imide are stronger acids than glutethimide. K h e n plasma samples containing these compounds were carried through the extraction procedure, there was no interference in the determination of glutethimide. Two alkaline n-ashings with 0.5S sodium hydroxide removed all the megimide and over 90% of the 2-phenylglutariniide. Urine may contain high concentrations of the metabolite, and small but significant amounts may appear in the final extracts. The characteristic rate of hydrolysis will distinguish the metabolite from the glutethimide. Determination of Glutethimide in Experimental Animals and Humans.
T h e results of a study on a dog given 200 mg. per kg. of glutethimide orally are shown in Figure 4.
(1) Bernhard. Karl. Just. M . . Vuillumier.
(5) Schreiner, George E., Berman, L. B., A . M. 8 . Kovach, R., Bloomer, H. A4.J Arch. Internal Med. 101, 899-911 (19.58). (6) Shaw, F. H., Simon, S. E., Cass, M. M., Shulman, A., Suture 174, 402-3 (1954). (7) Sheppard, Herbert, D’.\saro, Barbara S.,Plummer, Albert J., J. Ani. Pharm. ASSOC., Sci. Ed. 45, 681-4 (1956). (8) Shulman, A,, Shaw, F. H., Cass, Ii. M,, Khyte, H. >I., Brit. X e d . J . 1955, I , 1238-44. (9) Tagmann, E., Sury, E., Hoffmann, K., Helv. C h i m Acta 35, 1235-9; 1541-8 (1952).
39,596-606 (1956). (2) Gross, F., Tripod, J., hleier, R., Schweiz. med. Wochschr. 85, 305-9 (1955). (3) Kebrle, J., Hoffmann, K., Experientia 12. 21-2 (1966). (4) -iZcBa n (more viare lengths observed than the number of components), n-e have more equations than unknon-ns and can obtain a Trariety of solutions for the matrix by using different sets of equations. I n the presence of experimental errors in both the a and D matrices, it will not ordinarily be possible to satisfy Equation 3 or 4 exactly. However, it is possible to obtain matrix c n-hich will minimize the quantity
where the D, come from the experimental absorbancies and the D; are values computed using Equation 3 with the a matrix and the E matrix obtained. ;12 is the sum of the squares of the individual deviations. Equation 6 can also be written in matrix notation as _c_
where D, is then the absorbancy per unit length of cell. I n practice it sometimes proves convenient to work with equations of the form of Equation 2, in which D,, a,,, and e , represent functions derived from the absorbancies per
and columns. ;1*is a single number, so is not underlined in the matrix Equation 7 . Selection of c to minimize L2 is the familiar least squares criterion for obtaining the best set of concentration values, and can be seen to correspond to obtaining the closest fit of a calculated absorbancy curve to the experimental absorbancy curve. I t can be shon-n ( I , 4 ) that the least squares criterion is satisfied by solving Equation 3 in the following manner. First, multiply both s i d e of 3 by the transpose of the matrix (L (generally nonsquare). Then
A' =
(D'
- D )( D '
- D)
(7)
in which the matrix (D'- D) is the transpose of the matrix (D' - D)Le., it is obtained from the original matrix merely by interchange of rows
=
ggc
(8)
Xatrix 4 a will be a square matrix, m-ith dimensions n x n, because it results from multiplication of t h r n X m matriv d by the m x n matrix a. The matrix similarly will be 72 X 1. The multiplication by the transpose matrix to obtain the best least squares fit is a consequence of the form of Equation 7 , in which A* is itself a product of a matrix m-ith its transpose. The matrix Equation 8 may be regarded as a nelv set of n simultaneous linear equations in the n unknown concentrations. This may be solved by the usual methods of solution of simultaneous linear equations, n here the number of equations is equal to the number of unknowns. Unfortunately, the solution of a set of simultaneous linear equations would be necessary for each sample analyzed if Equation 8 were to be used. The matrix inversion method described below requires a more difficult operation than solving a set of simultaneous linear equations, but the difficult step needs to be performed only once, and the result can be used in all subsequent analyses. v,-ill (if nonAs the square matrix singular) h a r e a n inverse, both sides of Equation 8 may be multiplied by this inverse, (a (I) - * , Then (cj g)-1
g _D
(C g)-1 (G a)5
=
=
2 (9)
This is the solution to the matrix Equation 3, for it prescribes how to obtain from it the concentrations c best satisfying (by the least squares criterion) the experimental data. It is convenient to define a new matrix, by
x,
-If
=
(??).-I!
(10)
where is an n X m matrix which can be obtained directly, by suitable computations, from the known matrix a. nil1 be a matrix characteristic of the system studied and the wave lengths selected, and will facilitate calculation of the concentrations c b y
a
2
=
1pg
(11)
The individual concentrations then are given b y VOL. 32, NO. 1, JANUARY 1960
85
in which each concentration is eupressible as a linear combination of the absorbancy values a t the set of wave lengths selected. APPLICATION OF METHOD T O ERGOSTEROL IRRADIATION SYSTEM
had heen calibrated using an alkaline solution of potassium chromate, as described by HauPt ( 2 ) . The wave length scale was calibrated using a lowthe source Of pregsure mercury arc the reference spectrum, and the wave length dial ~ y a s found accurate to \yitllin 12 -4,in the Jvave length rarlge used. Dial settings ivere noted at regular intervals on the chart paper to eliminate errors due to alignment or
Materials and Experimental Procedure. Commercial grades of er-
gcqterol obtained from Parke, Dayis and Co. and Sutritional Biochemical Corp. were employed in t h e iiradiation work without further purification. Three of t h e components of t h e irradiation mixture TI ere utilized in the preparation of t h e synthetic mixtures: ergosterol, lumisterol2, and calciferolz. Isopropyl alcohol was the solvent in all of the work. The alcohol was purified by shaking with sodium hydroxide, separating the aqueous layer, and fractionally distilling the alcohol layer. Solutions of ergosterol n-ere subjected to the radiation of a cylindrical low p,ressure mercury lamp. The irradiation apparatus consisted of three concentric cylindrical quartz chambers. Tap water was circulated through the inner chamber to cool the system; a copper sulfate solution iyas circulated through the middle chamber to filter out undesired ultraviolet radiationadjustment of the concentration of the copper sulfate solution permitted a variation of the wave length of cutoff of the radiation; the ergosterol solution that was to be irradiated was circulated through the outermost chamber. The irradiated solution n-as cooled by passing it through a heat exchanger through which ice -rater Jyas circulated. -4 centrifugal pump that contained Teflon packing was employed to circulate the cell solution. K i t h the exception of the steel pump, the irradiation circuit consisted entirely of quartz and glass and Teflon tubing. Samples of the irradiated solution were withdralyn periodically and the ultraviolet absorption spectra were determined for several dilutions of the irradiated solution. Solutions of known comaosition consisting of ergosterol, lunyisterolz, and calciferol2 in varying proportions were prepared from the pure components. The ultraviolet absorption spectra of the synthetic mixtures and of the pure components were determined. The latter spectra were in good agreement with those reported by Shaw, Jefferies, and Holt (7, 8 ) despite the difference in solvent used. The Beckman Model DK-2 spectrophotometer was employed for the determination of all spectra. The automatic slit-control mechanism of the instrument was used, providing constant reference energy a t all wave lengths. Slit settings ranged from 0.08 to 0.15 mm., from 2960 to 2520 $., respectively. The absorbancy scale of the instrument 86
ANALYTICAL CHEMISTRY
a
calibration of the payer. ITncler these conditions, agreement with the results reported by Sham for the pure components 7sit1,ia 2% of those reported by him, despite the solvent and instrument differences; this small discrepancy is in such a direction as to indicate that it arises from the use of somenhat narrower slit u-idths in the present investigation.
Table I. Verification of Beer-Lambert-Bouguer Law (Additivity of absorbancies of pure components in synthetic mixtures) S.D.,a Composition of Solution, G./100MI. Solution .ldsorbnncy Unit Ergosterol Lumisterol2 Calciferol? =t0.006 0 001380 0.000444 0 =o. 009 0.000920 0.000888 0 +o. 009 0.000460 0.001332 0 0 001380 0 0.000392 ~0.005 i o .005 0 0.000784 0.000920 *0.008 0 0.001176 0.000460 AO. 008 0.000444 0.001176 0 10.010 0.000888 0.000784 0 +0.007 0.001332 0.000392 0 10,015 0,000444 0.000392 0,001380 ~0.006 0.000920 0.000888 0,000784 S.D. Equation 13.
A,
x.
2300 2320 2340 2350 2360 2370 2380 2400 2420 2440 2450 2460 2470 2480 2490 2500 2510 2520 2530 2540 2550 2560 2570 2580 2590 2600 2610 2620 2630 2640 2650 2660 2663 2670 2680 2690 2700 2710 2715 2720 2730 2740 2750 2760 2770
Ergosterol 44.7 45.2
51.4 65.8
88.8 97.1 104.8 111.0 116 2 121.8 128.5 136.6 148.0
Table II. Ultraviolet Absorption of E,'Z, Values in Absolute Ethanol Lumisterol2 Tachysterol2 Calciferol? PrecalciferoL 32.3 247 156 260 33.1 34.8 154 282 37.7 298 41.4 324 169 46.4 136 330 53.0 60.4 189 172 361 69.5 377 80.3
229
399
210
107 6
258.2
412
217
121.3 128.3 136.2
303 320.0
434 439.5 445
225
96.9
226.7
152.4 175.8 189.5 197.4 202.5 203.5 204.5 208.2 216.7 220.5
169.7
396
463
230
184.3 189.0 194.9 200.2 205.3 207.5
451.5 469.6 494
470 472 3 474.0 475
227.3 225.8 224
4i3 8
--
246.1
218.5
460
214.9
281.5 290.2 290.5 289.2 280.0 265.5 252.7 245.6 247.0
232.3 235.6 237.0 238.0 236.4 233.1 231.0 226.9 224.i
458
207
447.1 444
199.5 197.1
412 397.5 384
182 li6.8
525 551 583.6 590 602 609 611 614 620 631 668
'91)O. 5
Verification of Beer-LambertBouguer Law. Because the analytical method utilizes the Beer-LambertBouguer law, it \vas first necessary to establish the applicability of this law to the system studied. T h e linearity of absorbancy tis. concentration for single components has been established for several components of the irradiation mixtures (3, 10). The applicability of the lam was further verified in tn-o additional respects. 1. For irradiation mixtures, the linearity of the absorbancy us. over-all concentration of the entire mixture was established. 2. For synthetic mixtures prepared from pure components, the additivity of absorbancies of the pure components to give the absorbancy of the mixture was verified.
RESULTS.Plots of absorbancy a t various wave lengths us. concentration were obtained from the ultraviolet absorption spectra of the irradiated
solutions. A linear relationship was found between absorbancy and over-all concentration of the irradiation niisture. The standard deviation from linearity was found to be only k0.012 absorbancy unit. The spectra of synthetic mixtures of known composition mere compared with absorbancies calculated from the spectra of the individual components and the composition of the solution to establish the additivity of absorbancies of the pure components. This comparison was made a t intervals of 5 nip in the wave length range 230 to 300 mp and a standard deviation, S.D., was calculated for each synthetic mixture. The standard deviation was calculated on the following basis.
- J Z(absorbancy
S.D. =
of mixture
a t n hich the comparisons were made. The values of the absorbancies of the solutions ncrc about 0.45 to 0.70 at the masima. The data verify the additivity of absorbancics of components in a mixture n-ithin the limits of exupcrimental error. It was believed that deviations from the Beer-Lambert-Bouguer law would be most likely to occur in solutions containing calciferol? and lumisterol?, as these compounds form a crystalline molecular addition compound-i.e., the old vitamin D1. However, the data indicate the absence of such an interaction in solution, a t least a t the concentrations employed. Specific Modifications of Method for System Studied. I n the ergosterol irradiation system, all four of the products are isomeric, so t h a t the initial concentration of starting material (ergosterol) is always the total concentration of the five species present in the system. Designating ergosterol as component 1, we have
where n is the number of wave lengths
Ergosterol and Irradiation Products E:?,, Values in Absolute Ethanol A, A, Ergosterol Lumisterola Tachysterola Calciferol2 Precalcifero12 2780 258.4 223.7 718 2790 272.2 223.7 737 353 158.8 2800 288 2 223.7 745 340 152 2810 301.5 222.5 742 2820 306.0 219.3 728 306.0 137.? 2830 296.4 213.5 290 2839 275.5 203.7 679.5 275.5 124.2 2840 271 7 202.5 677 273.9 123 6 2850 240.3 258 117 2860 209 0 177.9 631 2870 617 227 2880 165.9 152.4 609 210.1 98.1 2890 157.7 608 2895 157.2 137.2 608 188.7 89 0 2900 158.3 133.3 608 181 86 2910 162.9 607 2912 164.0 125.8 606 79 164.0 2920 168.4 121.3 599 153.5 i4.3 2930 172.6 2935 174.0 112.8 572.5 134. 6 66.0 2940 110.1 173.6 56 1 2950 167.0 119 58.5 2960 92.3 148.4 481 107.7 53 2 2980 89.3 69.1 386 3000 42.5 46.4 307 70 37.5 3010 29.8 3030 8.2 3050 12.0 177 38 22 3070 2 6 3 100 0.8 4.1 118 18 12.5 3125 3.3 98 3150 2.9 82 The above values were obtained from la1'ge scale plots drawn from tabulated data kindly furnished by ShaF and Jefferies ( 7 ) ,and used for calculation of matrix X in cases 1 t o 4 2500 97.0 95.0 225 0 399 0 209 0 2600 186.0 170.0 394 0 461 0 230 0 2650 213.0 200.0 492 5 475 0 223 0 2700 276.0 231.0 601 0 459 0 208 0 2750 257.5 228.5 408 0 655 0 181 0 2800 282.5 224.5 743 0 340 0 153 0 2850 250.0 189.0 257 5 657 5 118 0 2900 159.0 133.5 607 0 182 5 85 0 The values listed above were obtained from enlarged plots of the figure& presented h ~ Shaw et nl. (8) and were utilized for calculation of matrix -11 in cxse 0 -
Substitution of the value for c1 from Equation 14 into Equation 1 gives
Because of the practical difficulty in making dilute solutions accurately up to known concentration by weighing, it is convenient to normalize the results t o put them on the basis of the initially observed ergosterol concentration, as determined spectrophotometrically. Equation 15 is therefore divided by Equation 16, which applies to the initial conditions, before irradiation.
The division gives
Equation 17 is put into the form of Equation 2 by defining
and
j=l
C', is seen to be the fraction of compoVOL. 32, NO. 1, JANUARY 1960
* 87
nent j in the irradiation products. Then 5
j=2
I
I
I
!
K h e n m wave lengths are considered, we obtain a set of m simultaneous equations in the four unknown concentrations. The resultant set of equations has the matrix form of Equation 3.
I
The best values for the concentrations by the least squares criterion are then given by Equation 9 or 11, which here have the form
-C = (-k E)-1 - D
(23)
or c
Kith
0
L
2
3
W
Matrix can no\\ be calculated from available data on the absorptivities of the components a t whatever set of v-ave lengths is selected for the analysis. This calculation requires setting up the E matrix. elements of nhich are defined by Equation 19, multiplying this matrix by its transpose obtained by interchanging the r o m and columns of _E, and then finding the inverse. (4' $)-I, of the square product matris, E E . The matriy inversion is the only tedious step, and here involves inversion of a 4 x 4 matrix When the inverse matrix, E)-1, is obtained, it is to be postmultiplied by to give the desired 3 niatris. Calculations of Matrix .I1 = [ ( E E ) - ' E ] . The data used n-ere those of Shaw et al. (7, 8)and are tabulated in Table 11. Although these data were obtained in absolute ethyl alcohol, the spectra have been found, within the error of the determination, unaltered by change of solvent. They have been used in this work in isopropyl alcohol, and in later kinetic studies in n-hexane, 207, mineral oil in n-hexane, and 2053 glycerol in isopropyl alcohol. The niatriy inversion TT as performed for several different cornhinations of wave lengths in an attempt t o find the .E)-l@] = d i i c h would matrix give the best results in the calculation of the coinpositions of synthetic mixtures. The following choices of wave length were carried through the matrix inversion :
B,
B I
1
(s
[(E
0. Eight wave lengths-2500. 2600, 2660, 2700, 2750, 2800. 2850, and 2900 A. Data for this case were taken
88
ANALYTICAL CHEMISTRY
directly from enlarged plots of the figures from Shavi et nl. (8). For caSes 1 to 4, data were taken from large scale plots drawn from tabulated data furnished by Shaw (7‘). 1. Tn-elve iyave lengths at intervals of 40 A . from 2520 to 2960 A. 2. TTT elve wave lengths including the maxima and minima of the components of the m i x t u r e 4 . e . . 2500, 2600. 2630. 2650. 2715. 2720. 2760. 2790. 2800, 2820, 2895, and 2935 A. 3. Tiyelve 11ave lengths including points of intersection of the ergosterol curve with thosc of the other components and masima and minima of the other components-Le., 2500, 2550, 2600. 2650, 2663, 2720, 2790, 2800, 2820, 2839. 2895. and 3000 A. 4. Tn elve Ivave lengths including the masima and minima of ergosterol and points of intersection of the ergosterol absorption curve with curves of the other components-Le., 2500, 2550, 2630. 2663, 2715, 2760, 2820. 2839. 2895, 2912. 2935. and 3000 A. AIatriccsJi anddett.rminants1 ( & E ) - l ~ are prcwnted in Table I11 for cases 0 and 1. The matrix inversions were, with the exception of case 0, carried out on the AJISTIC Computer at Michigan State University. Applicability of Calculated Matrices. T h e calculated matrices, &, \\ere checked b y applying them t o spectral d a t a obtained on synthetic mixtures consisting of ergosterol, lumisterol2, a n d calciferol2 in isopropyl alcohol in varying propoi tions. These were the same solutions employed t o verify the Beer-Lambert-Bouguer law. Each of t h e matrices described above was applied t o six synthetic mixtures and t h e calculated compositions were compared n i t h t h e known values for the composition. T h e results a r e summarized in Table I V Although all five components were not prcwnt in the synthetic mixtures, matrix JJ n as calculated from ultraviolet absorption data for the five components. Therefore, compositions for components at zero concentration also serve to establish the validity of the coniputational procedure. An orcr-all standard deviation def i n d as
*
c
a
Table IV.
Calculated Composition of Synthetic Mixtures
Matrix Ma Component hIixture 1 Lumisterol2 Tachysterol2 Precalciferolz Calciferol? Ergosterol Mixture 2 Lumisterol? Tachysterol2 Precalcifero12 Calciferol2 Ergosterol Mixture 3 Lumisterol2 Tachysterol2 Precalcifero12 Calciferol2 Ergosterol Mixture 4 Lumisterol? Tachysterol2 Precalciferol? Calciferol? Ergosterol hlixture 5 Lumisterol2 Tachysterol? Precalciferolz Calciferol? Ergosterol Mixture 6 Lumisterol2 Tachysterol2 Precalciferol? Calciferol? Ergosterol a
Calcd. yo Composition of Synthetic hlixture 4 0 1 2 3 79.3 -0.4 -5.7 25.7 1.2
83.4 -0.2 -7.2 25.7 -1.7
80.0 -0.7 -7.8 26.2 2.3
81.3 -0.8 -8.3 26.4 1.4
56.4
60.1 -4.4 -12.6 52.1 3.9
50.9 -3.9 -7.7 50.1 10.5
53.1
-4.9 49.4 1.2
57.4 2.0 -4.0 48.1 0.5
45.3 -1.7 -15.1 80.2 -8.7
32.6 0.1 -4.3 76.5 -5.0
35.0 -0.8 -8.7 77.0 -2.6
51.4 -2.2
33.0 -1.5 -10.4 78.2
27.4
0.7
0
73.9 -2.2 -8.2 3.0 33.5
78.8 -1.0 -6.5 2.1 26.5
81.8
74.3
-8.0 2.2 25.0
25.7
42.2 -0.7 1.0
80.5 -0.9 -8.1
26.7 1.7
61.0 2.5 -7.1 50.0 -1.1
-2.1
-18.1
81.3 -12.4
77.3 0 0
22.7 0 0 0
46.9 0
0 0
72.6
-8.2 2.2 24.6
84.3 -1.1 -9.1 2.6 23.2
50.2 -0.8 -2.2 -0.5 53.3
47.6 -0.5 -0.4 -0.3 53.5
50.2 -0.6 -1.5 0.2 51.7
49.1
58.0
53.8 -1.6 -4.5 2.1 50.2
8.3 -1.7 -2.7 1.9 94.3
31.6 -0.6 -9.1 3.2 75.0
31.1 -0.7 -7.6 3.0 74.2
24.6
31.2 -0.5 -7.5 2.9 73.9
24.3
-0.5
82.5
-1.1
-0. I
-4.9
1.8
78.7
0 0 0
-1.0
0 0 0
50.9 0 0 0
75.7
See text for description of origin of matrices 0, 1, 2, 3, and 4.
mental data-because with the exception of the results obtained from matrix 3, the results obtained from the matrices based on 12 wave lengths were superior to those obtained from the matrix based on eight wave lengths. Of the 12 wave length matrices, matrix 1-which was based on equally spaced wave length intervals-yielded the best results. I t was originally believed that more significant information would be obtained by use of wave lengths a t which
calculated fraction of component - (known fraction of component i in z in mixture) mixture) ]a 30
-where index i refers t o a summation over the five components, c refers to a summation over the six mixtures. and 30 is the total number of “determinations”-v-aq employed as a basis for selecting the best matrix g. The values for the standard deviation are givcn in Table V. The results demonstrate the effectiveness of employing a larger number of ware lengths-Le., utilizing more experi-
Known % ’ Composition
the absorptivities of other components intersected that of ergosterol, because at these intersections the difference from the initial ergosterol absorption is attributable entirely to the nonintersecting components. However, matrices 3 and 4, which were based on the intersection points (plus other wave lengths) yielded results inferior to those obtained from matrix 1 (equal wave length intervals) and matrix 2 (based on
Table V. Standard Deviations O b tained for Each M_ Matrix
Matrix 0 1
2
3
4
Standard Deviation, yc of Component in Mixture fi i
f 38 A4 4 f 7 7 3Z-i 9
the maxima and minima of components). Apparently, this artificial weighting of the data leads to a loss of information. IIatrix 1, which was based on equal intervals of wave length in the significant region of the spectrum, was selected as the matrix capable of yielding the best results on the basis of the above comparison. This matrix, was applied to five more synthetic mixtures to establish the validity of the procedure further. Results obtained with matrix 0 are also included for comparison purposes, and the results are presented in Table VI. An “over-all” standard deviation was VOL. 32, NO. 1, JANUARY 1960
89
Table VI.
of Synthetic Mixtures
Component Lumisterol? Tachysterol2 Precalciferolz Calciferol2 Calculated Per Cent Comwosition
Matrix LZ Mixture 7 True comp. 0 1
Mixture 8 True comp. 0 1
3Iixtui-e 9 True comp. 0
1 Mixture 10 True comp. 0 1
Mixture 11 True comp. 0
1
Table
Calculated Composition
0 -8.9
8.2
0 -0 6 -1.0
0 1.8 0.9
-1.2 -0.5
0
0
0
0
46.0 48.9 48.4
54.0 55.2 54.5
0
71.9 79.1 77.4
28.1 22.1 26.5
30.2 34.3 34.3
35.5 53.4 37.9
17.7 19.0 17.6
62.3 75.2 70.8
-4.8 -3.2 -13.0
34.3 22.2 37.2
-3.0
0
0 -6.9
-1.8
-7.7
0
14.3 l5,6
-2.3
-0.8
of
Standard Deviation Individual Components
Matrix
Lumisterol2 Tachysterolz Precalciferolz Calciferol2 Ergosterol
77.9 85.6 73.4
1.7 -5.7
-2.4 -1.5
VII.
Component
22.1 22.3 25.1
0
14.3 4.1
‘0.0
0
1
Standard Deviation, % Component in llixture i103 ~ t 84 I 1 0 i l 2 i s 0 i 5 8 - 4 0
+32 +3 4
-10 1
Ergosterol
-6.4
0
-6.2 -3.1
The data presented in Table VI1 demonstrate further the improvement effected by the utilization of data from a large number of wave lengths. As one would expect, higher deviations were obtained for lumisterol?, precalcifero12, and ergosterol than for tachysterol2 and calciferol2 since the absorption curves of the former are similar. The lorn deviation obtained for tachysteroll is probably the result of its uniquely high absorptivity, rather than the absence of tachysterol? in the synthetic mixtures.
where A , is the absorbancy a t A, and Ai” is the initial (ergosterol) absorbancy a t this wave length. The values of X i t are the components of the matriv .Ij tabulated according to component i and wave length j . Several matrices were determined, using different wave length combinations; the best results obtained correspond to matrix 1 in Table 111, based on absorbancy data a t 12 evenly spaced wave lengths in the absorption region of the five components. K i t h this matrix, a standard deviation of 1 4 % in the percentage of each component was obtained. This accuracy has been found sufficient for use of the method in kinetic studies currently in progress in these lahoratories (9). The method is derclopcd in detail, and should be applicable t o the analysis of other coniplev mixtures. especially those involving coniponcnte with highly overlapping spectra. ACKNOWLEDGMENT
The authors express gratitude to Shaw, Glaxo Laboratories, Ltd., for furnishing purified samples and spectra. The assistance of Susann Brimmer in carrying out the calculations on the MISTIC computer is gratefully acknowledged.
W. H. C.
LITERATURE CITED
calculated for matrices 0 and 1 utilizing the data for 11 mixtures or 55 dcterminations. The values for the standard deviation are:
Matrix
2
0 1
Standard Deviation, 5 of Component in Mixture h7.6 +4.0
These results are comparable to those obtained by the use of the values from only six mixtures. I n addition, a standard deviation for individual components defined as
.&
(calculated fraction of component in mixture)
7 -
ANALYTICAL CHEMISTRY
A lrast squares matrix method has been employed as a n analytical curvefitting technique to provide analyses of the complex ergosterol irradiation mixtures using ultraviolet spectrophotometric data. The final expression obtainpd has the form of Equation 12:
where Ci is the weight fraction of the ith component in the sample (the ratio of
- (knownfraction of component in mixture)?
11
-where index c refers t o a summation over the 11 mixtures for a given component-was calculated for matrices 0 and 1. The data are summarized in Table VII.
90
SUMMARY A N D CONCLUSIONS
the TTeight of the component to the initial weight of ergosterol before irradiation). D, is the fractional change in absorbancy at wave length hi, and is given by
(1) Dwyer, P. S., Ann. Math. Stat. 15, 82 (1944). (2) Haupt, G. W.,J . Opt. SOP.Am. 42,
441 (1952).
hT..Ewing, G. W.,Iiriger. J.. J . Ana. Chem. Soc. tj7,609 (1945). (4) Kenney, J. F., Keeping, E. S., “hlathematics of Statistics,’’ Part 2, Chaw. X. Van Xostrand. S e x York. 195i. (5) Raoooldt. 11. P.. Keeterhof. P.. Han&ld, ’K. H., Buisman, J. A. K.: Kec. trav. chim. 77,241 (1958). (6) Yharpe, L. H., Ph.D. thesis, Michigan State University, 1957. ( 7 ) Shaw. W. H. C., Drivate comniunication. (8) Shaw, W.H. C., Jefferiee, J. P., Holt, T. E., Analyst 82, 2, 8 (1957). (9) Stillo, H. S., Ph.D. thesis, Michigan State University, August 1959. (10) Yates, R., Ph.D. thesis, Michigan State University, 1952.
1 3 ) Hiber.
> - ,
RECEIVEDfor review May 7, 1959. Accepted September 25, 1959. Work supported by a grant from the National Institutes of Health. From a thesis submitted to the graduate school of Michigan State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
