Solution of Systems of linear Equations in Analytical Chemistry
N. Y.
David J. Wilson, Department of Chemistry, University of Rochester, Rochester 20,
laboratory work sys0 temsinofcontrol n linear simultaneous equaFTEN
tions in m unknowns must be solved, where m < n. For example, a mixture may be known to contain five unknowns, and seven elemental and group analyses are performed on it. The “best set” of values is to be computed for the per cent composition of the mixture, either with or without the restrictive condition that the sum of these values be 100%. There are n equations
ing X is included to introduce the restrictive condition given in Equation 2. dF If one forms - and sets these quantidxr ties equal to zero to minimize the error, m linear equations are obtained CW,Uk,b,
+ ;= ?X%
( ~ W , a k 1 a t , )
3
k = 1, 2,
m (4)
in the m unknonns 2%and in A. Equations 4 and 2 are then solved simultaneously for A and the x, as follom:
one can then solve Equations 4 explicitly for the xi’s:
X is then determined by summing Equations 7 over i, since
Let where the b, are the values of the analyses, the z1are the percentages of the components in the mixture, and a,, is the fraction of element (or group) j occurring in component i. The restrictive condition is
If the matrix C is inverted ( 1 )
xW,(lk]atl
=
-
Ckr
100.
X’ X i I-
3
and
(5) WlaklbI
= dt;
i, k
=
1,2, .
. tu
j
m
EXi
= 100%
Table II.
i-1
Elements of the Matrix A
i
The existence of experimental errors precludes the exact solution of Equations l ; therefore, an error function, F , is defined, which is then minimized.
j 1 -~ 2 3 4 5 6
1
n 47.56 0 0.4756 0.4756 0 0
Table 111.
k 1 2 3
where the w,are weighting factors. w,, for example, may be taken proportional to the square of the reciprocal of the average drviation in the analysis for group 1 , and so on. The term involv-
4
-~
2
3
4
n
0
n
Therefore
0 0.3916 0 0 0.3916 0 0.3916 0.2893 0.2559 0 0.2559 0 0.3916 0.2893 0.2559
Elements of the Matrix
1 0.6786 0.3725 0.1376 0.1217
S
__ i
2 0.3725 0.6134 0.2266 0 2004
3 0.1376 0.2266 0.1674 0 1481
C
4 0.1217 0.2004 0.1481 0 1966
L
where the s k s , dk, sk, and s are defined above. For a given type of sample the sh, s k , and s can be computed once for all, as can the quantities w,akl appearing in the definition of dk. When this method is used for control-laboratory work involving many samples of the same type, it involves little additional work. If the condition t h a t x x , = 1009; IS I
Table 1.
omitted, one merely sets A = 0 in Equations 7 . X, incidentally, is directly proportional to the discrepancy between E z ’ a n d 100% when the Z, are com-
Sample Analyses
c1- (as Cl’i
C10- (as C1) C1CIO- (as CI) Total C1 (2101- (as C1) ClodCIO1C1O- (as C1)
+
+
6 65%
+
23 70%
= =
bi bs
Substance
% (21
IiCl IiClO IiC103 KClOo
47.56 39.16 28.93 25.59
r .
I here are therefore R i s etlucttionR in four unknowiia: = 0.4756 XI h? = 0,3916 x i h, = 0.4756 XI 0.3916 Xy b4 = 0.4756 Xi 0.3916 X2 U ,289:2 S 2 0.2559 .Yd b, = 0,2559 Xc be 0.3916 X2 0.2893 X ? 0.2559 Xd hi
+ +
1578
ANALYTICAL CHEMISTRY
+ +
+ +
z
puted without applying the restrictive condition. A numerical example of the use of this method might be of interest. Consider the following artificial and rather simple case. A sample containing potassium chloride, potassium hypochlorite, potassium chlorate, and potassium perchlorate is found to have the analysis shown in Table I. Assume that all of the analyses are of roughly equal precision; the w,’s are then all equal and may be set equal to one. The a,,’s are given in Table 11.
The chi’s are then computed from Equation 5 , and are given in Table 111. The C-matrix is then inverted-the abbreviated Doolittle method ( I ) can he used to good advantage here, or, if the probleni is sufficiently complicated, a high-speed computer may be used and the element’s of the inverse matrix t’abulated (see Table IV).
Table IV.
Elements of the M a t r i x
C-’
i k
1
2
3
4
1 2 3 4
2.21047 - 1,34243 0.00048 -0.00033
- 1.34243 4.07616 - 4.41768 0.00392
0.00048 - 4.41768 23.9121 - 13.5172
- 0.00033 0.00392 - 13,5172 15.2730
The quantities S I and s are t,hen computed: SI = 82
The fact t h a t x z i ti 100% is due to
0.86819
= -1,68003
ss = s4 =
s =
a
rounding-off errors in t’he calculations. This general approach has been applied to a \vide variety of problems ( 2 , 3).
5.97770 1.75939 6.92525
The &’E are computed from Equation 5: dl = 32,764 dz = 35.377 da = 17.048
&
(i I3
z
x is computed from Equation 8
( I ) Anderson, R. L., Bancroft, T. .4., “Statistical Theorv in Research.” 1st ed., chap. 15, McGraw-Hill, New Tork, 1952. (2) Burnham, H. I]., Jones, L C., AISAL. CHEM.29, 82i-34 (1957). ( 3 ) Ijeniirig, JV. L , “Statistical Adjuutment of Data,” chap. 4 et sey., Riley, Sew >-ark, 1913.
= 16.782, -
2 = -0 0642
) and lastly, the
I , ’ S :LITcoinpiited
fiom k?qiiation 7 :
24.880% = 25.080~0 = 24.15670 26.052oj,
ZI = .CP
LITERATURE CITED
=_
-
100.16370
1
Micro-Parr Bomb Assembly Suitable for Microdetermination of Fluorine in Organic Compounds AI Steyermark and Frank P. Biava, Hoffmann-La Roche Inc., Roche Park, Nutley 10,
microdetermination of fluorine in organic compounds, a number of inrestigators (2-8, 10, 11) recommended iiision with metallic potassium or -odium in a sealed microbomb a t ele\-ated temperatures (up to ’700” C.) for periods as great as 2 hours. Belcher and Tatlow (6, 12) described a special nickel bomb that could be used, but the regulation micro-Parr bomb as-embly ( 1 , 3 ) (Parr Instrument Co.. Moline, Ill., Series 2300, KO. AlMB) 15 not suitable, even when the lead gasket is replaced b y a copper one (3-5, ‘7, a), because of breakdown of the clamping device u-hich holds together the parts of the bomb (cup and cover head of 98% nickel). This device was not designed for prolonged heating a t such elevated temperatures. The one described here is suitable, and, when a soft copper gasket is used between cup coyer, the assembly may he treated as above nithout loss of the fusion mixture. Thew units hold iip under continued UT. TOR
Figure 1 shows the details of construction of the nut and bolt clamping device (which should be machined from Type 303 free machining stainless steel), rlimmsions of the copper gaskets, and the assembly showing a micro-Parr bomb ( 1 , 9 ) in position. A stainless steel block may be drilled out as a support for the assembly. For a tight seal, before use, the gaskets should be
HEXAGON HEAD
9/32 DlA D R I L L THRY CHAMFER 1/32 X 4 5
,/
I 1/4 A C R O S S FLAT,;
N. J.
,
A
5/8 --1-14
_ -.
1
MACHINE
‘-CC’SINK
FLAT+
,-
,
HEXAGON NUTI 1/4 ACROSS FLATS
NF-2
6 F U L L THDS
/
x
I / B DP
600
.~ ASSEMBLY
- I - 14NF-2 4 FULL THDS
- 1
5/8
t
~
1/8-L
ic
-I
1
1/16 DIA
Figure 1. Details o f construction of clamping device for micro-Parr bomb All dimensions in inches. Use Type 303 free machining stainless steel
:iiinealed a t 700” C. (preferably in an atmosphere of carbon dioxide or nitrogen), and any oxide coating removed with dilute nitric acid. Annealed coppc’r gaskets of the correct dimensions arc’ commercially available (Parr Instrument Co., Moline, Ill., KO, 73IR copper, or Arthur H. Thomas Co.. Philadelphia, Pa., No. 2196-E). Figurib 1 shows the dimension of both the hexagon head and hexagon nut to be 1’ inches across the flats. For greatw durability this may be increased t o l 1
inches, all other dimensions being left a. qhown. To prevent leakage around the gasket during fusion, the cup and cover should be cleaned after each determination n ith fine emery cloth, while being turned in a lathe. For closing, the nut portion of the unit should be held in a vise while the bolt portion is tightened using a long-handled (20 inches or more) socket w e n c h . After fusion the cup and lid may be separated by holding the former in a vise xhile VOL. 30,
NO. 9, SEPTEMBER 1958
1579
