Data Linearization Activity for Undergraduate Analytical Chemistry

Activity pubs.acs.org/jchemeduc

Data Linearization Activity for Undergraduate Analytical Chemistry Lectures James K. Harper and Emily C. Heider* Department of Chemistry, University of Central Florida, 4111 Libra Drive, Orlando, Florida 32816, United States S Supporting Information *

ABSTRACT: Throughout the undergraduate curriculum, students utilize linearized forms of nonlinear equations from the Clausius−Clapeyron equation in general chemistry to the Michaelis−Menten equation in biochemistry. Presenting the linearized forms of equations as a fait accompli may be a lost opportunity to empower students with understanding the general process of linearization as an analytical tool. This work describes a series of activities that can be implemented and spaced throughout the analytical chemistry curriculum. The activities are low-cost and chemical-free, so they can be implemented in a lecture setting. Distributed practice allows students to attain familiarity with linearization, and the practical guided application in class provides real experience with the otherwise abstract mathematical idea. KEYWORDS: Upper-Division Undergraduate, Analytical Chemistry, Hands-On Learning/Manipulatives, Mathematics/Symbolic Mathematics

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INTRODUCTION Fitting linear data is a cornerstone of chemical analysis and is commonly used to extract constants and parameters from linear portions of data which may be nonlinear. The data are plotted so the model equation describing it is arranged in the slope− intercept form (i.e., y = mx + b). Thus, a least-squares fit to these data provides the slope and intercept, which are parameters of a given model equation. A well-known example is the linear Beer−Lambert law: a plot of absorbance versus molar concentration can yield the molar absorptivity parameter (obtained from the slope), providing an equation for determining concentration given a measured absorbance. The power and utility of fitting linear data can also be applied to nonlinear functions in chemistry, but to do so requires a mathematical model and/or algebraic rearrangement in order to obtain a linear relationship. Table 1 summarizes some linearized versions of chemical relationships that are employed throughout the undergraduate chemistry curriculum. Despite many published limitations of nonlinear data linearization,1−3 graphical analysis of linearized data remains a powerful tool. However, the general concept of linearization is often neglected from chemistry texts and can be difficult to fit into the curriculum. Depending on the nonlinear equation, revising it to resemble the linear slope−intercept form of an equation can present a challenge to students. We describe activities that were used in an analytical chemistry lecture as a platform for introducing the concept of linearization in general and the application of it in particular to Bragg’s law. They were implemented in an active-learning setting over three semesters © XXXX American Chemical Society and Division of Chemical Education, Inc.

at a large undergraduate university. These courses generally have an enrollment of 70−100 students and have been implemented by two different instructors. The activity can be completed either in small groups of students or with a class demonstration, followed by analysis of the class data in small groups. Many outstanding articles on educational approaches to teaching the Bragg equation have been published previously, ranging from simple demonstrations with ambient light,4 lasers,5 and paper cut-outs6 to detailed experiments with crystallography7,8 and spectroscopes.9 This activity does not replace thoseparticularly because it does not explicitly describe the application of Bragg’s law in crystallography. Rather, this is intended to demonstrate how linearization can be applied to glean data using linear least-squares fitting. The learning objectives of the activity are (1) To understand the purpose of linearizing data (2) To collect data that is graphically displayed in a linear way (3) To apply linearization methods to multiple chemical scenarios Received: September 7, 2016 Revised: February 27, 2017

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DOI: 10.1021/acs.jchemed.6b00687 J. Chem. Educ. XXXX, XXX, XXX−XXX

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Table 1. Linearized Equations for Graphical Analysis throughout Undergraduate Chemistry Curriculum course

concept

linearized equation

graph

Freshmen Chemistry

vaporization curves and Clausius−Clapeyron equation

ΔH vap ⎛ 1 ⎞ ⎜ ⎟ + ln β ln Pvap = − R ⎝T ⎠

ln Pvap vs 1

ΔHvap (slope) or Pvap at given T

Physical Chemistry

equilibrium constant and temperature with the van’t Hoff equation

ln Keq = −

ln Keq vs 1

ΔH° (from slope) and ΔS° (from intercept)

Biochemistry

enzyme reaction rates and Michaelis− Menten kinetics modeled with Lineweaver−Burke plots

K 1 1 1 = m + v vmax S vmax

1 v

Km (from slope), vmax (from intercept), and comparing competitive, noncompetitive, and uncompetitive inhibitors

Analytical Chemistry

fluorescence quenching and Stern−Volmer relationship

I 0f

If0

If

ΔH ° ΔS° + RT R

= 1 + kqτ0[Q ]

measured parameters

T

T

If

vs

1 S

vs

kqτ0

[Q]

(slope)

Table 2. Questions Used for Group Practice in Linearization number

part

practice question

1

a

2

b a

Snell’s law (which we will use later in this course) is n1 sin θ1 = n2 sin θ2. A student measures θ2 as a function of θ1. The value of n1 is known and constant. How would you plot the data so that you can extract the value of n2 from a least-squares analysis of a straight-line plot? What do you predict would be the value of the intercept from the plot in 1a? The decomposition of compound A is known to be first order with respect to rate. This means it follows the integrated rate law: [A] = [A]0 exp(−kt). If a student measured the concentration of [A] as a function of time, how would he/she plot the data so that he/she can extract the value of the rate constant, k, from a least-squares analysis of a straight-line plot? What information could be obtained from the y-intercept in 2a?

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b

INTRODUCING LINEARIZATION IN THE COURSE (30 MIN CLASS TIME REQUIRED) To introduce the concept of linearization, a simple scenario involving the familiar ideal gas law was first presented to the class, for which linearization provides a useful quantity. Students were asked the following questions: how can the gas constant, R, and the value for absolute zero be measured? The students were given a set of data for pressure (P) and volume (V) with constant moles (n) of gas at varying Celsius temperatures (T). A plot showing PV/n on the y-axis and temperature on the x-axis yields a straight linethe slope of which is the gas constant (R) and the x-intercept is the value for absolute zero. When asked to describe and plot the data using Kelvin temperature and determine the equation of the line in slope−intercept form, the students quickly recognize the ideal gas law. In our classrooms, the above information is presented as an in-class discussion, and plots of data are constructed in real time using a projector and a computer. For ease of use by others, a worksheet-style version of this information has been included in the Supporting Information. This simple exercise shows students the potential for how linearization can be used, but students still need to learn how to make a plan to linearize nonlinear equations. For practice, the students then work in groups to complete the problems in Table 2. After the students had approximately 10 min to work on these problems, the solutions were discussed with the class as a whole and the students informed that the activities in subsequent class periods would require knowledge of data linearization.

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Diffraction grating material (1000 lines/mm) was purchased from an online supplier10 and divided into smaller units for the groups. A stand and clamps were used to mount the laser pointers, and a picture frame with the glass removed was used to mount the grating. Activity Description

Activity instructions as they were provided to the students are provided in the supporting material. On the day of the activity, the students were organized into groups of 3−4 and were provided a set of instructions that provided guided inquiry into the form and variables of the Bragg equation (eq 1):

mλ = d(sin θ + sin ϕ)

(1)

Here, m is the diffraction order, λ is the wavelength of light, d is the distance between rulings in the grating, θ is the angle of incident light relative to the grating normal, and ϕ is the angle of exiting light relative to the grating normal. These variables are illustrated for a transmission grating in Figure 1. Students may recognize the simplified form of the equation mλ = 2d sin θ from general chemistry discussion of

LINEARIZATION ACTIVITY WITH THE BRAGG EQUATION (50 MIN CLASS PERIOD REQUIRED)

Materials and Supplies

Sets of three laser pointers were utilizedred, blue/violet, and green. Caution: Light from laser pointers should never be directed into the eyes. Stray reflections from pointers pose a hazard, and students must be instructed to exercise caution.

Figure 1. Illustration of a light impinging on a transmission grating at incident angle θ, with respect to the grating normal, and diffraction angle ϕ. Equation 1 shows how the angle ϕ depends on the distance between rulings of the grating and the wavelength of incident light. B

DOI: 10.1021/acs.jchemed.6b00687 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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Figure 2. (A) Diffracted light from laser pointers, showing zero-, first-, and second-order spots. The violet, green, and red laser pointers are arranged vertically stacked, clamped to a buret stand, which is why the spots are separated vertically in the image. The three spots that appear to the right of the zero-order spots are also first-order spots. (B) Illustration of the diffracted light showing the exiting angle (ϕ) relative to the grating normal. The angle ϕ can be determined from measuring “r” and “s” and applying the trigonometric relationship tan ϕ = s/r.

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ACTIVITY FOLLOW-UP (30 MIN CLASS TIME REQUIRED) Much later in the course, when the electrochemistry unit was presented, the students applied data linearization to another scenario with an activity requiring them to utilize the Cottrell equation. This equation described the current (i) measured with time (t) under potential step conditions, where F is Faraday’s constant, n is the moles of electrons, A is the area of the electrode, and D is the diffusion coefficient (eq 3):

crystallography, and the connection can be pointed out during the classroom discussion. Students were also provided a set of three laser pointers (red, green, and purple), a diffraction grating mounted on a stand, and rulers. Students constructed an experimental setup with the laser pointers normal to the grating and the grating parallel to the wall so that the distance between zero- and first-order beams could be measured (Figure 2) and the distance between the grating and the wall also measured. These two measurements can be combined to yield the exit angle (using the arctangent of the ratio). The different colored beams can be measured one at a time and shared between groups so fewer laser pointers are required. The students were advised to construct a plot of the data to determine the distance between rulings of the diffraction grating (or “d”). When plotting the sine of the exit angle (ϕ) on the y-axis and the wavelength of light on the x-axis, the slope provides the inverse of the distance between rulings of the diffraction grating. The rearranged Bragg equation (as plotted in the activity) is given by eq 2. (sin θ + sin ϕ) =

m λ d

⎛ nFA[reactant]bulk D ⎞ ⎜|i| = ⎟ π t ⎝ ⎠

(3)

The students were provided a set of data with current and time, and they linearized the equation by plotting current on the yaxis and t−1/2 on the x-axis. The slope can be used to determine the concentration of the reactant in bulk solution. At this point in the semester, the linearization does not seem to trouble the students, and many can apply the technique as a matter of routinethis task can be made more challenging if the students are not supplied with the labels of the axes indicating that the current should be plotted on the y-axis and the inverse square root of time on the x-axis. Complete instructions for this activity are provided in the Supporting Information.

(2)

Incident angles other than 0° may be used, but the process is greatly simplified if the incident beam is parallel to the grating normal (because sin(0) is 0). An incident angle of 0° can be verified by ensuring that both first-order spots are equidistant from the zero-order beam. The students plotted the data manually during the 50 min lecture period and then electronically as homework. They fit the data to the linear least-squares best fit equation and reported the distance in the following lecture period. Using a diffraction grating with 1000 lines/mm, student responses were averaged to yield a spacing of 1030 ± 90 nm (N = 37 trials). A simple variation on this experiment could be the use of a set of gratings with various rulings and a single laser pointer. For example, a set of three gratings (100, 300, and 600 lines/ mm) can be inexpensively acquired11 and a single laser pointer (e.g., red, which can be inexpensively purchased) used to measure the diffraction. In this scenario, plotting the sine of the angle and the inverse of the grating spacing yields a straight line, from which the wavelength of the laser can be acquired.

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ASSESSMENT OF LEARNING OUTCOMES The three learning objectives for these linearization exercises were for students to understand the purpose of linearizing data, to collect data that are graphically displayed in a linear way, and to apply linearization methods to multiple chemical scenarios. To determine if students could report the purpose of linearization, an extra credit assignment included the following question after completing the Bragg equation exercise: “It is sometimes advantageous to plot nonlinear functions in a way that yields linear data that can be fit to an equation. Why would this be done?” Responses that could reasonably be expected might include, for example, using a set of data to derive an empirical value or to graphically visualize relationships between variables. Of the 51 students that completed the question, 44% were able to give a complete relevant explanation of the purpose of linearization, 45% gave a partial explanation that indicated at least superficial understanding, and 11% gave either C

DOI: 10.1021/acs.jchemed.6b00687 J. Chem. Educ. XXXX, XXX, XXX−XXX

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Figure 3. Final exam questions used to evaluate students’ abilities to apply the concept of linearization to an unfamiliar scenario.

here should complement a well-rounded discussion of linearizing data and fitting data, which includes mention of the limitations in linearization. Limitations are particularly pronounced when the spread of the data is inadequate or errors can be magnified when linearizing an equation to produce a double-reciprocal15 or logarithmic plot.1 For these situations, nonlinear curve fitting is a superior approach. For the activities discussed herein for the analytical chemistry lecture setting, students learn and revisit the linearization concept over three separate lecture periods to exploit the retention advantages of spaced practice. Under these circumstances, the application of linearization extends beyond calibration curves with the goal of expanding the analytical tools students may apply in subsequent laboratory and lecture courses.

a wholly incorrect or no response. This indicates partial success in achieving the first stated learning objective. The second learning objective, to collect data that are graphically displayed in a linear way, was accomplished by the completion of the Bragg equation activity. Assessment of the third learning objective, to apply linearization methods to multiple chemical scenarios, would ideally be accomplished by completion of all the activities described hereinnamely, application to the ideal gas law, Snell’s law, kinetic rate laws, the Bragg equation, and the Cottrell equation. Exposing students to these scenarios does not necessarily ensure that they understand and can independently apply the concepts. To determine if students could successfully apply the concept of linearization to a new scenario, answers to two linearization questions from the final exam were surveyed, including both students who had completed the linearization activities and those that had not. The questions are shown in Figure 3. For the first question regarding activation energy, 30% more students answered correctly if they completed the linearization activities (N = 51 students) than if they did not (N = 36 students). For the second question regarding the pre-exponential factor, 70% more students answered correctly if they completed the linearization activities than if they did not. The average final exam scores for the students who completed the linearization activity was 64.2%, which is indistinguishable from the average final exam score (63.9%) of those that did not.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.6b00687.

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CONCLUSIONS The primary goal of this project was to develop students’ understanding of the purpose and ubiquitous application of linearizing nonlinear equations. It is worth noting that these activities do not address the equally valuable application of nonlinear curve fitting, which also merits a place within the analytical chemistry curriculum. Several suggestions on using nonlinear curve fitting in an instructional setting can be found in the educational literature including, for example, kinetics,12 biochemistry,13 and electrochemistry.14 The activities suggested

Complete instructions for the activities described in this paper (PDF, DOCX)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1 407 823 5950. Fax: +1 407 823 2252. ORCID

Emily C. Heider: 0000-0002-9728-6863 Notes

The authors declare no competing financial interest. D

DOI: 10.1021/acs.jchemed.6b00687 J. Chem. Educ. XXXX, XXX, XXX−XXX

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ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation under CHE-1455159. REFERENCES

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DOI: 10.1021/acs.jchemed.6b00687 J. Chem. Educ. XXXX, XXX, XXX−XXX