Exploring Interactive and Dynamic Simulations Using a Computer

Jul 23, 2014 - The article describes the use of Mathematica, a computer algebra system (CAS), in a high school chemistry course. Mathematica was used ...
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Exploring Interactive and Dynamic Simulations Using a Computer Algebra System in an Advanced Placement Chemistry Course Paul S. Matsumoto* Galileo Academy of Science and Technology, San Francisco, California 94109, United States S Supporting Information *

ABSTRACT: The article describes the use of Mathematica, a computer algebra system (CAS), in a high school chemistry course. Mathematica was used to generate a graph, where a slider controls the value of parameter(s) in the equation; thus, students can visualize the effect of the parameter(s) on the behavior of the system. Also, Mathematica can show the steps in the solution to an algebraic expression (e.g., solving equilibrium problems), which can benefit students with weak algebra skills. The CAS was used to create interactive and dynamic simulations in support of an inquiry-based advanced placement chemistry Beer’s law, kinetics, and solubility of ionic compounds laboratory activities. A primer in the use of some Mathematica commands is included. This contribution is part of a special issue on teaching introductory chemistry in the context of the advanced placement (AP) chemistry course redesign. KEYWORDS: High School/Introductory Chemistry, First-Year Undergraduate/General, Curriculum, Interdisciplinary/Multidisciplinary, Computer-Based Learning athematics is used in high school chemistry and first-year college chemistry courses, where a weak mathematics background often impedes success in chemistry.1 The use of mathematics to describe a chemical system’s behavior is abstract, which can impede student understanding of chemistry. Although the use of a graph (of an equation) describing the chemical system’s behavior may improve student understanding, a deeper understanding may be achieved by using a dynamic graph, where one explores the effect of changing the value of the parameter(s) in the equation describing the system’s behavior; such an exploration can be facilitated by a computer algebra system (CAS2). Another potential student impediment to succeed in high school and first-year college chemistry is the tedium in solving algebraic equations, which can be remedied by a CAS by freeing up time for students to explore and deepen their understanding of chemistry. Similar arguments for a CAS were made in learning mathematics3 and physics.4 Mathematica5 is the CAS used in my class. There are alternative proprietary and free versions of CAS software.6 Other chemists7 developed f ree interactive and dynamic chemistry content using Mathematica, which requires the f ree cdf player plugin8 installed into a web browser, thereby minimizing the cost to implement Mathematica in a high school chemistry course. The limitations of using the free cdf player plugin are that (i) input to the simulation must use a slider, (ii) there is no access to the Mathematica “show your work” feature (see Supporting Information for examples of this feature),9 and (iii) students would not have an opportunity to develop their own Mathematica simulations. Although there are articles in the Journal of Chemical Education advocating for the use of CAS in college chemistry courses10−12 and there is a column entitled JCE SymMath: Symbolic Mathematics in Chemistry, which provides CAS-based content for college chemistry courses,13−18 there are no articles in the use of CAS in high school chemistry courses.

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© 2014 American Chemical Society and Division of Chemical Education, Inc.

The common core high school mathematics standards19 are adopted by most states in the United States and it mentions to use a CAS, so high school mathematics teachers may be familiar with using a CAS in the classroom. Such a situation is an opportunity for collaboration among high school chemistry and mathematics teachers, which reflects the interdisciplinary nature of science.20 The redesigned advanced placement (AP) chemistry course began in the fall 2013 and seeks to develop a deeper understanding of chemistry in high school students.21 The inquiry nature of the laboratory portion of the course22 and the various science practices21 supports this goal. For example, science practices 2, 4, and 5 state that students can (i) use mathematics appropriately, (ii) plan and implement data collection strategies, and (iii) perform data analysis, respectively. The science practices are not restricted to a laboratory setting, for example, science practice 2, the use of mathematics, can be developed in students in the nonlaboratory portion of the course and a CAS can support this goal. I used Mathematica to make an answer key for chemistry tests in equilibrium, one of the most computationally extensive topics in the course because it is much more efficient than doing it by hand. A few years ago, when I taught a second semester first-year college chemistry course at a local community college during the summer, a handful of students requested to use Mathematica during their exam. Even a few of my AP chemistry students preferred to use Mathematica (rather than their calculator) during a test on equilibrium. As such, it seems that using a CAS to do homework problems can free up time for students to explore and deepen their understanding of chemistry. Special Issue: Advanced Placement (AP) Chemistry Published: July 23, 2014 1326

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Figure 1. Screenshot of a CAS-based simulation of the range and uncertainty in [CV+] as a function of the wavelength with respect to the dye’s simulated absorbance spectrum. The range is defined as the [CV+], where the absorbance is less than 1.0 (due to the lower reliability of absorbance values greater than 1.0) and the uncertainty is the difference in [CV+] due to an uncertainty in the absorbance of 0.1. The reason the horizontal axis (in the lower plots) has [CV+] rather than [Cu2+] is that initially, the simulation was used in a kinetics lab activity but was later adapted for an earlier Beer’s law lab.

appropriate wavelength in the data analysis for the laboratory report. The CAS-based simulation in Figure 1 is a guided-inquiry activity for students to discover the appropriate wavelength to use in a Beer’s law laboratory activity. This simulation illustrates the trade-off between maximizing the range while minimizing the uncertainty in the absorbance measurement. As such, the ideal wavelength would be to minimize the uncertainty, while maintaining an appropriate range in the measurement, which is the maximum anticipated concentration of the chemical during the experiment. In this inquiry laboratory activity, students use a colorimeter to measure the absorption of different concentrations of aqueous solutions of Cu(NO3)2 at different wavelengths, then in their laboratory report, students decide upon the appropriate wavelength of light to use in their analysis. The simulation’s purpose was to support students’ exploration to discover the basis of the selection of the appropriate wavelength in the Beer’s law lab activity. Although a more efficient use of the simulation would be for only the teacher to use it during the prelab session in the Beer’s law lab, providing students with an opportunity to work with the simulation without a teacher’s guidance can develop students’ inquiry skills and their critical thinking skills, a major goal of the revised AP chemistry course. In support of the proposition that students learned this concept, most students were able to answer a subsequent test question on this concept. Another lab utilizing the same concepts, but with different chemicals, would be the bleaching of a food dye,24

The remainder of this article will be divided into two sections; the first section will be a description of using a CAS in my high school chemistry classroom and the second section will be a primer on some Mathematica commands.

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EXAMPLES OF THE USE OF CAS IN A HIGH SCHOOL CHEMISTRY CLASSROOM In this section, there will be six examples of using a CAS in a high school chemistry classroom: (i) Beer’s law lab, (ii) kinetics lab, (iii) equilibrium constant lab, (iv) ionic solubility lab, (v) simple kinetics simulation, and (vi) class lecture. The Mathematica code for the simulations in Figures 1−5 are available online,23 which requires either the cdf plugin8 or Mathematica. Beer’s Law Lab

One of the laboratory activities in the AP chemistry lab manual22 is a spectrophotometric assay of copper ions based on Beer’s law (investigation 2) absorbance = abc

where c = concentration, b = path length, and a = absorptivity constant. The value of the absorptivity constant is a function of the wavelength of light absorbed by the chemical; the absorption spectrum (in Figure 1) illustrates this relationship. The calibration curve is described by Beer’s law and its behavior depends on the wavelength of the absorbance measurement. As such, one aspect of this laboratory activity is to select the 1327

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Figure 2. Screenshot of a simulation of the [CV+] and [NaOH] as a function of time, rate constant, [NaOH], and the order of the reaction with respect to crystal violet (graph, where α = 2 is not shown). In case 1, [NaOH] = [CV+] and the concentration of hydroxide is not a constant, so this simulated experimental condition does not satisfy the condition for the pseudorate law, but in case 2, [NaOH] is 10 times [CV+] and the concentration of hydroxide is approximately a constant, so this simulated experimental condition does satisfy the condition for the pseudorate law.

replace the original rate constant, k, by a new constant, k′. Once this condition is met, the exponent in the rate law associated with crystal violet can be determinedassuming the exponent is zero, one, or twoby using graphical analysis (i.e., [CV+] versus time; ln [CV+] versus time, or 1/[CV+] versus time) to determine the value of the exponent (and the value of the pseudo-order rate constant). The simulation in Figure 2 was used in the kinetics prelab session by the teacher to illustrate the experimental condition, whereby the pseudorate law approximation would be valid. Alternatively, this simulation can be used in a prelab assignment for students to discover the appropriate experimental condition, [NaOH] ≫ [dye], whereby the pseudorate law approximation would be valid. In support of the proposition that students learned this concept, most students were able to answer a subsequent test question on this concept.

where the preceding simulation can be used in discovering the basis of selecting the appropriate wavelength. In a subsequent kinetics laboratory activity using a colorimeter, the teacher uses this simulation in the prelab session of the kinetics lab to reinforce the basis of selecting the appropriate wavelength in the upcoming lab activity. Kinetics Lab

Another of the laboratory activities in the AP chemistry lab manual22 is to determine the rate law for the reaction of crystal violet, CV, and sodium hydroxide (investigation 11) CV + + OH− → crystal violet−OH

To simplify the rate law of the reaction rate = k[CV +]α [OH−]β

Equilibrium Constant Lab

to the pseudorate law

The purpose of the lab25 is to determine the value of the equilibrium constant for the reaction: Fe3+ + SCN− ⇌ FeNCS2+, where a colorimeter was used to determine the concentration of FeNCS2+. Although this laboratory activity is not a part of my AP chemistry course, it is done in my honors chemistry course and it illustrates the use of a CAS-based simulation (shown in Figure 3)

rate = k′[CV +]α where the pseudo-order rate constant, k′ = k[OH−]β, is valid occurs at [OH−] ≫ [CV+] because at such a condition, the concentration of hydroxide is (approximately) a constant, so the product, k[OH−]β, is (approximately) a constant; thus, one can 1328

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(ii) develop student’s critical thinking skills, for example, the “synthesis/application” level in Bloom’s taxonomy.26 Though a flowchart supports writing computer code, from a pedagogical perspective, a flowchart can illustrate (or model) the strategy in solving a problem, a part of science practice 4 and 5to plan and implement data collection strategies and data analysis, respectively. Simply telling students to “apply the solubility rules” is insufficient in developing science practices 4 and 5 in students. The flowchart is a visual representation that summarizes and describes the algorithm that illustrates science practices 4 and 5; the CAS was used to implement this algorithm. The flowchart’s content and the CAS program based on this algorithm was used in a prelab session by the teacher for a laboratory activity to identify a group of unknown ionic compounds (from a list of possible ionic compounds) based on the results of mixing the unknown ionic compound with various known ionic compounds. The “success” of this presentation is supported by students using a flowchart to answer a subsequent exam question about identifying an unknown ionic compound based on its solubility rules. The prelab assignment for the ionic solubility lab asks students to generate a flowchart to describe how to identify an unknown ionic compound based on observations of mixing it with various known ionic compounds, but it does not provide an example of such a flowchart. Figure 4 was shown in the prelab session af ter students had an initial attempt to develop a flowchart for the lab activity and the CAS-based simulation was demonstrated in the prelab session to illustrate/simulate science practices 4 and 5. This simulation is an example of a simple expert system.27

Figure 3. Simulation for the reaction: A + B ⇌ AB, where the concentration of the product, AB, is a function of the equilibrium constant and the concentration of the excess reactant; the concentration of the limiting reactant is 1.

to rationalize the experimental protocol to generate the calibration curve, where the assumption was made that equilibrium [FeNCS2+] = initial [SCN−], which was based on the experimental condition that [Fe3+] ≫ [SCN−]. The equation in this simulation was based on using an ICE table to describe the system and Mathematica’s Solve command was used to get the solution (not shown).

Classroom Presentation

The use of Mathematica in the classroom could use any of the preceding stand-alone content or it can be integrated into a slide presentation28 or as a lecture notebook.29 There are many screen casts describing the use of Mathematica in a classroom setting.30 Currently, Mathematica is used primarily to support the laboratory portion of my course, not in the lecture portion of the course. Nonetheless, Mathematica can be useful in the lecture portion of a chemistry class. For example, in the kinetics unit, a teacher might describe a simple first-order reaction

Ionic Solubility Lab

A dichotomous key is used to identify minerals or organisms in an earth science or biology course, respectively. A similar system was developed to identify an unknown ionic compound based on its solubility rules for a laboratory activity in my AP chemistry course. The flowchart in Figure 4 describes the program’s algorithm, which (i) provide a scaf fold to apply the solubility rules of ionic compounds to identify an unknown ionic compound and

Figure 4. Flowchart describing the algorithm to identify an unknown ionic compound based on the solubility rules of an ionic compound and the results of adding a known ionic compound to the unknown sample. The high-lighted ionic compounds are the potential unknown ionic compounds that students have to identify in a laboratory activity. 1329

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A→B

simple to use, will be described in this article. Given the time constraints33 for students to learn the content in a high school chemistry or first-year college chemistry course, it is my opinion that it is not a worthwhile expenditure of time to teach students to develop programs using a CAS except the NSolve command due to its simple syntax and potential to reduce the tedium to solve algebraic equations.

which is described as an exponential decay (see Figure 8), where the concentration of the reactant approaches zero, which is incorrect due to the simplification in the analysis. Mathematica is useful to more accurately describe a system,4 which in this case is to describe the system as

A⇌B

NSolve

which does complicate the analysis.31 That is, it is more accurate but more complicated to describe the relationship of the concentration of a reactant as a function of time when the backward reaction is taken into account in the analysis of the system. The result of such an analysis was used to produce the simulation in Figure 5.

The NSolve command

NSolve[lhs rhs,var] is used to “solve” (or “isolate the variable” in) an algebraic expression, which is found in the classroom assistant palette, where lhs, rhs, and var are left-hand side of the equation, righthand side of the equation, and the variable in the equation, respectively. The NSolve command produces a numeric solution, whereas the Solve command produces a more exact solution, for example, NSolve[3x == 1, x] = 0.333333, whereas Solve[3x == 1, x] = 1/3. After entering the values for the command, press the “enter” key (on the number pad or shift + enter on the keyboard). To access Mathematica’s feature to “showing the steps” to solve a problem,9 move the cursor to the “+” sign, then select the “Wolfram |Alpha Query” option (or type “==”), enter the NSolve command, fill-in the prompts, then press “enter”. To view the steps to solve the problem, select “Step-by-step solution”. This feature requires that your computer is connected to the Internet36 and you are using Mathematica version 8 or later. Plot

The Plot command is used to generate a graph. Using the classroom assistant palette, the syntax for the (i) Plot command Plot[function,{var,min,max}]

(ii) Sin command,

Figure 5. Screenshot of a simulation showing the concentration of reactant (blue) and product (purple) in the reaction: A ⇌ B as a function of time and the value of the forward and backward rate constants.

Sin[expr]

and (iii) ex command, e expr

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An example of a lecture notebook presentation was used in the nuclear chemistry unit in my honors chemistry course (the redesigned AP chemistry course does not have a nuclear chemistry unit). The presentation used Mathematica’s feature to open/close sections of the notebook during the presentation. Figure 6 shows the opening screen shot for a presentation on nuclear chemistry.

can be used to quickly generate a plot of these functions by entering the requested information in the template of the various commands. In the Plot command, (i) f unction refers the function of the graph to be generated, (ii) var refers to the independent variable in the function, and (iii) min and max refer to the minimum and maximum value of var, respectively. In the Sin and ex commands, expr refers to the expression to be evaluated by the Sin and ex functions. Using the classroom assistant palette, the AxesLabel option

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MATHEMATICA COMMANDS−A PRIMER As a CAS is a programming environment, a potential difficulty for the novice is the time33 needed to learn its commands and syntax. To lower the (teacher’s) learning curve, Mathematica’s classroom assistant palette34 generates a template that contain the syntax to use various commands so that the user does not have to remember the syntax of the command, thereby increasing the ease in using the CAS software. The software vendor’s Web site35 provides additional support in the form of screencasts and textbased tutorials. The purpose of this section is to provide a very limited introduction in the use of Mathematica for readers unfamiliar with this CAS, which will provide enough of a background to begin exploring its use in the classroom setting. In this section of the article, only three Mathematica commands, NSolve, Plot, and Manipulate, which are relatively

AxesLabel → {x label,y label}

generates axis titles on the plot and x label and y label refer to the text to be associated with the horizontal and vertical axes, respectively. Using the classroom assistant palate, sequentially select the Plot command, then the Sin (or ex) command, then the AxesLabel option. After entering the values for the command, press “enter”. The result is shown in Figure 7, which shows the graph of the sin and exponential functions, as well as, the Mathematica commands to generate these plots. Manipulate

The Manipulate command can be used to provide interactivity and dynamic content in conjunction with the Plot command. The 1330

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Figure 6. Screen shot of a lecture notebook on radioactive decay and radioactive dating unit, which contains some interactive content. Various sections or subsections can be opened or closed by clicking the cell bracket symbol on the left with a computer mouse.

continuous line of code, to increase its readability, each section of the options in the command will be shown on a new line. First, using the classroom assistant palette, sequentially generate the template with the syntax for the Plot, PlotRange, and AxesLabel options, which will generate the following code Plot[function,{var,min,max}, PlotRange→{y min,y max}, AxesLabel→{x label, y label}] In the PlotRange option, y min and y max refers to minimum and maximum value on the vertical axes, respectively. Second, using the classroom assistant palette, generate the Manipulate command, Manipulate[expr, control]

where expr and control is the expression and the parameter(s) to be manipulated in the simulation, respectively. Third, select, copy, then paste the content of the preceding Plot command into the “expr” portion of the Manipulate command, Manipulate[Plot[function,{var,min,max}, AxesLabel→{x label,y label}, PlotRange→{y min,y max}], control] Fourth, using the classroom assistant palette, generate the template for the control element to be manipulated in the simulation {{var, init, label}, min , max}

where the var, init, label, min, and max refer to the parameter(s) in the function (in the preceding Plot command), the initial value of the parameter, the name of the parameter (which will be shown near the scroll bar controlling this parameter), the minimum and maximum range of the value of the parameter, respectively. Move the cursor to between the control box and the comma associated with the preceding entry, generate a second template for the control element, delete the control box, then add a comma after the bracket associated with the Plot command, Manipulate[Plot[function,{var,min,max},

Figure 7. Illustration of using the Plot command to generate a graph of the sin function and exponential decay are shown in the top and bottom of the figure, respectively. Notice that text within the double-quote marks (bottom) in the AxesLabel option are printed exactly on the axes, whereas without the double-quote (top), the content is printed in alphabetical order.

following example will describe the sequential steps to generate an interactive and dynamic content based on the exponential decay function. While the input into the command will be a 1331

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PlotRange→{y min,y max}, AxesLabel→{x label,y label}], {{var,init,label},min,max}, {{var,init,label},min,max}] Finally, fill-in the requested information, then press the “enter” key; the results are shown in Figure 8, which includes the Mathematica commands to generate this simulation.

Article

ASSOCIATED CONTENT

S Supporting Information *

The content in this section includes various technical/legal issues in using the software and a series of screen shots to illustrate the use of Mathematica’s “show your work” feature. This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The reviewers’ comments significantly improved the quality of the article. REFERENCES

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Figure 8. Screenshot of a simulation of radioactive decay using the Manipulate command to generate an interactive and dynamic content to explore the behavior of a reaction described by first-order kinetics as a function of the initial concentration of the reactant and the value of the rate constant.

The result shown in Figure 8 generates an interactive and dynamic graph to explore the effect of the values of A, initial concentration, and k, rate constant, in the exponential decay function, which describes a reaction following first-order kinetics. Like the Plot command, the Manipulate command is evoked by pressing the “enter” key. The values of A and k are adjustable using the slider and its value can be seen by pressing the “+” sign next to the slider, where students would be able to observe how the graph changes as a function of the value of A and k, which can support students understanding the effect of the various parameters in describing the behavior of the chemical system. Pressing the “+” sign next to the slider exposes additional features to control the dynamic behavior of the simulation (shown below the rate constant slider in Figure 8). Pressing the solid triangle starts an animation, where the parameter changes its value and the consequence are shown in the graph. The arrow on the far right controls the direction of the preceding animationthe value of the parameter under study would repeatedly increase, decrease, or oscillate between an increase and decrease. The upward or downward arrowheads control the speed of the animation. The + and − signs allows the user to incrementally change the value of the parameter under study. 1332

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