A SIMPLE TEST for STRAIGHT LINE RELATIONSHIPS DONALD E. BABCOCK Mellon Institute of Industrial Research, University of Pittsburgh, Pittsburgh, Pennsylvania
M
ANY students have at times desired a rapid method for determining whether or not the relation sought in the physical chemistry laborstory was a straight line fundion. The long and tedious plotting of data in various types of coordinates to obtain the solution of problems in physical chemistry or other branches of Or other sciences' is exasperating when graph paper is not available. Through any two points s straight line may he drawn, but more than this amount pf data is necessary to determine whether a straight line exists in a graphic plot of physico-chemical data. The introduction of any third point will suffice to complete the necessary conditions required for the solution of the problem involved. Because any three points determine a trangle, the only condition in which these three points can lie in a perfectly straight line is where the area of the triangle formed by them is equal to zero. To accomplish the solution, once this condition is from analytical recognized, one need only geometry that the area of a triangle is obtained by taking one-half the value of the determinant made UP of the coo~dinatepoints of the triangle. If four or more points are used, the area of the polygon is still zero, if the points lie in the same straight line, and the value of the determinant formed by them is also zero. In order to show the method of testing described in this note, let us solve a typical problem. P R O B L E ~ : In a Order process the following observations were made on the product x of a reaction:
~ o ~ ~ ~ , " ~ ; ~I $
r
O-Z
SOLUTION
o0 60.8 10 20 30 40 ~l 97.7 119.9 133.4 153.8 153.8 93.0 56.1 33.9 20.4 0.0
TAE PROBLEM, O ~ - T E S T
LINEARITY
TIE
In first order reactions the reactant controls the rates of the reaction. The concentration of the reactant is then determined by the valuer. of (a - x ) , for the x values found in the problem are the values for the product only. First order reactions are linear if log (a-z) values are plotted against 6. .. (4 Lag (a-%) I GUFEIC
PLOT?
.
98.0 56.1 83.9
-,
1.9865 1.7490 1.5302 1.9865 1.7490 1.5302 1.9865
- DitIerence = 0.19.
10 20 30 1D.
10 20 30
Plus voluar 17.4900 80.6040 59.5950
107.6890
Minur onlwr 39.~300 52.4700 15.3020 107.5020
Area = 0.095 or 0.10, which for all practical purposes is a straight line, if one considers the possible errors in measurement. In tests of other points more widely spread apart the areas obtained
be
to he
A case may be found wherein the area obtained will increase progressively as a larger number of points are taken. If such is found to be the case, the graphic plot of these points may be a curve or made up of two different straight lines which some further testing will reveal. If graph paper is not available, the foregoing test is of considerable utility. With the aid of a slide rule, it is readily applied.
