A Simple First-Order Conwcutiva Rate Reaction. A Dlflerent Method for Its Solutlon
To the Edltor:
which can be rearranged to give eq 3
A recent article in this Journal [1989,66, 461 describes a series solution for the differential eq 1 d[B]ldt = k,[A],e-"'
- kdB1
(1)
resulting from the mathematical analysis of a simple firstorder consecutive chemical reaction
A~BSC the method has a twofold purpose, I think it is inadequate for an undergraduate student since the author does not resolve eq 1;instead, he obtains the series expansion of the answer to the solution given by eq 8. Here, I present a different way to solve eq 1that does not appear in physical chemistry textbooks available to students. I start by multiplying eq 1by the term ehzi,to get eq 2
L eh'd[B]ldt
+ k2[B]ekzf= kl[~]oeik2-k"i
(3)
The left side of this equation can be replaced by eq 4 k t
e ? d[B]/dt
+ k2[B]ekzf= d[B]eb"/dt
(4)
Equation 3 then becomes the simple differential eq 5 d([B]ek")= k , [ ~ ] ~ e ~ ~ ~ l " ~ " d t
(5)
Whose solution can be obtained in just one integration step
-
[B]ek2'= k , [ ~ ] ~ e ~ ~ ~ - k,) ~ ~+' I~ / ( k ~
(6)
for the simpler case, [B] = 0,at t = 0, I is equal to I = -k~[Al~l(k, -k ~ )
(7)
If onesubstituteseq 7 intoeq 6 and divides bothsides by ek2',
Volume 68
Number 2
Februaly 1991
179
one obtains, after collecting terms, the solution for [B], given by eq 8 and valid for k2 # kl [B] = k,[A],(e-'I' - e-&" )/(kz - k ~ )
(8)
For the case of theoretical interest with kl = kz, eq 5 reduces to = k,[AlOdt d([~]e'~'
(9)
Whose solution corresponds to eq 10 [B] = k,[~],te-'I'
(10)
for [B] = 0, a t t = 0. This solution is not given by the series method cited before. A computer program for the solution of a larger sequence of successive first-order reactions, using the power series method, is discussed by J. G. Kay in this Journal 1988,65, 970-973. Carlos Castlllo S. and Gerrnanla Mlcona S. Universidad del Vaiie Allartado A h 0 25360. Cali. Colombia
To The Editor
I would like to take this opportunity to thank Castillo and Micolta for sharinc! their solution toea 1. I have not seen this method previousli, and it appears td be a very reasonable way to solve the equation.
- kJB] d[B]/dt = k,[&~e-~~"
(1)
I understand that Castillo and Micolta are of the opinion that the series method that I developed in a recent article [1988,66,46] does not develop (resolve) a solution to eq 1. Instead, they state that the solution obtained by the series method is the series expansion of the answer, eq 2. This appears to be a question of interpretation. In some instances, series expansions cannot he expressed in a closed form. In this case, the series solution can be expressed in closed form, the traditional solution given in textbooks and presented in eq 2. [B] = [~~]k,(e-'" - e-'~')l(k, - k,)
(2)
Under no circumstances should the reader conclude that the solution is nothine more than the answer exnressed in a powerseries. The intention of the article was to demonstrate how a simole Dower series solution could be develoned and expressed& ciosed form. Power series method are commonly used in physics, mathematics, and finance. They form the mathematical models for computing the future values of ordinary annuities and annuities due. They are also used to determine monthly payments that are needed t o amortize a liability in a given period of time. I am of the opinion that series should be used whenever possible and try to integrate a small amount of the mathematics of finance into some chemistry courses. A brief exposure to series methods is a convenient vehicle for this exposure.
Since statistics d a v a ereat role in all sciences. i t is necessary to understand w i l a t ' k the advantages and'limitations s (.1..) . auestions of statistics. In a recent letter to t h ~ Journal concerning linear regression and causality-correlation relationship arose. When concerned with linear regression with an independent variable (let's say x ) and a deoendent variable (Y), calculators and computers with statistical programs trfto find the best line through the points by minimizing the distances in the y direction between this "hest estimated line" and each point by letting d 2 ( A ~ ) ~ / = d k0 for the leastsauares method. This means that the nroeram tries to limit tce errors due to the uncertainty in y, while assuming that the uncertainty of x is negligible, even though this may not be the case. The line obtained should he used to predict values of y given x (for example, when dealing with Beer's law, if the concentration of a cell is 5.0 mM, what will the absorption be?), hut not the other way around (what should I make the concentration to get an absorption of 0.10?). If students are reauired t o oredict the values of the indenendent variable (ckcentraGon) given the values of dependent variables (ahsorotion), they need to find the "best estimated line" through the points himinimizing thedistances in thex direction hetween the line and each point: a2(Ax)?ldk' = 0. Because most calculators do not perform this routine readily, students should enter the dependent variable y as "x", the independent variable x as "y", and perform the linear regression. Admittedly, these two "best estimated lines" are usually so similar that students are justified in using them interchangeably, but when correlation is not very high (due to slonnv techniaues. instrumental imnrecision. or other large ;&om errois, fbr example), students should draw the "best estimated line" that will bemost useful for their needs. A common misconception when dealing with statistics is that high correlation automatically implies causality. This in fact is not the case. For example, over the past 25 years both the number of golf courses and of divorces have increased (2); there is also a high correlation between the number of television sets and the rate of colon cancer from one country to the next (31. Yet it wouldseem foolish forafamily toavoid gulf courses and to throw out their television sets in order to Dreserve their familv and their health. One needs to look at knderlying factors, berhaps population growth, that would increase both the number of golf courses and divorces or a t increased per capita income, which could increase both the quantity of fatty food intake and the number of television sets. Often it is rather difficult to find all the underlying causes, and frequently this leads to inappropriate interpretations and had decisions, or worse, to abuse of statistics. The advent of ~owerfulcalculators is certainlv a blessine. but we need to be careful not to punch in datablindly and jump t o conclusions when looking a t the graphs that are produced.
.
-
sis,3.d ed.: Macmillan: Ner York. 1989;p 104. 3, Ames, B. N. Careinopena. Anlirorcinogew, and Riak Asaoa,manf: Jon Miller PmducfionStudios: Bath. PA, 1988.
Elvln Hughes, Jr.
Peter D. Mlynek
Southeastern Louisiana University
University of Wisconsin Madison, WI 53706
Hammond. LA 70402
180
The Advantages and Llrnltatlons of Statlotlcs TO the Editoc
Journal of Chemical Education
