Simultaneous Determination of Several Elements Spectrochemical Add i t ion Method . . . (11) (Tabulation of Analytical Value)

by

Isoo Masuda and Tomon Inouye Central Research Laboratory, Tokyo Shibaura (TOSHIBA) Electric Co., Ltd., I Komukai Toshiba-cho, Kawasaki, Japan (~eceived 26 February 1968; revision received 8 May 1968)

An improved method for the tabulation of analytical data, obtained by addition and successive dilution procedures for spectrochemical analysis, is presented. The author's previous work shows that the solution of the first approximation diverges at some dilution factor smaller than unity when the slope of the working curve of added series is greater than that of unadded series. By ob- taining the distance between this position and the origin, and taking it as a correction factor for zero-order approximation, tabulation of the analytical value, in the case of ~ >~, is carried out. ()ne parameter of the calculation is deleted by normalizing the spectral intensity; therefore, the tal)ulation can be simplified.

I N T R O D U C T I O N

A new addition method tha t can be applied to emis- sion spectrochemical analysis has previously been re- por ted2 However, the numerical method was ra ther complicated and t ime consuming. Tabula t ion of this numerical method is, therefore, useful for the actual applicat ion of this technique. I t is done by obtaining the zero-order approximat ion and modulat ing it to fit the relationship between two curves tha t belong to added and unadded series, respectively, if the slope of former is greater than tha t of the latter. The relation- ship can be represented by using two values, i.e., the relative intensi ty at the origin ( k = k ' = l ) and the difference between the slope of two working curves.

I. N O R M A L I Z A T I O N OF T H E I N T E N S I T Y

I f two working curves are parallel, it is obvious tha t the relationship between these curves is not affected by adding equal logarithmic intensi ty on any points of each curve. Therefore, the two curves can be shifted for the purpose of the normalizat ion of the intensity.

The intensi ty of the added series at k ' = 1 is normal- ized 100% and the intensities involving unadded series are expressed by using' the relative value. Thus, an analytical value is deduced f rom the normalized pair of curves by the following formulas :

relative intensi ty of I(k=l) unadded series at k = 1

X (1) I ' (k '=l)--I(k=l) 100-relative intensi ty of

unadded series at k = 1 o r

k' X -

~ ' - k '

dilutioff factor of added series giving equal in tens i ty to the in tensi ty of

unadded series at k = 1 (2)

1-dilution factor of added series giving equal intensi ty to the intensi ty of

unadded series at k = 1

In Fig. 1, a pair of nonparallel working curves are shown• Addition of constant logari thmic intensity to the whole range of two curves, results in the imbalance of the linear intensi ty at any dilution factor. For ex- ample, at k=k '= 1, Is is 60 and Ibl is 29• However , a t two dilution factors k aI~d k' giving equal intensities on the normalized pair of curves, the two original linear intensities are always equal, i.e., Ibl = I a l = 29. I~2 = I~2 = 5•8, etc. This Proper ty is expressed by the fundamen- tal equat ion as follows:

alog (kx) + y = ¢~log [k ' (x-t- 1) ] Jr- Y,

or (3)

alog (kx) + loglb = ~logEk' (x + 1) ]-I- logla.

Because L=I~, Eq. ( 3 ) i s reduced for any paired dilution factors to

.log ( k x ) = Nog[k' (x + 1) ], (4)

which is the fundamenta l equat ion for the unnormal- ized series. The procedures of the tabula t ion are much simplified in this case.

leo --

80 //% 60 / / Io=60

40 ,-'I /~ / / / br=29

/o55. ~

s ~ "- /~%' IbJ o:.~o

4 j~P" f o n t

2 " / f ~ f

, 1 , '?lk 1 . . . . . , . . . . . . .

O.Ol 0.02 005 O.I 0.12 05 I k,k'

Fro. 1. Normalization of paired working curves.

Volume 22, Number 6, 1968 APPLIED SPECTROSCOPY 749

II. CORRECTION OF ZERO-ORDER APPROXIMATION

If two working curves are parallel, as mentioned above, the analytical value can be deduced from Eq. (1) which is the zero-order approximation of Eq. (3) or Eq. (4). I t is equal to the value deduced from Eq. (2). However, if the two curves have different slopes and if the condition/3 > a is satisfied, the two analytical values obtained from Eq. (1) and Eq. (2) do not coin- cide and the two curves cross at a dilution factor smaller than unity. Also the solution of the first ap- proximation,

X = ~k' + (~ - ~) / (~k -- ~k'),

diverges at somewhat larger value of dilution factor than the value at the crosspoint. Such dilution factors are found by extending the pair of curves into the region of small dilution factor and applying the equa- tion

a k - ~ k ' =0. (5)

The dilution factor tha t gives the diverged solution de- creases the difference between the slopes or decreases the relative intensi ty of unadded series at k = 1. The k factor, therefore, can be used for the correction of zero-order approximation. In this case, the relation- ship between k and a tha t satisfied Eq. (5) is expressed as

dk / k = - - da/a

where k and I are related by the equation

dk = dI /ax .

Namely, the increment of k is the reciprocal of the slope of the unadded series and proport ional to the increment of the intensi ty of unadded series. How- ever, its dependency on the intensi ty can be expressed by the zero-order approximation so tha t only the de- pendency on the slope should be regarded as a cor- rection factor. The correction te rm must have a reciprocal form of the k value and must converge to uni ty when k has small values. Therefore,

c = 1 / ( 1 - ~ ) (6)

IO0 -- 80

4 0

3 0 ~

2o ', I

I '~ 8 P I "- 7 I I

61- ~

3 I I [ I I

I I 2 i I t

I I 1 I

I I I I I I 2 3 4 6 8 I0 20 30 40 6080100 200 300400600800

amount of impur i ty (~g}

FIG. 2. Paired working curves by addition method.

I00 80 F / / / 4060 ~ / / / / /

I0 / / / / / 8 / /

:L / " y-.

0.3

0.2

I I ~ I I I I I I I

QOI 002 0.03 Q06 OI 02 030.4 O6 0.8 I k.k'

F~G. 3. Paired working curves by addition method.

is taken as the correction term, where k~ implies the ~ factor of unadded series which gives the indefinite- analytical value. Thus, the analytical value for t h e case of ~ > a can be deduced from the following: equation :

k' k' 1 Xo = ×~ = × . (7)'

1 - -k ' 1 - -k ' l - k s

The table of analytical value for the parallel pa i r can be calculated immediately from Eq. (1) or Eq. (2) but the table for the nonparallel pair of curves can only be calculated by taking into account of Eq. (6) for each pair and then tabulat ing Eq. (7) for each intensi ty and slope difference.

Before discussing the practical procedures, t he reliability of this method should be examined by using a simulated model. Figure 2 shows a usual working curve of an element in a certain material mixed with some diluent. Suppose tha t 200 mg of the test sample is taken and divided into two equal parts, i.e., 100 mg each, and 50 t~g of the test element is contained in each fraction. If four parts of a fract ion are excited after the dilution of ½, ~, and ~, intensities of 5.0, 2.6, and 1.2 would be observed corresponding to 50 gg, 25 ug, 12.5 gg, and 6.75 t~g, respectively, which are shown as round points in Fig. 2 and Fig. 3. These in- tensities can be plot ted for the dilution factors of k = 1, 0.5, 0.25, and 0.125 though the quanti t ies of the ele- ment in each step are unknown. Similarly, intensities of 30, 15, 7.5, and 3.8 would be observed and plot ted for the dilution factors of k ' = l . 0.5, 0.25, and 0.125 as shown in Fig. 3 by tr iangular points, af ter 250 gg addit ion of the test element and dilution on the re- maining part . These curves may be called as added series. Full lines in Fig. 2 are conver ted to broken

750 Volume 22, Number 6, 1968

FIG. 4. Typical working curves in usual method.

IO

5 I0 20 50 50 I00 a m o u n t of impuri ty (~g)

lines by the normalization. From the pair of broken lines

Slope-unadded working curve Slope-added working curve From Fig. 3

a=0 .687 8=0 .990 k ' i n Eq. (2)=0.152 k~=0.007

are determined so tha t the analytical values are ex- pressed as follows:

0.152 1 X o = X - -

1 .0 -0 .152 1 --0.007

=0 .179X 1.01 =0.181.

The absolute analytical weight is, therefore,

250 #g X0.181 = 45 #g, error : - 10%.

In this example, the correction on the zero-order ap- proximation is not needed because k= is too small.

If the amount of 100 mg of the material is taken and one a t tempts to determine 25 #g of the same element by a 20-#g addition, the usual working curve would be shown in Fig. 4. In contrast the new addition method would afford a pair of working curves as shown in Fig. 5. In this case, analytical values

a=0.416, fl=0.644, k' in Eq. (2)=0.435, k~=0.435,

are deduced, and the following analytical value is obtained

0.435 1 X o - - X

1--0.435 1--0.435

= 0.770 X 1.77 = 1.36,

20 #gX1.36=27 .2 #g, error: + 8 . 8 % .

FIG. 5. Paired work- ing curves by addition method.

IOO 8o

6O

40

5o

2O

8

6

4

3

2

- - / / /

/ / /

/ / / /

_ / /

&

I I OI 025 05 I

k.k '

The correction factor of 1.77 cannot be neglected even in the rough calculation.

III. PROCEDURES OF TABULATION

In Fig. 6, variations of k~ for fixed values of ,8 of 0.70 and a of 0.60, 0.50, 0.40, and 0.30 are shown as functions of intensi ty of unadded series at k = 1. T h e curves are straight and their slope is determined as , 8 - a . The constant te rm of these lines cannot be de- termined because k values giving zero intensi ty can- not be found.

Table I shows the data for the slopes of paired work- ing curves and slopes of the (k~--Ik=l) curves for several values of ~. In practice, k~ in each case is determined separately by Eq. (5), then the c factor of Eq. (6) is calculated and finally Eq. (7) is calculated. This is the tabulat ion procedure.

This method is based on a pair of averaged straight lines tha t are determined by eight observed points for

Table I. Slope of I - -k curves.

~ - a ~ 1.0 0.90 0.80 0.70 0.60 0.50 0.40 average

0.10 0.095 0.110 0.107 0.098 0.100 0.100 0.099 0.101 0.20 0.196 0.216 0.200 0.196 0.194 0.194 0.212 0.200 0.30 0.298 0.338 0.308 0.298 0.293 0.293 0.297 0.304 0.40 0.400 0.435 0.408 0.392 0.392 0.420 0.407 0.50 0.500

FIG. 6. Relation between k~ and /;k=l) for fixed ~ and several a values.

I 00 80

60

40 30

20

I0 8 6

4 3 0.001

- - slope = 0 . 0 9 8 ~ = ~ = ~

I [ I I I I I , I I I I' 0.003 0.004 0.01 0.05 0.06 0.1 0.2 0.4 0.6 0.8 I

km

APPLIED SPECTROSCOPY 751

Table II. Analyzed results--Spark exc2

Tes t sample N B X - - 1 9 8 .Element (line in .~) 5~g (2802.7) b M g (2802.7) Ct~ (4226.7) Fe (2599.4)

0.95 0.358 0.43 fl 0.95 0.504 0.507 k ' 0.80 0.55 0.58 k 0.35 0.065 Xo (anal.) % 0.067 2.6 0.41 Xo (real) % 0.07 2.7 0.66 re la t ive error (%) --4.3 - 3 . 7 - -38%

See Fig. 4. 5, Ref. 1 b 0 n l y one component addit ion.

an analysis. Reduct ion of the number of observed points would be accompanied by some increased error in the practicM applications of the discharge excitat ion method because it has considerably unstable char- acter. As far as the tabula t ion is concerned, two-figure table would be adequate for practical application.

IV. E X P E R I M E N T A L

A table is used to analyze the experimental l obta ined working curves. Some cases of ~ < a are observed in the experiments, however, tabula t ion has not been done for these cases. Table I I shows a result of the spark excitation. Calcium and iron have the characterist ics of $ > a and, therefore, can be analyzed t lmugh iron has considerable error of - 3 8 ~ v . Magnesium curves have the characterist ics of t3 < a and are not analyzed.

Table I I I shows the results of the dc-arc excitation. Calcium and manganese have the characterist ics of

> a and the table can be applied. The error is smaller than tha t of the spark excitation.

Table Ill. Analyzed results--dc arc exc."

Test sample BCS--267 Element (line in ~) Ca (3518.9) Mn (2801.1)

0.81 0.67 1.09 0.90

k' 0.64 0.49 k 0.45 0.20 Xo (anal.) % 1.6 1.2 Xo (real) % 1.25 1.2 relative error (%) --t-28 0

a See Fig. 7, Ref. 1.

V. DISCUSSION

The nonlinear propert ies of the working curves in the concentrat ion region has been observed often. Principal reasons for this are as follows:

1. Noise level of the detector. 2. Contamina t ion of test sample. 3. Interference of other emission line. 4. In te rac t ion between ions or a toms in exciting

processes.

Contamina t ion is not considered in this work. Some of the remaining sources can be evaluated by more or less laborious work; for example, Fassel and others report m a n y cases of nonlinear working curves? They showed tha t E r 3906.32/Ho 3910.30 and Ho 3 % 6 . 0 0 / D y 3443.46 curves are still curved af ter the correction of background intensity. An addit ion me thod tha t takes into account the natura l slope change of working curves is therefore, of pract ical importance.

I f the intensi ty measurements are more reliable as in flame pho tomet ry , the number of observed points can be reduced. Thus the solution can be restr icted to a narrower region of dilution factors which provides a more precise analysis. For such applications, a three- figure table would have practical meaning. However , this work has been l imited to an applicat ion to dis- charge excitation.

The analyt ical error of the methods seems to be within :t: 10% in the model case but it will increase in practice because of instabi l i ty of exciting processes and the matr ix effects.

Though the method is originally based on the analysis of the arc-excitat ion technique, the experi- menta l results show tha t the method can be applied to the spark excitation if some increased error is per- mitred. The value of k , depends only on the difference between two slopes. This is a favorable factor because it permits a small error of slope verification which is substant ia l ly related to the error of sensi t ivi ty calibra- t ion of light detector.

1. I. Masuda and Y. Onishi, Appl. Speetry. 20, 305 (1966). 2. V. A. Fassel, B. Quinney, L. G. Krotz, and C. F. Lents, Anal.

Chem. 27, 1010 (1955).

752 Volume 22, Number 6, 1968