W. E. Wentworth
University of Houston Houston, Texas
Rigorous Least Squares Adjustment Application to some non-linear equations, I /
In part I of this paper1 the theory of Deming was presented in a condensed version for the general least squares adjustment to non-linear eqnations. This was followed by a discussion of various points concerned with the use of a digital computer for these problems: Finally a few topics on the interpretation and value of the treatment to an experimenter were presented. The primary purpose of this paper, part 11,is to apply the general theory of least squares adjustment to problems in physical chemistry. The calculations follow the procedure described in part I. Equations (1) through (47) are in part I; equations (48) through (91) are in part 11. Application to Some Equations of Physical Chemistry
A few examples will help to clarify this type of calculation. (These problems have been included in the undergraduate physical chemistry course a t the University of Houston.) Simple equations which are frequently encountered are used to illustrate the value of the additional information that can be derived from this rigorous treatment. The data are actual experimental figures taken from the literature. In all cases the procedure used here differs from the treatment in the original papers. This should not be construed to be a criticism of the original papers, however, since these calculations are generally long and tedious and are not always warranted from a practical standpoint. Leost Squores Determinotion of Rote Constant and Order o f Reaction-Application to the Decomposition of Acefoldehyde
Of utmost importance in kinetics is the determination of the order of a reaction and subsequently or simultmeously the reaction rate constant. Numerous graphical or numerical techniques may be used. One of these methods is to integrate the rate expression for a given order and then to graph various functions, dictated by the integrated expression, versus time until a linear or best linear relationship is obtained. One distinct disadvantage in this method was recently pointed out by Swinhourne (IS). Frequently, in order to manipulate the integrated expression into a form such that two functions of the variables will yield a linear relationship, it is necessary to include final or initial values of the variables as exactly known quantities. These generally are not known with absolute certainty and an error in these values could lead to a distortion of the expected linear graph. This could then be misleading in regard to the decision of the order of the reaction and an error in the desired rate constant.
' WENTWORTH, W. E., THIS JOURNAL 42, 96 (1965). 162
/
Journol of Chemical Education
Swinbourne suggests that the experimental observations can be differentiat.ed with respect bo time by a numerical technique and a graph of these reaction rates will not be influenced by errors in the initial or final conditions. From the viewpoint of carrying out a least squares adjustment, it is most difficult to consider the reaction rates as observations since estimates of errors in these variables cannot be readily obtained. From this standpoint it would be more desirable and possibly essential to work with the original ohsewations directly. Here the general method outlined in this paper becomes useful since it is not restricted to a linear or polynomial equation and can readily be applied to equations containing more than two parameters. Represent the decomposition of acetaldehyde into methane and carbon monoxide by The rate expression for the forward react,ion is then
where n = unknown order of the reaction. At constant volume and temperature the pressure is directly proportional to the concentration and we can write the rate expression as
Integrating this expression from PA = Poa t t to the value of PAa t time t, assuming n # 1,
=
0,
In order to follow this reaction the total pressure of the gaseous mixture was measured as a function of time. Relating the partial pressure of acetaldehyde, PA, to the total pressure, P, P
=
PA
+ P s + Pcl
Then equation (51) becomes (2Po - P)-*+I - Po-^+'= (n - 1)kt
(55)
The function comparable to equation (17) is F&,
F+,Pa,n,k ) = -(2Po - Pi)-"+'+ (n - 1)kli + PO-'+'
=0
(56)
The partial derivatives evaluated a t ti, Pi with the first approximations to the parameters P$, ko,no are
Table 1.
Least Squares Computation of Kinetic Expression for the Decomposition of Acetaldehyde
F.
