AN ALTERNATIVE METHOD FOR THE CALCULATIONS OF INDIRECT GRAVIMETRY B. A. FIEKERS, S. 1. College of the Holy Cross, Worcester, Massachusetts
ONE
of the commonest numerical problems in indirect gravimetric analysis requires the. solution for two chemically related components in a binary mixture. The data include the weight of the mixed sample, the weight of a mixed product from a reaction which is common t o both components, and the gravimetric factors involved. Solutions for such problems given in most of the textbooks are invariably based on the use of sirnultaneous equations. I n this paper an alternative method is suggested. It has remotely the same algebraic basis, hut it adapts this type of problem t o a schematic technique, known as alligation alternate, familiar to many, no doubt, through the work of Pearson' and others.= I n many cases this alternative has the advantage of being more rapid than the algebraic method and might thus be of value in routine calculations. It is a concise method and so it tends t o reduce the chances of erroneous calculations. It lends itself to the rapid diagnosis of data which are physically false, and hence it is of value t o the instructor who composes his own exercises. A typical exercise follows. A mixture, containing only sodium chloride and sodium iodide, weighs 1.0238 g. It is treated in solution with silver nitrate. The weight of the precipitated mixture is 1.8979 g. Calculate the percentages of the components in the original mixture. (Once such a problem is solved, calculations can then be made for the weights of the components and for other chemically related quantities.) Solution. wrr.ol
+
WNSI
=
1.0238 = W.
(1)
where w is the weight of the components designated; W., the weight of mixed sample; and W,, the weight of the mixture of products. Formula subscripts identify the components in each mixture. It is well also t o write equation (2) in terms of equation (1): by applying appropriate factors. The values of the factors are given by: AgCI/NaCI = 2.4521
AgI/NaI
=
1.5662
(4)
(5)
Then take the ratio, R, as follows: R
=
Wp/W8 = 1.8979/1.0238 = 1.8535
(6)
A square is then constructed as shown in the figure, in which the value of R is written a t the center and the values of the factors which appear in equation (3) are written in the corners t o the left. Differences between the value of R and the values of the factors are taken diagonally and written in the corners t o the right. These differences comprise the ratio of the weights of the components in the order sodium chloride, sodium i o d i d e n o t necessarily the actual weights of the components. The percentage of a given component is then calculated : % NaI
=
(0.5986 X 100)1(0.2873 f 0.5986) = 67.5%
(7)
The percentage of sodium chloride can then be calculated in a fashion analogous t o equation (7), or by difference from 100:
100% - 67.5% = 32.5% NaCl (8) 'LANQE,N. A,, "Handbook of Chemistry," 7th ed., HandOther problems can often be reduced t o this form by book Publishers, Ine., Sandusky, Ohio, 1949, p. 775: "Standardization of milk and cream. Pearson's square methd.'Orl Univer- means of chemical factors, for example, in cases where sity of Illioais Bulletin No. 323. the analysis for some common atomic constituent in the 'BwLEY, W. T.,C . B. GUSTAPSON, AND M. 3. ~ K L O S A , "Phsrmsoeutical Calculations," 2nd ed., Lea and Febiger, Phila- sample might be given. The value of R could be confused with its reciprocal. delphia. 1952, p. 155.
The choice is correct when the value of R lies within the values of the two factors. If this condition is not fulfilled, the value of the reciprocal of R might be tried. If both R and 1 / R are not of intermediate value, then the data are physically impossible. An equivalent criterion demands that the two terms of the ratio of the weights of the components. should have like algebraic u'vna "-m..u.
An algebraic justification of the method follows. W.3
Paw.
+
Wb =
w.
+ Fszos = W ,
(9)
Substituting in equation (12) the values of W , and W. from equation (10) . . and equation (9) . . resuectivelv: F-wa
+ Fbwb = Rw. f Rwb
Equation (13)is also given by: w./wa
=
(R
+ Fsws/W. = W,/W. = R
-
RW.
(14)
a and b cannot be
AgCl/NaCl 2.4521
(10) 1.8535
(11)
and letting the ratio of the weight of product t o the weieht of s a m ~ l ebe eoual t o R. it is seen that this calklated factor R is 'the weighted average of the chemical factors appearing in equation ( l o ) , and that its value has t o lie between the values of these factors. By equation ( 1 1 ) : Wp
- Fd/(F. - R)
. in which the component weights of
wherein the subscripts a and b identify the components and F is the appropriate factor. Dividing all terms in equation (10) by W,, F.w,/W.
(13)
(12)
1.5662 A~I/NOI Q
~ :~ wNOI ,
:: 0.2873 : 0.5986
t d e nseparately, but only as the ratioof the weights of the two components. Equation (14) is schematized in the figure.
