Technology Report Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
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Employing Spreadsheets for Applying Calculus in Upper-Level Chemistry Courses Paul D. Cooper* Department of Chemistry and Biochemistry, George Mason University, 4400 University Drive, Fairfax, Virginia 22030, United States
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S Supporting Information *
ABSTRACT: This article describes the use of a spreadsheet to reinforce basic calculus that is expected of all university-level chemistry majors. The example provided shows a calculation using Excel to estimate, using a Riemann summation, the radiant exitance of a hot object using Planck’s Law of Blackbody Radiation. The approach reinforces the elementary calculus topic of integrating an area under a curve in an applied chemistry setting and, additionally, requires students to understand how units propagate through integrals in order to calculate correct numerical values. This manual calculation in Excel is pedagogically more satisfying than using black-box type software to calculate areas under curves, and is widely available to almost all students through the use of Excel (or similar spreadsheets software packages). KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Computer-Based Learning, Mathematics/Symbolic Mathematics
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BACKGROUND Given the ever-increasing use of mobile personal electronic devices (laptops/tablets/phones) by students, utilizing technology to enhance the education of students in chemistry has become ever more present across almost every institution. There are already a large number of published examples advocating the use of spreadsheet software dating back to the first introduction of personal computers at all levels of chemistry curricula. Such examples cover data analysis,1,2 simulations,3,4 and quantum chemical applications,5,6 all of which are applicable to physical chemistry, the principle subdiscipline within chemistry that requires a strong background in calculus. However, none directly address the direct student application of basic calculus to chemistry. It has been this author’s experience that although students in chemistry degree programs are required to take calculus, they are not always well prepared to apply calculus to chemistry problems. A survey of the literature reveals, in general, an absence of research in undergraduate student understanding of calculus as it applies to chemistry. A commentary7 from this Journal from nearly 30 years ago calls for the better incorporation of application-based calculus, yet there are many examples of approaches to teaching chemistry with the intention of minimizing student exposure to calculus.8−11 While the latter certainly has some benefits in an introductory chemistry course, it perhaps does not serve the students who require the academic rigor to be successful in physical chemistry. In addition, as the focus of this article, it has been observed that students are also unaware of how to use spreadsheets, in which today’s students are generally reasonably proficient in © XXXX American Chemical Society and Division of Chemical Education, Inc.
basic operations, to estimate areas under curves for the purposes of evaluating an integral of a mathematical function, a basic calculus concept. Additionally, many students are not comfortable with unit conversions involved with such calculations. While there are a number of black-box type software packages available that perform calculations of areas under a curve, many students do not have access to such software through personal or even school/university software licenses. Most students, however, do have access to spreadsheet software, such as Excel. While this article describes the use of Excel to perform numerical integrations of Planck’s Law, the simple technique is easily transferable to other mathematical functions they may encounter. The process reinforces their understanding of basic calculus principles, and more importantly a real application of calculus in a chemistry setting. Students are usually exposed to the use of Excel in the construction of graphs and simple regression of linear data throughout their undergraduate degree.1 Spreadsheet-based multiple linear regression can also be used for spectroscopic analysis12 in undergraduate physical chemistry. SDAT, an Excel add-on, is also available13 and is a comprehensive suite of tools that includes an integration function that could be used for this exercise. However, over the past four years that I have implemented the present activity, it has become evident that many students struggle to independently determine how to manually integrate an area under the curve of a graph using Received: March 14, 2018 Revised: July 17, 2018
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DOI: 10.1021/acs.jchemed.8b00193 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Technology Report
Excel, despite acknowledging after the exercise they possessed all of the Excel and calculus knowledge to do so. Although there is value in having software packages calculate areas under curves, it does not address the bigger issue of how students learn how to apply calculus to chemistry.
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PLANCK’S LAW OF BLACKBODY RADIATION ACTIVITY The activity can be presented to students very early in a traditional quantum/spectroscopy physical chemistry lecture as an in-class exercise but also naturally works well in flipped classrooms. Instructions are given to students to use Planck’s Law of Blackbody Radiation (eq 1) to calculate both the spectral exitance Mλ(λ, T) across a range of wavelengths, and radiant exitance, Me(T), from a blackbody object. M e (T ) =
∫ Mλ(λ , T ) dλ = ∫
2πhc 2 λ 5(ehc / λkT − 1)
dλ (1) Figure 1. Method of a Riemann summation to integrate the area under the blackbody emission distribution curve.
In my experience, as I have led this discussion in my class, most students do not think of a spreadsheet solution to the integral, with most thinking an algebraic solution to the problem is the only path to take. When the students learn that Excel is to be used to solve the problem, most students are not aware of how this could be done. Comments from students echo the same question of “where is the integrate function?” in the hope for some quick Excel function that will provide an answer. Leading the discussion, I will then draw on the whiteboard something similar to blackbody emission distribution and ask the students how I might manually determine the area under the curve. This then naturally leads to the process of formulating a Riemann summation14 to approximate the integral under the curve and then in the ensuing discussion students determine how to go about performing this summation in Excel.
the SI unit calculations, units of micrometers are also needed for creating the graph. Not only is the use of micrometers here more practical, but also it introduces an important unit conversion that students will need to think about. Students may also want to include a column of nanometers for use of a unit of wavelength that they should be very familiar with. In this example, wavelength values start at 0.200 μm (in the UV) and extend out to 40.0 μm (in the far-IR) in intervals (Δλ) of 0.0200 μm. Given the negligible emission beyond these limits at a temperature of 700 K, these wavelength limits are acceptable for this calculation, but may need some adjusting if different temperatures are chosen. This yields approximately 2000 data points, which is sufficient for the integral calculation but easily manageable with a spreadsheet. The spectral exitance in SI units (W m−3) at each wavelength at 700 K is shown in column C. Given the typical wavelength of emitted radiation of hot blackbody objects, it is usually more convenient to display on plots wavelengths in micrometers rather than meters. However, in the case of the Riemann sum in eq 2, the choice of wavelength interval unit is important as it sets the numerical value of Δλ. Column D is subsequently calculated through conversion of m−1 to μm−1, necessary for the plot of the spectral exitance (W m−2 μm−1) versus wavelength, λ (μm). Students will often overlook this step because dimensional analysis in integrations is not something commonly (if at all) addressed in traditional calculus courses, but yet are critically important in application. Figure 2 shows this plot of Column D vs Column A and shows the classic Planck distribution that students see in their textbooks, and is an additional product of this exercise. Finally, Column E, the radiant exitance at each wavelength interval is calculated by multiplying the value in Column D by the interval width (Δλ) of 0.0200 μm. The process is complete when the values in Column E are summed to calculate the total radiant exitance across all wavelengths for a 700 K blackbody.
Spreadsheet Calculations
Students are asked to construct a graph displaying the spectral exitance from a blackbody as a function of wavelength (in μm) for a blackbody at a given temperature (e.g., 700 K). Then they are asked to determine the radiant exitance (in W/m2) of the object. Once the process of a Riemann summation has been determined, students will then have to consider what wavelength range to plot, what units of wavelength to use, and how many subdivisions or data points to calculate in order to attain a reasonably accurate numerical value of the integral. In order to get students thinking about units, I ask them to perform a dimensional analysis of eq 1, with many students not immediately recognizing the dλ term as having units itself. 2
M e (T ) =
∫ Mλ(λ , T ) dλ ≈ ∑ λ 5(ehc2/πλhckT − 1) Δλ i
i
i
(2)
Equation 2 shows the Riemann summation approach to this estimate of the integral that students follow where Mλ is calculated at each wavelength λi. Figure 1 illustrates this process of calculating the Riemann summation that students will perform in Excel. The instructor can at this point let students work on the calculation independently or with some guidance to ensure students stay on the right path during this initial phase of the calculation. Table 1 shows the first few rows of data that students typically calculate. Students will be required to input two columns (A and B in Table 1) of wavelength. While units of meters are required for
Comparing Calculations
The value of radiant exitance calculated through the spreadsheet-based Riemann summation is 1.35 × 104 W/m2. In comparison, the value obtained directly from the Stefan− Boltzmann equation (eq 3), which students should also B
DOI: 10.1021/acs.jchemed.8b00193 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Technology Report
Table 1. Spreadsheet Example for Blackbody Radiation Calculation at 700 K A, Wavelength/μm
C, Spectral Exitance/W m−3
B, Wavelength/m −7
2.00 × 10 2.20 × 10−7
0.200 0.220
D, Spectral Exitance/W m−2 μm−1
−27
−33
2.73 × 10 1.93 × 10−23
2.73 × 10 1.93 × 10−29
E, Radiant Exitance/W m−2 5.46 × 10−35 3.87 × 10−31
In addition, a calculation of the wavelength of maximum emission (λmax) can be performed using the spreadsheet calculations and compared to Wien’s Law. λmax =
b T
(5)
Using Wien’s constant, b = 2.898 × 10−3 m K, a peak emission wavelength at 700 K of 4124 nm is calculated. To use the spreadsheet, students can calculate the difference between successive calculations of spectral exitance as a proxy for the derivative of the curve. Students then determine λmax by determining where the difference (derivative) changes from a positive to a negative value (i.e., where the slope is zero as in eq 6). d(Mλ) =0 dλ
In the Supporting Information this calculation is included and with a spreadsheet calculated value of ∼4130 nm, in excellent agreement with Wien’s Law.
Figure 2. Spectral exitance (W m−2 μm−1) of a 700 K blackbody as a function of wavelength (λ).
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CONCLUSION The reinforcement of the basic principles of calculus, as it applies to chemistry, is an area of importance for all chemistry students, but especially students in physical chemistry. Using Excel to facilitate an investigation of these calculus principles is a hands-on way for students to gain confidence in both Excel and the application of calculus to chemistry. Planck’s Law of Blackbody Radiation is an appropriate example for use in developing such a hands-on experience for students, as it requires students not only to perform spreadsheet-based calculus through an estimate of an integral under a curve, but also to consider the result of calculus on dimensional units within an equation. Instructors of physical chemistry could also potentially adapt this exercise to other topics of interest, such as integrating radial wave functions or Maxwell−Boltzmann distributions. These are valuable tools that students should acquire during the course of an undergraduate chemistry degree.
perform to verify their result, is 1.36 × 10 W/m , where the Stefan−Boltzmann constant, σ, is 5.670 × 10−8 W m−2 K−4. 4
M e (T ) = σT 4
2
(3)
The derivation of this equation from eq 1 is shown in the Supporting Information. The excellent agreement (∼0.5%) between the Riemann summation value of radiant exitance and the directly calculated Stefan−Boltzmann value, confirms to the students the validity of their spreadsheet calculation. Furthermore, students might be directed to refine their calculation using smaller wavelength intervals to improve the agreement between the two values. The process of spreadsheet-based integration is proven to them to be a relatively simple task based upon their already existing spreadsheet and calculus knowledge. As an additional task, students can then use their spreadsheet to calculate the fraction of emitted electromagnetic radiation that is produced in the visible region of the spectrum for an incandescent light bulb. Assuming blackbody emission and using a temperature of 2400 K for a standard incandescent lamp, students can determine the percentage, ε, of the total electromagnetic radiation emitted by the light bulb over the 400−750 nm range through eq 4, which is easily calculated from the existing spreadsheet, where λj are the wavelength intervals between the 0.400 and 0.750 μm summation limits. 0.750μm
∑0.400μm ε=
40μm
∑0.2μm
2πhc 2 λj5(ehc / λjkT 2πhc
− 1)
λi5(e hc / λikT − 1)
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Δλ
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.8b00193.
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Δλ
2
(6)
× 100%
Handout for students and Stefan−Boltzmann derivation (PDF) Blackbody spreadsheet (XLSX)
AUTHOR INFORMATION
Corresponding Author
(4)
*E-mail:
[email protected].
Alternatively, an examination of thermal emission from the Sun (∼5780 K) and the Earth (∼288 K) can be explored to drive discussion of Earth’s energy budget or global warming.
ORCID
Paul D. Cooper: 0000-0002-5573-3549 C
DOI: 10.1021/acs.jchemed.8b00193 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Technology Report
Notes
The author declares no competing financial interest.
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REFERENCES
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DOI: 10.1021/acs.jchemed.8b00193 J. Chem. Educ. XXXX, XXX, XXX−XXX
