The Ratio Method in Spectrophotometric Analyses JOHN A. PERRY, ROBERT G. SUTHERLAND, A N D NINA IIADDEN Monsanto Chemical Company, Texas City, Tex. A method has been developed for normalized spectrophotometric analyses of multicomponent mixtures in which only the mutual relationships of absorption coefficients are determined rather than the absolute values. The method is intrinsically rapid and accurate, and has been shown to effect a considerable saving in time and materials relative to older methods.
A
Y\- system of analysis of a mixture depends on differences
among the constituents. To determine the component distribution when all components are practically identical chemically, it is necessary to resort to physical methods of analysis such , . t ~mass spectrometry and absorption spectrophotometry. The practice of making spectrophotometric analysis depends on the accurate and precise knowledge of the coefficients of optical absorption involved in the analysis. These absorption coetficients are usually obtained by measuring several solutions of accurately known strengths for optical absorption at the wave lengths that have previously been selected for the analysis. I n order to know the strengths of these solutions, the operator weighs in the pure component separately for each solution. If Beer’s law is under test, several different strengths must be madc up. Once these data have been acquired, measurements on synthetics and unknowns may commence.
Table I. Ratios for m-Xylene 2613 A. 7.28 7.64 6.52 0.74897 0.74031 0.72444 0.73791
2709 A. 4.88 5.16 4.48 0.50206 0.50000 0,49778 0.49995
2723 A. 9.72 10.32 9.00 1,00000 1 ,00000 1 .OOOOO 1.00000
2742 A. 4.50 4.60 4.00 0.46296 0,44874 0,44444 0.45105
This standard approach to obtaining absorption coefficients has several disadvantages: I t is time-consuming. It involves a good deal of handling, which permits errors to be introduced. I t requires, for assurance of accuracy, preparation of two or several standard solutions. The absorption coefficients determined are unrelated except by the consistent accuracy and precision of the operator as he proceeds through a variety of independent operations. The ratio method permits calibration data to be bbtained not only more quickly and with less reliance on technique than more usual methods, but also with intrinsic accuracy. The basis of the method described depends on the fact that the absolute values of the absorption coefficients need not be known in order to effect a normalized multicomponent analysis. What is desired in such an analysis is to establish the concentration relationships of the various components making up the mixture to be analyzed. This can be done merely by establishing the correct relationships of the absorption coefficients, without regard to absolute values.
the optical densities of each pure component a t the wave len@lls selected for the analysis. Table I reproduces, for instance, the data sheet obtained for m-xylene. (Five significant figures are usually carried in order not to lose data in calculations on mntrices.) The column headings (Table I ) are the wavelengths in h g s t r o m units, The first three rows correspond to chart readings (which are ten times the optical densities) obtained from measuremrnts on rn-xylene. The solution of rn-uylene is made up to a desired concentration as follows: The drawn-out end of a capillary tube is dipped into the msylene, and subsequently dipped into the absorption cell which is already filled with solvent (n-heptane). The concentration of the resulting solution is then adjusted (by dilution with solvent, or by addition of more m-xylene) until the optical densities a t the measured wave lengths are between 0.4 and 1.0, This procedure is repeated until enough data to furnish a good average are obtained; here, three separate entries of sample are generally used. The second three rows show the readings in the first three rows expressed as ratios to the readings for one arbitrary wave length. The last row, being the average of the preceding three rows of ratios, is the object of the first set of operations. This method is used to obtain all the optical data. There are no standard solutions to make up; only a minute amount of the material is used; there is no call for highly skilled technique; the data are related; and the time required is about 10 to 15 ’ minutes per component or per mixture of components. Data similar to those shown in Table I are obtained for all the components in the analysis, and for as many synthetic samples as are desired to set up and check the analysis. These data are then arranged into a “raw” matrix, as in Table 11. For this demonstration, one synthetic was used as an “adjustment” synthetic. The Second Object. ARRIVINO AT CORRECT COLUYNRELATIONSHIPS. The raw matrix shown in Table I1 is thensolved, using the method of Crout ( 4 ) , and the answers are normalized. The percentages found and known are shown in Table 111,and the percentage deviations of the found from the known based on the known. The percentage deviations are used to change each figure in the corresponding columns by the amount and in the direction indicated. This correction will tend to establish the correct relation-
Table 11. Raw Matrix A. 2613 2709 2723 2742
SElTING UP THE ANALYSIS
Ethylbenzene 1.00000 0.15988 0.080349 0.037748
1,2-Xylene 1.18083 1,00000 0.71189 0,25457
1,3-Xylene 0.73791 0.49995 1,00000 0.45105
1,l-Xylene 0.46791 0.35648 0 28759 1.00000
Table 111. Results from Raw Matrix
The First Object. OBTAININGABSOF+PTIONRATIOS. An analysis for the xylenes and ethylbenzene, which was set up in this laboratory as a demonstration, can serve to illustrate the ratio technique. The first object is to measure quantitatively the relationships of
Compound Ethylbenzene l12-Xylene 1,3-Xylene 1,4-Xylene
1122
Found 16.38 14.90 20.78 47.94
Known 24.87 24.23 24.90 26.00
% Denstion -34.14 -38.51 -16.55 +84.38
Syn. L 1.59427 1.00000 1.03323 1.37168
V O L U M E 2 2 , NO. 9, S E P T E M B E R 1 9 5 0
1123 .
ships between the columns. The result of the first rorrection to the raw matrix is shown i n Table IV. This matrix, which converged with A. one adjustment, is uscd to treat the data from 2613 2708 synthetics L and 11. Table V presents these 2723 results. 2742 - ._ A perfect match was obtained for synthetic L, which was used as an adjustment synthetic; synthetic M was analyzcd its :in unknown, with an aversge absolute error of 0.57,. Had results froin both synthetics been used in the adjustment process, no perfect match would have been obtained, but the errors would have been halved. The ratio method can he summarized symbolically, using mat,rix algebra notation. Let a matrix be expressed by A, a column in that matrix by { A i 1, and a row by [Ai]. Let a matrix be formed with elements in the rows corresponding to optical densities at analytical wave lengths, and elements in the columns corresponding to optical densities for components. I n the raw matrix and in the final matrix, the internal relationships of the column elements remain unchanged:
Let a proportionality constant, Ki, be defined as
Ki = .f/i/fkni where f ~ i = fraction of component i found from normalization of answers from solution of raw matrix a n d f x = known fraction of component i. Then the final matrix is related to the raw matrix by the relationship
{Ailfinsi
Ki ( A i l raw
APPLICABILITY OF, TH.E RATIO METHOD
The criteria which are used to judge whether or not a given analysis can be done by the ratio method :Ire general and, for the most part, apply equally to any spec~tro~~liotometric multicomponent analysis. Obviously, the first requirement is that the syst.em to be analyzed absorb radiation. T h r ratio method, however, requires that all components obey Beer's law a t least to within experimental error, while other methods ran be set up which require only that components exhibit reproducible hehavior. Secondly, absorption curve shapes of t,lie componcnt.s t o be determined mu& show mutual differences *wfficient to allow the desired precision in results. Thirdly, the general level of light absorption of the various components a t their analytical wave lengths should hr as nearly alike as possible. These last two criteria ni'e matters o l choice and of dcgrce; they can be abused or ignored to the extcnt that the recipient of the results is willing to accept lower pntrision. Obviously, if two components have nearly identical absorption curves, no measurement depending on differences between the curvcs is going to differentiate between these compoiicnts. If the uncertainty of a measurement is a substantial fraction of the total absorption of one of the components, that component v:tn 'be determined only with vrry poor precision. ERROR
os.
COMI'ONEYT DISIIIIHIJTION
It is it general charitctcristic of :til!. iii(~;~surcnicnt tli:it for a given at~soluteerror, the relative r i w r increases inversely as thcs magnitude of the measurement. I t follows that relative errow iiicr(ww as the region near zero is :tppro:tclied. At timcs, for one rcabon or m o t h r r , such a s poor I(.vc,r:ix
