription

1 Two Suggestions for Improving Chemical Equilibrium Instruction Journal: Journal of Chemical Education Manuscript ID: ed--000h Manuscript Type: Article Date Submitted by the Author: -Jan- Complete List of Authors: Yaron, David; Carnegie Mellon University, Department of Chemistry Davenport, Jodi; WestEd, Greeno, James; University of Pittsburgh, Learning Research and Development Center Karabinos, Michael; Carnegie Mellon University, Chemistry Leinhardt, Gaea; University of Pittsburgh, Learning Research and Development Center Keywords: First-Year Undergraduate / General < Audience, Chemical Education Research < Domain, Problem Solving / Decision Making < Pedagogy, Acids / Bases < Topics, Equilibrium < Topics, ph < Topics, Precipitation / Solubility < Topics

2 Page of Submitted to the Journal of Chemical Education 0 ed-xx-xxxxxx Two Suggestions for Improving Chemical Equilibrium Instruction David Yaron*, Jodi L. Davenport, James Greeno, Michael Karabinos & Gaea Leinhardt Carnegie Mellon University, Pittsburgh, Pennsylvania, WestEd, Oakland, California,, United States University of Pittsburgh, Pittsburgh, Pennsylvania, ABSTRACT Two suggestions for instruction in chemical equilibrium are presented along with the evidence that supports these suggestions. The first suggestion relates to the indirect connection between chemical reactions, written as rules for how reactant species are converted to product species, and the effects of reactions on concentrations, which result from application of this rule to collections of chemical species. This connection is obvious for experts and so instructors may not recognize the need to make this connection explicit to students. The second suggestion is the use of the majority and minority species (M&M) strategy to analyze chemical equilibrium systems. The argument is made that reasoning about equilibrium systems necessarily involves a twophase analysis, and the M&M strategy makes these two phases explicit for students. INTRODUCTION Chemical equilibrium and its applications is one of the most difficult topics in an introductory chemistry course and consumes a considerable portion of instructional time,. Prior work has focused on the dynamic nature of equilibrium, wherein concentrations reach a steady state at equilibrium despite the fact that chemical reactions are still occurring. The dynamic nature of equilibrium is indeed an important and central concept. However, a substantial set of additional knowledge is needed to reason about the concentrations of chemical species in an equilibrium system. This paper focuses on this essential additional knowledge. Below, we propose two evidence-based instructional suggestions. One type of evidence is student data, including interviews with students as they solve problems and the results of a design experiment that compares instructional approaches across years in the same class. Another type of evidence is analysis of textbooks, which we use to verify that the targets of our instruction align with those of current courses. In particular, we show that the proposed approach to analyzing equilibrium systems can be used successfully on the tasks posed to students in an introductory chemistry text. We further show that the proposed instructional approach frames the knowledge in a way that supports qualitative reasoning about the key ideas of acid-base chemistry. Our first suggestion is to make the progress of reaction coordinate explicit in instruction. The progress of reaction coordinate describes the amount of reactants that have been converted to products. Figure shows this coordinate for conversion of NO molecules to N O molecules. Domain experts associate motion along this coordinate, with a chemical reaction representation, NO N O () Journal of Chemical Education // Page of

3 Page of ed-xx-xxxxxx For instance, in LeChatelier s principle, the phrase shifts to the right (left) is often referred to the chemical reaction (), whereas the shift is in fact more directly connected to the progress of reaction coordinate in Figure. For experts, the connection between these representations is obvious. However, prior work suggests that teachers and students incorrectly interpret the meaning behind the language of shifting to one side of a reaction. Our research provides additional evidence that suggests the connection between the chemical equation and reaction coordinate is not obvious for novices. The progress of reaction is not necessarily difficult to learn, however it is not made sufficiently explicit in current instruction. A working knowledge of the concept of progress of reaction is essential, since it links chemical reactions to concentrations. The reaction is a rule regarding conversion of reactions to products. Concentrations are an outcome of the application of this rule to collections of molecules. Our second suggestion is the use of the majority/minority species (M&M) strategy to frame instruction in both qualitative and quantitative analysis of equilibrium systems. The M&M strategy breaks analysis of equilibrium systems into two phases. In phase one, reactions with large K are identified and run to completion. Chemical species that have non-zero concentrations after phase one are majority species. Chemical species whose concentrations are zero after phase one are minority species that will be present with small concentration at equilibrium. In phase two, laws of mass action (Q=K at equilibrium) are used to determine the concentrations of these minority species. We argue that experts use a two-phase approach when analyzing equilibrium systems, although this is typically done implicitly. Making these two phases explicit to students, by using the M&M strategy to frame instruction, has a number of potential advantages. The M&M strategy integrates qualitative reasoning ( what is present in large versus small concentrations in this solution? ) with quantitative reasoning ( what are the values of the concentrations at equilibrium? ). When framed within the M&M strategy, quantitative problem solving can reinforce qualitative reasoning, and vice versa. The M&M strategy also provides an organizing structure for reasoning across applications of equilibrium, including acid-base and solubility chemistry. This structure can be introduced early in the course and used to frame further instruction, independent of a particular order of covered topics. A variety of evidence is provided in support of the above instructional suggestions. For our first suggestion, our research demonstrates that the progress of reaction coordinate is not obvious to novices and requires explicit instruction. We identify common types of student errors that suggest students fail to flexibly understand chemical reactions as rules that are applied to collections of molecules. For our second suggestion, our research demonstrates that the M&M strategy is both easier to learn than current instructional approaches and universally applicable to analysis of equilibrium systems. In addition to providing new modes of explanation, our instructional suggestions provide a lens through which to view and react to student reasoning. Certain student errors, described below, may indicate shallow understanding of chemical reactions that can be addressed by connecting to the progress of reaction coordinate. The two-phases of the M&M strategy also provide a frame for analyzing students work on equilibrium problem solving. Journal of Chemical Education // Page of

4 Page of Submitted to the Journal of Chemical Education 0 0 ed-xx-xxxxxx THE PROGRESS OF REACTION COORDINATE Q=0 Q= Figure : Schematic for the reaction NO N O, that shows the progress of reaction coordinate (from all reactants on the left to all products on the right), along with the connection to the reaction quotient Q, the rates of the forward and reverse reactions (blue and green arrows), and the equilibrium position (Q=K). Our first instructional suggestion is that the progress of reaction coordinate be made explicit in instruction. A chemical reaction is a rule describing the conversion of reactant to product molecules. When applied to a collection of molecules, this rule generates a series of states that can be ordered according to the degree to which reactants have been converted to products. Explicit instruction is needed, since the connection between the reaction rule and the states of a collection of molecules is quite indirect. We have observed a class of student errors in a virtual lab activity that provides evidence that the connection between reactions and collections of molecules is not straightforward. In the activity, students are asked to determine the reaction between four chemical species (A, B, C, D), using a virtual lab simulation containing M solutions of each species. A commonly applied strategy is to mix pairs of solutions until a reaction is observed. For instance, the following may be observed when mixing solutions of A and B: 0 ml of M A + 0 ml of M B 0 ml of 0. M A, 0.0 M C () The information from this experiment is sufficient to determine that the reaction is A + B C. Interpretation of this experiment is based on (i) dilution (leading to initial concentrations of 0. M A and 0. M B), (ii) identification of B as the limiting reagent, and (iii) extraction of the stoichiometric coefficients of the reaction. Over a fouryear period with students, 0% of them entered a reaction in which the same species appears as a reactant and product. For the above experiment, this class of error may lead students to write the reaction as: A + B A + C () A student who makes this error is using the arrow of a chemical reaction to indicate the effects of time on a collection of molecules, i.e. A and B were present before the reaction while A and C are present after the reaction. Indeed, this is how the arrow is used in Eq. (). In a chemical reaction, the arrow has the much more subtle use of indicating a rule that gets applied to a collection of molecules. The ability to solve limiting reagent problems is, apparently, not a strong indicator that students understand this aspect of chemical reaction notation. The class average on limiting reagent problems before undertaking this virtual lab activity was above %. Journal of Chemical Education // Page of K

5 Page of ed-xx-xxxxxx Another class of student errors, that suggests the progress of reaction coordinate is not obvious to students, arises in equilibrium problem solving. The law of mass action (K=Q) applies only when the reaction has reached equilibrium. A common student error is to insert given, as opposed to equilibrium, concentrations into K=Q while solving equilibrium problems. The lack of a distinction between these two very different positions on the progress of reaction coordinate provides further evidence that students do not have a robust understanding of this coordinate. A classroom study suggested that instruction that explicitly taught the progress of reaction strategy along with molecular diagrams of multiple possible states of the reaction along the continuum improved student outcomes for low-performing students. Finally, we note that the common instructional practice of attaching Lechatlier s principle to chemical reaction notation may be problematic for learning. The shift of LeChatelier reasoning is best referred to a collection of molecules, as at the bottom of Figure. The common practice of attaching the shift to the reaction rule itself may be problematic, since students may not be sufficiently aware of the indirect connection between the rule and the concentrations of collections of molecules. THE M&M STRATEGY Our second instructional suggestion is that the M&M strategy be used to frame instruction. The goal of the M&M strategy is to support quantitative and qualitative reasoning for situations, such as those that arise commonly in acid-base and solubility chemistry, where the equilibrium position of a reaction lies far to the right or left of the progress of reaction coordinate of Figure. In such situations, a clear distinction exists between species with a large (majority) versus small (minority) concentration at equilibrium. Minority species have concentrations that are at least times smaller than majority species and typically to times smaller. Recognition of these large differences is an important part of qualitative reasoning regarding the nature of the solution. In addition, the reasoning used to determine the majority species is of a different character than the reasoning used to determine the minority species, and these differences play an important role in both qualitative and quantitative reasoning. M&M strategy for systems with one reversible reaction We begin by considering situations where equilibrium lies at an intermediate position along the reaction coordinate, such as the case shown in Figure. Although the M&M strategy is not applicable to such situations, this provides a good starting point for both our discussion and for equilibrium instruction. Consider a generic binding reaction with an intermediate value for the binding constant, K=, and initial concentrations of M for both components, A and B. The equilibrium concentrations may be determined by using an Initial-Change-Equilibrium (ICE) table to capture the constraints imposed by the stoichiometry: A + B A:B K = 0 initial -x -x +x change -x -x x equilibrium Journal of Chemical Education // Page of

6 Page of Submitted to the Journal of Chemical Education ed-xx-xxxxxx The law of mass action generates a second-order polynomial K A: B x A B x () with a single positive root x=0.. For such cases of intermediate K, further algebraic simplification is not possible, and qualitative reasoning is limited to LeChatelier s principle. For situations involving multiple reactions, obtaining a quantitative solution becomes considerably more involved and is outside the scope of introductory courses. Thus, while a single reaction with intermediate K is a useful starting point for instruction in equilibrium, additional means for reasoning about equilibrium systems are necessary for progress to be made on more complex systems. Such additional forms of reasoning are available when K is large. In current instruction, this additional form of reasoning takes the form of a small-x approach. First, the reaction is pushed all the way to completion (see better start below). Displacement from this position is described with a variable x (see change and equilibrium ). Since K is large, the displacement x, away from this position, may be assumed to be small (see assume x<<.* - ). Protein + Drug Protein:Drug K = initial better start +x +x -x change x.0 - +x. - -x equilibrium x x assume x<<. - With this small-x assumption, the algebra simplifies to.*- x.0* - This can be easily solved to give x= -. The assumption that x<<. - must be checked and is accepted if it applies to within some tolerance, typically taken as %. The M&M strategy is mathematically equivalent to the above small-x approach, but brings qualitative reasoning to the foreground while simultaneously simplifying the algebra. The M&M strategy involves two phases. The first phase determines the identity and concentration of the chemical species present in large amounts at equilibrium, i.e., the majority species. This is phrased as a thought experiment about what would happen if K were infinitely large, corresponding to a reaction that goes to completion. Thus, phase of the M&M strategy leads to a familiar limiting reagent computation. Journal of Chemical Education // Page of

7 Page of 0 0 ed-xx-xxxxxx Phase : Determine the Majority Species Protein + Drug Protein:Drug K = initial assuming K= At the end of phase, any species with non-zero concentrations are majority species, and the concentrations of these majority species are now known. Species with zero concentrations are minority species. We note that the concentrations of these minority species are in fact small but non-zero, since K is large but not infinite. Phase of the M&M strategy uses the law of mass action, K=Q, to determine the concentrations of the minority species. From phase, the concentrations of the majority species are already known, and may be substituted into K=Q to give: Phase : Determine the Minority Species K Protein : Drug. Protein Drug Protein.0 This is easily solved to give [Protein] =.0 -. Since the M&M strategy is applicable only when the minority species are present in smaller concentrations than the majority species, one should check that this is indeed true to within the % tolerance used in the small x approach. Here, the Protein concentration is much smaller than the concentrations of the other species, and so the M&M strategy is applicable. For reactions with small K, phase can pose the thought experiment what would happen if K were not just small, but zero? Alternatively, the reaction can be reversed to give a reaction with large K. The M&M strategy is mathematically equivalent to the small-x approach. Step of the M&M strategy is identical to finding the better start position of the small-x approximation, but phrased as a thought experiment that clearly connects this step to an assumption regarding the magnitude of K. Step of the M&M strategy is equivalent to introducing the variable x and then ignoring it when it is added or subtracted from a majority species. In the M&M approach, we rephrase this by saying that the majority species concentrations are known from step, and one need only use the law of mass action to determine the minority species concentrations. Finally, the check that the minority species concentrations are small compared to the majority species concentrations is equivalent to the check of the assumption that x is small. Although the M&M strategy and small-x approach are equivalent, there is evidence that the M&M strategy is considerably easier to learn. The use of the M&M strategy more than doubled student performance on a challenging equilibrium problem, compared to a semester with conventional instruction. Journal of Chemical Education // Page of ()

8 Page of Submitted to the Journal of Chemical Education ed-xx-xxxxxx A number of factors may contribute to making the M&M strategy easier to learn than the small-x approach. In the small-x approach, the move to the better start position is motivated mathematically, to support the algebraic simplifications arising from the assumption of small x. Phase of the M&M strategy replaces this with a thought experiment that focuses attention on the chemical consequences of having a large K. The thought experiment asks what would happen if K were not just large, but infinite? This question helps the transition to phase, when we ask what are the consequences of K only being large, as opposed to infinite? The small x approach may obscure this chemical reasoning by introducing unneeded algebraic complexity. In particular, the various algebraic approximations (for example, x is ignored when added or subtracted from a large number but not when multiplied by a large number) are difficult. This may distract students from the key chemical idea that a reaction with large K lies to the far right of the progress of reaction coordinate. In addition, students may attach significance to the sequence of actions in the small-x approach, and misinterpret the reaction as proceeding to completion and then relaxing back to an equilibrium state. The phrasing of this as a thought experiment in the M&M strategy may help prevent this misinterpretation. M&M Strategy for systems with Multiple Reactions The M&M strategy may be especially powerful when applied to systems in which multiple reactions are occurring, such as in acid-base and solubility chemistry. In such cases, the M&M strategy improves coherence by providing a single unifying approach to analysis of equilibrium systems. This unifying approach can be viewed as a planning net, that: breaks the analysis into subgoals, provides a means for achieving these subgoals, and provides a means for verifying that the subgoals have been met. For systems with multiple reactions, the two phases of the M&M strategy expand to form the two subgoals of a planning net (see Figure ). Journal of Chemical Education // Page of

9 Page of 0 ed-xx-xxxxxx Figure. Planning net for application of the M&M strategy to systems with multiple reversible reactions. Journal of Chemical Education // Page of

10 Page of Submitted to the Journal of Chemical Education ed-xx-xxxxxx Figure. Reactions with K>> for acid-base and solubility chemistry. Acid-Base Chemistry (HA and A- indicate any conjugate acid-base pair of a weak acid/base) H + + OH - H O HA + OH - A - + H O A - + H + HA HA + A - A - + HA (K a of HA << K a of HA ) Solubility (MX is assumed to be a slightly soluble salt) M + + X - MX(s) (precipitation) M + + nl ML n + (complex ion formation, K f ) MX (s) + nl ML n + + X - (if K f K sp >>) The M&M strategy provides a systematic approach to the initial analysis stage of equilibrium problem solving, which is often left implicit and unverbalized in instruction. Given an equilibrium problem, experts will naturally select the correct chemical reaction with which to begin the analysis, without being explicit about how this reaction is selected. The criterion for this selection process is made explicit in the M&M strategy. In phase, the criterion is strong reactions between species that are present in high concentrations. If a strong (K>>) reaction exists between two species, they cannot both be majority species in a solution. The strong reaction will progress until one or more of the reactants becomes a minority species. The choice of the reaction is therefore coupled to processes actually occurring in the solution. In phase, the criterion is reactions for which the law of mass action allows one to obtain the desired minority species concentrations. The selection of the reaction is based on which minority species are of interest. We return to this important distinction between phase and phase reasoning following a concrete example. The supplemental materials show application of the M&M strategy to common problem types encountered in current instruction. Here, we examine two cases. The first is the challenging problem of determining the ph of a polyprotic buffer, after addition of a strong acid. Consider a case where, after dilution, the concentrations of the initial reagents are: [H PO - ] = 0. M [HPO - ] = 0. M [H + ] = 0.0 M In phase, the subgoal is to determine the identity of the majority species and their concentrations. The process is to identify strong (K>>) reactions between species with nonzero concentrations and run these to completion (assuming K= ). The only such reaction is HPO - + H + H PO -, so we assume K= and find: [H PO - ] = 0. M [HPO - ] = 0. M [H+] = 0 M At this point, we can verify that no strong (K>>) reactions exist between species with nonzero concentration and thus phase is complete. The identity and concentrations of the majority species are now known. In phase, the subgoal is connected to our intent of determining ph, for which we need the concentration of the minority species [H + ]. The process is to select any reaction for which the law of mass action allows us to determine [H + ] from the majority species concentrations. The most natural such reaction is the Journal of Chemical Education // Page of

11 Page of ed-xx-xxxxxx second acid-dissociation reaction for which K a = [H + ][HPO - ] /[ H PO - ]. Once a minority species concentration is known, it can be used in laws of mass action to determine the concentrations of other minority species. If the concentration of [H PO ] and [PO - ] are desired, they can be determined by using the known values of [H PO - ], [HPO - ] and [H + ] in the law of mass action for K a and K a, respectively. Note that the selection of the chemical reaction is based on different criteria in phase versus phase. In phase, the selection of the reaction HPO - + H + H PO - is based on a process that actually occurs in the solution. Solutions in which HPO - and H + are both majority species are not stable, and their reaction will progress until one or both becomes a minority species. In phase, the choice of reaction is based on the practicality of determining the desired minority species concentration from the available concentrations. The available concentrations may form a daisy chain: when [H + ] is determined from the majority species concentrations, the determination of [H PO ] and [PO - ] relies on both the majority species concentrations and the previously determined minority species concentration, [H + ]. The criterion used to select the reaction changes substantially between phase and phase and the M&M strategy has the advantage of making this explicit for students. We next consider a complex situation in which the planning net of Figure may lead to different pathways to the same conclusions. Consider a mixture of multiple reagents such that, after dilution, the concentrations are: [H + ] = 0. M [OH - ] = 0. M [Ac - ] = 0. M [NH ] = 0. M An efficient application of phase reasoning is selection of the following series of strong (K>>) reactions: H + + OH - H O leading to: [H + ] = 0.0 M [OH - ] = 0 M H + + NH NH + leading to: [H + ] = 0 M [NH ] = 0. M [NH + ] = 0.0M A less efficient sequence is: H + + OH - H O leading to: [H + ] = 0.0 M [OH - ] = 0 M H + + Ac - HAc leading to: [H + ] = 0 M [Ac - ] = 0. M [HAc] = 0.0M HAc + NH NH + + Ac - leading to: [HAc] = 0 [NH ] = 0. M [NH + ] = 0.0M [Ac - ] = 0. M Both sequences, and any other sequence that satisfies the planning net of Figure, will lead to the same result for the majority species ([NH ] = 0. M; [NH + ] = 0.0M; [Ac - ] = 0. M). Alternative pathways may also arise in phase. One pathway to determine the minority species [H + ], [OH - ], and [Ac - ] is: Use K a of NH to determine [H + ] Use K b of NH to determine [OH - ] Use K a of HAc to determine [Ac - ] An alternative pathway is: Use K b of NH to determine [OH - ] Use K w to determine [H + ] Use K a of HAc to determine [Ac - ] Again, any pathway that satisfies the planning net of Figure will lead to valid results. Journal of Chemical Education // Page of

12 Page of Submitted to the Journal of Chemical Education ed-xx-xxxxxx Supporting qualitative reasoning The M&M strategy also supports qualitative reasoning by focusing attention on important distinctions between majority and minority species. Consider, for instance, the nature of buffer solutions and why control of ph through buffers is so important in chemical and biological systems. The Henderson-Hasselbalch (H-H) equation, A ph pka log, HA is a useful tool, provided its use is limited to phase reasoning. (Errors in use of the H- H equation arise when the concentrations used in the equation are not equilibrium concentrations. Within the M&M strategy, such errors can be avoided by restricting the use of the H-H equation to phase reasoning, by which point the equilibrium concentrations of the majority species are known.) In phase reasoning, the H-H equation operates in different directions depending on whether A - and HA are minority or majority species. When A - and HA are both majority species, the equation operates from right to left, such that the ratio [A - ]/[HA] establishes the ph of the solution. When A - and HA are minority species, the equation operates from left to right, such that the difference between the ph and the pk a sets the ratio [A - ]/[HA]. Buffer components are intentionally made to be majority species, such that the H-H equation operates from right to left and the ph is stabilized. The reason it is important to use a buffer to control the ph is that, for minority species such as proteins or reactants of an organic chemistry reaction, the H-H equation operates from left to right and the ph establishes the protonation state. This chain of qualitative reasoning is arguably the most important aspect of acid-base chemistry to include in an introductory course, since it captures both the reasons for controlling ph and the means to do so. As another example of the ability of the M&M strategy to support qualitative reasoning, consider the mechanisms through which buffers function. This relies on coupling of phase and phase reasoning. By phase reasoning, addition of acid or base to a buffer leads to strong reactions, A - + H + HA and HA + OH - A - that effectively absorb the added acid or base and keep both H + and OH - minority species. By phase reasoning, as long as both HA and A - are majority species with roughly equal concentrations, the ph will be stabilized to a value close to the pk a. This is a rather complex chain of reasoning that may be supported by drawing a clear distinction between the phase and phase components. CONCLUDING COMMENTS This paper presents two instructional suggestions for equilibrium instruction, along with evidence supporting these suggestions. The first suggestion is to provide explicit instruction on the progress of reaction coordinate. This is supported by evidence which suggests that the connection between the reaction rule and the effects of this rule on collection on molecules is not obvious for novices. The second suggestion is to use the M&M strategy to frame instruction. For systems involving a single reaction, there is classroom evidence that the M&M strategy is easier to learn than the mathematically equivalent small-x approximation. For systems involving multiple reactions, the evidence presented here relates to the ability of the M&M strategy to provide a single, consistent, Journal of Chemical Education // Page of

13 Page of 0 0 ed-xx-xxxxxx approach to the analysis of equilibrium systems. The evidence includes illustrations of the use of the M&M strategy on the problem types present in the current introductory course, and the use of the M&M strategy to frame understanding of key concepts in acidbase chemistry. The impact of the M&M strategy on student learning for systems involving multiple reactions is left for future work. ASSOCIATED CONTENT Supporting Information Use of the M&M strategy in acid-base and solubility chemistry is illustrated with a set of worked examples. The definition of strong reaction is refined from K>> to a criterion that works for dilute solutions. This material is available via the Internet at AUTHOR INFORMATION Corresponding Author * yaron@cmu.edu Notes Carnegie Mellon University Department of Chemistry 0 Fifth Avenue Pittsburgh, PA ACKNOWLEDGMENT The research reported here was supported by the Pittsburgh Science of Learning Center (National Science Foundation grant number SBE-0), the National Science Foundation (DUE-), and the Institute of Education Sciences, U.S. Department of Education, through Grant RA00 to WestEd. The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education. REFERENCES () Banerjee, A. C.; Power*, C. N. International Journal of Science Education,,. () Van Driel, J.; Verloop, N.; Vos, W. De Journal of research in Science Teaching. () Gorodetsky, M.; Gussarsky, E. European Journal of Science Education,,. () Harrison, A. G.; De Jong, O. Journal of Research in Science Teaching 0,,. () Wilson, A. H. Journal of Chemical Education,,. () Tyson, L.; Treagust, D. F.; Bucat, R. B. Journal of Chemical Education,,. Journal of Chemical Education // Page of

14 Page of Submitted to the Journal of Chemical Education 0 ed-xx-xxxxxx () Yaron, D.; Karabinos, M.; Lange, D.; Greeno, J. G.; Leinhardt, G. Science,,. () Davenport, J.; Yaron, D.; Klahr, D.; Koedinger, K. In International Conference of the Learning Sciences; B. C. Love; K. McRae; V. M. Sloutsky, Eds.; Cognitive Science Society: Austin, TX, 0. () Brown, T. E.; LeMay, H. E.; Bursten, B. E.; Murphy, C.; Woodward, P. Chemistry: The Central Science; th Ed.; Princie Hall,. () Hayes-Roth, B.; Hayes-Roth, F. Cognitive Science,,. () Sacerdoti, E. A structure for plans and behavior; Elsevier-North Holland,. Journal of Chemical Education // Page of

15 Page of If yes, repeat process Phase : Phase : Subgoal: Determine the identity and concentration of the majority species. Process: Identify strong (K>>) reaction between species with non-zero concentration (see Figure ). Carry out the limiting reagent computation arising from the thought experiment in which K is assumed to be infinite. Verification: Are there any strong (K>>) reactions between species with non-zero concentrations? If no Subgoal is met: The chemical species with non-zero concentrations are the majority species and their equilibrium concentrations are now known. If not, repeat process Subgoal: Identify and determine the concentration of the minority species. Process: Identify laws of mass action (K=Q) from which the minority species concentrations may be determined. This is done using currently known concentrations and stoichiometry constraints (e.g. [H + ] = [A - ] in a pure solution of HA). Verification: Are the minority concentrations of the relevant species known? If yes Subgoal is met: All desired concentrations have been determined. Test applicability of M&M strategy: Are the concentrations of all minority species less than % of the majority species? If yes The M&M strategy is applicable The M&M strategy is not applicable, and full algebraic approach must be used. If not