Letter pubs.acs.org/jchemeduc
Reply to “Comment on ‘Calculating the Confidence and Prediction Limits of a Rate Constant at a Given Temperature from an Arrhenius Equation Using Excel’” Ronald A. Hites* School of Public and Environmental Affairs, Indiana University, Bloomington, Indiana 47405, United States ABSTRACT: A rate constant calculation including propagation of errors with covariance is demonstrated. KEYWORDS: Second-Year Undergraduate, Physical Chemistry, Computer-Based Learning, Kinetics
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where these are standard errors. The values of the three variances in eq 3 are
ernot points out correctly that my propagation of errors did not include the covariance term, and he provides the correct equation (see his eq 2).1 In fact, there is nothing special about the propagation of errors calculation using the linearized form of the Arrhenius equation. The propagation of errors from any linear regression should always include the covariance term.2 Consider a line of the form y = a0 + a1x (1) ⎛ ∂y ⎞⎛ ∂y ⎞ 2 ⎛ ∂y ⎞2 2 ⎛ ∂y ⎞2 2 =⎜ ⎟⎜ ⎟ sa0 + ⎜ ⎟sa a ⎟ sa1 + 2⎜ ⎝ ∂a1 ⎠ ⎝ ∂a0 ⎠⎝ ∂a1 ⎠ 0 1 ⎝ ∂a0 ⎠
(7)
sa20a1 = −2.5 × 0.0086885 = − 0.021721
(8)
= 0.079642 + 0.078196 − 0.13033 = 0.027508
(2)
(9)
Note that the covariance term (≈−0.13) is about the same magnitude as the sum of the other two terms (≈0.16), showing how important this term is to the calculation and how important it is to carry along extra significant figures. Hence, the standard error at x = 3.0 is
(3)
The errors of a0 and a1 come out of almost any regression analysis computer tool, but the covariance needs to be calculated from
sa20a1 = −xs̅ a21
sa21 = 0.0932122 = 0.0086885
sy2 = 0.079642 + 32 × 0.0086885 − 2 × 3 × 0.021721
where s2a0 and s2a1 are the variances from the regression, and the covariance, s2a0a1, is in the last term. The partial derivatives are 1 and x, respectively. Hence sy2 = sa20 + x 2sa21 + 2xsa20a1
(6)
At x = 3.0, the propagated variance of y is
The propagation of errors gives sy2
sa20 = 0.282212 = 0.079642
sy =
0.027508 = 0.166
(10)
Thus at x = 3.0, y = (0.62381 + 1.5971 × 3) ± 0.166 = 5.41 ± 0.17. This is the same result one would get if one used a confidence limit calculation.4
(4)
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Note this is negative when x̅ is positive. Substituting this in eq 3 gives
AUTHOR INFORMATION
Corresponding Author
sy2 = sa20 + x 2sa21 − 2xxs̅ a21
(5)
*E-mail:
[email protected].
Note that if x̅ = 0 (or can be made to be 0 by subtracting x̅ from each x value3), then the covariance term can be omitted. Let us demonstrate the magnitude of the different terms using the following synthetic data: x y 0 0.9 1 2.2 2 3.8 3 4.8 4 7.0 5 9.0
ORCID
In this case, x̅ = 2.5. The regression analysis tool of Excel gives a0 = 0.62381 ± 0.28221 and a1 = 1.5971 ± 0.093212,
Received: June 28, 2017
© XXXX American Chemical Society and Division of Chemical Education, Inc.
Ronald A. Hites: 0000-0003-0975-5058 Notes
The author declares no competing financial interest.
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REFERENCES
(1) Pernot, P. Comment on “Calculating the Confidence and Prediction Limits of a Rate Constant at a Given Temperature from an Arrhenius Equation Using Excel. J. Chem. Educ. 2017, DOI: 10.1021/ acs.jchemed.7b00251.
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DOI: 10.1021/acs.jchemed.7b00469 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Letter
(2) Bevington, P. R.; Robinson, D. K. Data Reduction and Error Analysis for the Physical Sciences, 3rd ed.; McGraw Hill: New York, 2003; p 43. (3) de Levie, R. Collinearity in Least-Squares Analysis. J. Chem. Educ. 2012, 89, 68−78. (4) Hites, R. A. Calculating the Confidence and Prediction Limits of a Rate Constant at a Given Temperature from an Arrhenius Equation Using Excel. J. Chem. Educ. 2017, 94, 398−400.
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DOI: 10.1021/acs.jchemed.7b00469 J. Chem. Educ. XXXX, XXX, XXX−XXX
