In the Classroom
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Problem Solving in Physical Chemistry with the TI-89 Calculator Warren J. C. Menezes Department of Chemistry, Northeastern Illinios University, Chicago, IL 60625;
[email protected] Physical chemistry, one of the most challenging undergraduate courses in the field, requires more advanced math skills than either general or organic chemistry. Recent research indicates the results of the math diagnostic test significantly correlates with students’ grade in physical chemistry (1). Furthermore, within the math diagnostic test the item that most significantly reflected students’ success in physical chemistry was their ability to solve word problems. In addition, the GALT (Group Assessment of Logical Thinking) test has shown that students’ logical-thinking skills are directly related to their performance in physical chemistry (1). Logical-thinking skills are necessary for solving word problems, and enable students to connect their knowledge of math to solving the real world problems in physical chemistry (1). The primary challenge our students face is two-fold: not remembering the details of the math skills that they have acquired, and not having sufficient experience applying their math skills to solving complex science problems. My physical chemistry exams include both conceptual questions and numerical problems. Thus, to succeed students need decent logical-thinking skills and the required level of math. To maximize student success in physical chemistry I spend a large portion of my time fine-tuning their logical-thinking skills. However, mathematics is necessary to quantify science. Due to time constraints placed on the in-class exams, our students have difficulty balancing logical-thinking and math skills. I investigated various options, such as computer-based software (MathCAD, Mathematica, Maple, etc.) and advanced graphing and programmable calculators, which would help alleviate this problem. Several articles have appeared in this Journal discussing the uses of calculators for solving problems in general chemistry (2–4), and fewer articles discussing their uses in physical chemistry (5). In this paper I shall describe the use of the Texas Instruments TI-89 calculator, which I have used extensively to solve a variety of physical chemistry problems. I have selected physical chemistry problems with varying degrees of difficulty from several areas, such as thermodynamics, quantum mechanics, and kinetics. A handful of commands for the TI-89 calculator will be used to solve the problems. Type of Scientific Calculator The TI-89 is an advanced scientific calculator; it has graphing, programming, and algebra capabilities, and advanced mathematics software, which includes numerical and symbolic commands (6). The latter feature allows the manipulation of mathematical expressions and function, for example, one can factor, solve, differentiate, integrate, and more. In addition, unit conversion, solving single or systems of differential equations, linear algebra, and statistical regressions can be performed.
The files and data in the TI-89 can be transferred between calculators, and between the calculator and computer. The calculator is connected to the serial port of a computer via the TI-Graph Link accessory, which includes a cable and software. Furthermore, the flash technology in the TI-89 gives the user the flexibility to upgrade calculator software applications and add additional functionality by downloading the appropriate software via the Internet, thus providing longterm value to the user. Application of the TI-89 Calculator to Problems in Physical Chemistry Several problems in physical chemistry will be solved using the TI-89 calculator. These types of problems can be found in standard textbooks of physical chemistry (7–9). The problems have been chosen to cover a broad area of this subject matter. Due to space constraints only two examples applicable to quantum chemistry will be shown in print. However, the entire manuscript, which includes a wide variety of numerical-logical-analytic exercises in physical chemistry, may be viewed in this issue of JCE Online.W These exercises include detailed and stepwise illustrations of calculator commands used for a variety of mathematical techniques such as numerical integration, nonlinear regression analysis, analytical and numerical solutions of differential equations, and matrix manipulation. In addition, there are examples illustrating spreadsheet-type manipulation of data, graphing multiple plots, and using units in algebraic equations.
Example 1 The unnormalized excited-state wavefunction of the hydrogen atom is given as: ψ = (2 (r/a))e-r/2a Normalize the wavefunction to 1. Solution: * N 2 ψ ψ dτ = 1
therefore
N =
1
(1)
ψ* ψdτ
Enter the ∫ function by pressing the 2nd
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JChemEd.chem.wisc.edu • Vol. 79 No. 12 December 2002 • Journal of Chemical Education
keys and
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In the Classroom
enter eq 1 into the TI-89 as follows:
ENTER
1/( ( ( r^2*((-r/a+2)*e^(-0.5*r/a))^2,r,0,∞)|a>0)
Solution:
* (sin(θ),θ,0,π)* (1,φ,0,2π)))
6axeax
2
Therefore, eigenvalue is 6a.
ENTER
Solution: Conclusions
2
Many problems in physical chemistry can be solved using advanced scientific calculators. This convenience allows students to spend more time mastering the science of physical chemistry. In addition, more complex and interesting problems can be studied with relative ease, thereby generating greater student enthusiasm for learning, relieving math and problem-solving anxiety, and providing a glimpse into modern scientific techniques.
8 a3π therefore 1 N = 32 π a 3
1
2
The angle is set for radians on the TI-89 (use MODE key) and the value for a must be specified as positive, in order to get the correct result.
Example 2 Show that the function ψ, ψ = xe
Acknowledgments I thank Paul Poskozim and David Rutschman for reading and commenting on the manuscript. W
The entire manuscript, which includes a wide variety of numerical-logical-analytic exercises in physical chemistry, is available in this issue of JCE Online.
ax 2
is an eigenfunction of the operator,
Literature Cited
2
d − 4a 2 x 2 dx 2
What is the eigenvalue? Solution: The eigenvalue equation is represented as: ˆ = ωψ Οψ
(2)
where, ω is the eigenvalue of the operator Ô. The left hand side of eq 2 is input into the calculator by activating the differentiate function, d, which is obtained by pressing the keys. 2nd 8 d(x*e^(a*x^2),x,2)4*a^2*x^2*x*e^(a*x^2) The derivative above is specified as the second derivative.
Supplemental Material
1. Nicoll, G.; Francisco, J. S. J. Chem. Educ. 2001, 78, 99. 2. Kim, M. H.; Ly, S. Y.; Hong, T. K. J. Chem. Educ. 2000, 77, 1367. 3. Morgan, M. E. J. Chem. Educ. 1999, 76, 631. 4. Alberty, R. A. J. Chem. Educ. 1983, 60, 102. 5. Cortes, F.; Jose, E.; Moore, D. A. J. Chem. Educ. 1999, 76, 635. 6. Texas Instruments. TI-89 Guidebook, 1998; http:// education.ti.com/product/tech/89/guide/guides.html (accessed July 2002) 7. Atkins, P. Physical Chemistry, 6th ed.; W. H. Freeman: New York, 1998. 8. Alberty, R. A.; Silbey, R. J. Physical Chemistry, 3rd ed.; John Wiley & Sons: New York, 2001. 9. Levine, I. N. Physical Chemistry, 5th ed.; McGraw-Hill: New York, 2001.
Visit CLIC, an Online Resource for High School Teachers, at http://jchemed.chem.wisc.edu/HS/
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