Calibrating Commercial Test Equipment Scales for Results from linear Reactions GEORGE B. SAVITSKYI Division o f Tropical Research,
United Fruit Co., f a Lima, Honduras
.i device is presented for transforming linear reactions of the typey = mx b to the types y = x, y = m'x, or y = m'x b', where m' and b' are constants conveniently chosen to save time in computation. The transformation is achieved by varying the volume ratio of reacting materials in accordance with a generalized formula. This device is especially recommended for setting up instruments for routine test determinations involving a large number of samples.
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+
HEN routine analytical procedures are established for a large number of samples, care is usually taken t o diminish the amount of laboratory work. S'olumes of aliquots are chosen which can be conveniently and readily measured by available equipment, such as volumetric flasks and transfer pipets. Computation of results is then adapted t o the method. However, this usually results in repeated graph readings or prolonged calculations, which take almost as much time as the analytical procedure itself. The object of this paper is t o show how, by using fractions instead of whole volumetric units, final cnlculations of results may be simplified or even eliminated. Analyses of thousands of leaf samples for copper residue were carried out during extensive studies on the coverage and weathering of Bordeaux fungicide on banana leaves. Leaf samples A square inch in area were extracted with B ml. of 0.05N hydrochloric acid. -41-ml, aliquot of the extract and 1 drop of ammonium hydroside (1 to 1) were added to 20 ml. of sodium diethyldithiocarbamate reagent (0.01% aqueous solution). The resulting absorbance was measured in the Fisher Electrophot'ometer, using a blue filter (No. 425). This instrument is equipped with a logarit'hmic scale graduated into 100 units covering the range of absorbance from 0 to 1.0. Because the reaction followed Beer's law, a linear equation of the type y = mx b (Figure 1,3) was obtained, x being the copper concentration in the aliquot espressed in parts per million, and y the number of absorbance scale units (each scale unit = 0.01 absorbance unit). T h e blank reading varied slightly from day t o day and separate graph plottinps were necessary for each determination. Each value (zo) was determined directly from the graph or by substituting the corresponding absorbance ( y o ) in the equation y = mx b. This value ( 5 ' ) was further multiplied by the factor B / A in order to espress results in micrograms of copper per square inch of leaf surface,
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This method is applicable to other fields of instrumental analysis, as long as there is a linear relationship between concentration and the instrument dial reading. The only requisites for obtaining the y = x line are that the blank reading be equal to zero on the instrument dial, and the number of units of the standard be numerically equal to the scale reading. Should it become impossible to zero the blank, it may be adjusted to read some round number; such as 10 or 20. After the slope is transformed to unity, this number may be mentally subtracted from the instrument reading to give the required result. In certain instances transformation of the slope to unity may lead to lower sensitivity. In this case, the slope value may be adjusted to any whole number by which the instrument reading may be easily multiplied to obtain the final result. For example, such equations as y = 112 5 and y = 2.252 lose considerably in sensitivity when transformed t o the y = x line. These equations may he conveniently transformed into 21 = 10x and y = 22 without serious loss in sensitivity. The final results are ihen obtained by dividing the instrument scale reading by 10 and 2, respectively. A similar transformation simplifies computation for reactions which are represented by lines with a negative slope. For esample, colorimetric determination of some chemical constituent may depend on the degree of decoloration produced in the reagent, m hich is directly proportional t o the concentration of this constituent. The reaction line is then similar to the one shown in Figure 2 4 , represented by the equation y = b mx. In this case, the slope may be adjusted to -1, and the constant b t o any round number from which the scale reading is then subtracted to obtain the final result. Figure 2,B, illustrates a convenient form into which the y = b - mx line may he transformed By adjusting the blank to read 1.0 absorbance (100 scale units), and the standard of 50 p.p.m. to read 0.5 absorbance, the equation becomes y = 100 - x. I n this case, the answer is obtained by subtracting the number of scale divisions from 100. A general formula has been derived to eliminate the trial and error method for determining the new volume ratio of reacting materials for the desired transformation in slope. This general formula, based on a colorimetric reaction following Beer's law, is equallj applicable to any other linear reaction,
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The following modifications of the experimental procedure made these computations unnecessary.
Let T', be the volume of the aliquot of a solution in which the concentration of compound Cis to be determined colorimetrically. Let he the volume of the reagent or reagents t o which the aliquot is added, and with TT-hich compound C reacts t o produce a color proportional in absorbance to the concentration of this compound. Let the sum of the volumes V , and VF he denoted by V,. If absorbance is plotted on the ordinate and the concentration of compound C on the abscissa, the resulting equation may he expressed as
1. By trial and error t'he volume of aliquot was so adjusted that, when the instrument was "zeroed" wit,h the blank, a 50p.p.m. standard resulted in an absorbance of exactly 0.50. This was done by making use of a n adjustable automatic pipet (Schaar & Co., 754 West Lexington St., Chicago 7 , Ill.) 2. T h e factor B / A was brought t o unity by having the number of milliliters of extracting solution equal the number of square inches in the total area of sample, which was kept constant with a standard sampling device.
y=mx+b
(1)
with ?nbeing the slope and b a constant depending on the reading of the blank. Let V,' be the new volume of the aliquot which, when added to the same volume of reagents V,, results in a new equation with the slope equal t o unity, represented by
When t,he instrument readings were plotted against t.he final results, the y = zline, shown in Figure 1,B, was obtained. Under these conditions the scale reading on the instrument was numerically equal t o the final result and computation was eliminated.
y = x + b
Present address, Department of Chemistry, University of Florida, Gaineaville, Fla. 1
(2)
If the apparatus is initially zeroed with distilled water, let Vp ml.
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368
ANALYTICAL CHEMISTRY Y
I .o
Y
B
d
0
I
I
I
50
C
0
I X
50
0
100
. ..
100
COPPER MICROGRAMS / SQ. IN.
COPPER PPM
Figure 1. Typical linear reaction A.
Positive slope obtained in analysis of copper with electrophotometer
of the reagent be completed to V , ml. in the three following ways and. with these resulting absorbances: Absorbance equal to CY when completed with distilled water Ahsorbance equal to CY +9 when completed \T-ith the solution used for the preparation of the blank Ahsorbance equal to CY +9 y when completed xvith the same solution t o which compound C was added a t concentration c
+ + +
The meaning and relationship of CY, p, y , and c are illustrated graphically in Figure 1,A. I n this case, the value CY is due to the absorbance of the reagent. The value (3 is due to the color of impurities of the solution with which the blank ip prepared or to a certain amount of the compound C initiall. present in the blank. The value y repres?nts the net effect on the absorbance from the reaction of the compound C d h the reagent. The general Equation (1) of the initial reaction may now be rewritten by replac4ng the parameters m and b q-ith CY, p, y , and c, using the two-point form equation of the straight line:
Y x -
Yl
21
-
Y2 22
-
Y1
(3)
21
+ +
Substituting (c, CY 0 y ) for,(z2, YZ) and (0,a: the equation of the initial reaction becomes
+ P ) for
(XI) VI)
B.
Modified to eliminate computations of final results
due t o the addition of compound C. The final absorbance, however, is increased due to the more concentrated solution resulting from the reduced volume V i of the aliquot. The correction factor by which the final absorbance is multiplied to obtain the actual reading is consequently equal to
Va
+ V,
v: + v, The absorbance corresponding to concentration c for the reduced volume V : accordingly becomes
and the absorbance corresponding to the blank solution, when the neiv volume V : is used, is equal to
Having established these two points of reference of the transformed equation, this equation may now be expressed by substituting the coordinates of these points into the two-point form Equation 3. After replacing c by 21. according to Equation 5, this trans-
m
formed equation is finally expressed as follows The initial slope is then given by m =
-,Y
expressed in terms of the initial slope and
from which c may be y;'
thus:
Because this equation is equivalent to Equation 2 by hypothesis, its slope must be equal to 1. (5)
Consequently
If the volume of the aliquot is now reduced to V : ml. from the
original V , ml., the effect on the absorbance due to the solution used in the preparation of the blank is reduced in proportion, and is given by
s X' v
By making
I
v a
Similarly, the effect on the absorbance due to compound C a t the given concentration becomes equal to y
' x V -" Va
The constant CY, which depends upon the absorbance of the reagent, remains the same, because any change produced in the reagent is included in the value y , which represents the net result
after a simple transformation the follo\~-ingformula is obtained
(7) This formula permits computation of the new aliquot to reagent volume ratio, F ' , that results in the linear equation with the
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V O L U M E 28, NO. 3, M A R C H 1 9 5 6 Y
Y
B
W
0
2
a
m U
0
UJ
m
U
CON C E N T RAT ION PPM
CONCENTRATION PPM
Figure 2. A.
Typical linear reaction
slope equal to unity, when the initial slope, m, of any linear reaction and the initial aliquot to reagent volume ratio, F , are known. B y similar reasoning, it may he shown that the same formula is applicable to reactions represented by lines n ith a negativp slope. It is evident from this formula that the slope of a lirirar reaction is independent of the initial absorbance due to the blank reading. Because the slope of the line is not changed by adjusting the blank reading to zero or any other number, this formula is applicable t o any linear reaction. whether or not it passes through the origin. Relationship 7 also shows that the transformation of any slope to unity is possible, provided the denominator is positive. This condition is alwavs satisfied if the initial slope is greater than 1, irrespective of F value. For reactions with a negative slope this is true nhen the value m is algebraically less than 1. K h e n it is desired to change the slope to any number other than I-e.g., 2, 10, l / , ~etc.-the j following generalized formula is ohtained by equating the coefficient of x, in Equation 6, to in’ rather than to unity: ni’
F = m(l
+ );
- m’
where vi’ip any slope value desired, the values nz, F’, and F are as previously defined. The transformation made according to Relationship 8 is easily achieved when the initial slope m is slightly greater than m’ for reactions with a positive slope, and when it is slightly less than vi’ for reactions with a negative slope. This condition is not indispensable, provided that greater than (The+practical :.).is numerically application of Formula i may be illustrated by
nz 1
VI’.
~
the original example of the copper reaction. When the blank was adjusted to zero, the absorbance corresponding to 50 p.p.m. of copper r as 0.55. The initial slope was consequently equal to 5 5 / 5 0 . The ratio F was 1/20, since a I-ml. aliquot was used for every 20 ml. of reagent. In order to bring the slope to unity the new ratio, F‘! may now be calculated
F’
=
B.
With negative slope
1
=
~~
55 jo(1
+ 20)
-
1
0.04525
Modified t o simplify computations
Hence the exact new volume of the aliquot that gives the required transformation in the slope, all other conditions being kept constant, is given by
V:
=
TiF’ = 20 X 0 04525 = 0.905 ml.
If a considerable change in the F value is required to achieve the desired transformation of the slope in any given reaction, it may become imperative t o change the concentration of the reagents t o avoid excessive wastes or abnormalities that may result from offsetting the proportion of the reacting materials. Thus if in some reaction the aliquot to reagent ratio were the reverse of the ratio in the reaction just discussed-Le., if 20 ml. of aliquot nere added t o 1 ml. of reagent-the F’ value would be entirely different, even if the initial slope had been the same: F‘ =
1
50 E(l
+A)
= 6.45
- 1
This means that the volume of the aliquot must be 6.45 times greater than the volume of the reagent; or inversely, that the volume of the reagent must be 6.45 times less than the volume of the aliquot. For each 20 nil. of aliquot, therefore, 3.1 ml. of reagent must he added to bring the slope to unity. Since this quantity is 3 times more than is necessary for the actual reaction, the reagent may be diluted about three times and then 3.1 ml. of the diluted reagent may be used to achieve the same end. Once the volume ratio is found bi means of the foregoing formulas, the trial and error method may still be used for minor adjustments which are usuall! necessary because of the variation inherent in any reaction. If this technique is follon-ed, analytical procedures involving linear reactions ma>- be designed so that direct instrument reading or some easily calculated proportion of it becomes numerically equal to the concentration nhich is being determined. Only minor adjustments in the volume of reagents or solutions are necessary t o utilize the instrument readings conveniently. ACKNOWLEDGMENT
The author expresses his appreciation to S. R. Freiberg of the United Fruit Co., Division of Tropical Research, for his critical examination of the manuscript. RECEIVED for rei,iew June 30, 1955. Accepted December 19, 1955.
