ANALYTICAL CHEMISTRY
690 assistance. They also wish t o thank H. J. Osterhof and the Goodyear Tire and Rubber Co. for permission to publish this work. LITERATURE CITED
(1) Barrer, R. M., Nature, 140,106 (1937). (2) Barrer, R. M., Rubber Chem. and Technol., 15, 537 (1942). (3) Craig, D., Davidson, W. L., and Juve, A. E., J. Polymer Sci., 6, 177 (1952). (4) Davis, C. D., and Blake, J. T., “Chemistry and Technology of Rubber,” Chap. VI, New York, Reinhold Publishing Corp., 1937. (5) Griin, F., Ezperientia, 3,490 (1947); Rubber Chem. and Technol., 22,316 (1949). (6) Hildebrand, J. H., and Scott, R. L., “Solubility of Nonelectrolytes,” p. 279, New York, Reinhold Publishing Corp., 1950.
(7) Kemp, A. R., Malm, F. S., and Stiratelli, B., Ind. Eng. Chem., 36, 109 (1944); Rubber Chem. and Technol., 17,693 (1944). (8) Kemp, A. R., Malm, F. S., Winspear, 0. G., and Stiratelli, B., Ind. Eng. Chem., 32, 1075 (1940); Rubber Chena. and Technol., 13,807 (1940). (9) Libby, W. F., ANAL.CHEM.,19, 2 (1947). (10) Morris, T.C., Ind. Eng. Chem., 24, 584 (1932). ~ ~ ,35 (1943); Rubber Chem. (11) Williams, I., India Rubber W O T 108, and Technol., 16,863 (1943). RECEIVED for review October 27, 1953. Accepted January 11, 1954. Presented before the Division of Rubber Chemistry a t the 124th Meeting of the AMERICAN CHEMICAL SOCIETY,Chicago, Ill., September 1953. Work performed as part of the research project sponsored by the Reconstruction Finance Corp , Office of Synthetic Rubber, in connection with the Government Synthetic Rubber Program. Contribution No. 201, Reeearch Labcratory, Goodyear Tire and Rubber Co.
Rectification of Nonlinear Beer’s l a w Plots Application to Alkaline Chromate and p-Nitrophenolate Solutions HARRY GOLDENBERG Department o f Biochemistry, Hillside Hospital, Glen
Oaks, N. Y.
Two colored solutions that nominally obey Beer’s law, alkaline potassium chromate and alkaline p-nitrophenol, yield nonlinear absorbance, A, us. concentration, C, plots with filter photometers and wide-bandwidth spectrophotometers. These systems have been studied to determine whether linear relations may be established by an appropriate choice of axes. Plots of A / C us. A are shown to be linear for chromate solutions in a Coleman junior spectrophotometer at 400 mp and in a Klett-Summerson colorimeter using a No. 42 filter. The nonlinear A-C data for the p-nitrophenolate system are likewise rectified by a n A / C us. A plot for the Klett colorimeter and for the Coleman instrument at 430 to 440 mfi. The adherence to linearity is sufficiently rigid to permit establishment of the relationship with as few as two to four points. For purposes of calculation without reference to a graph, concentrations are computed directly from the observed A 1 absorbancesby means of the formula C = K - A X & where Kand mare constants.
C
OLORED solutions which adhere to Beer’s law in tests with a sensitive spectrophotometer are occasionally observed to
deviate widely with filter photometers or the less expensive spectrophotometers in general use ( 1 , 4, 6). The difficulty is characteristic of instruments possessing a low degree of monochromaticity and arises when the absorption of the solution changes rapidly through the spectral band transmitted by the monochromator. This is particularly the case where the transmittance curves of the solution and of the monochromator are not inversely symmetrical. Under these circumstances a calibration curve relating absorbance to chromogen concentration cannot be established with one or two standards. Instead a sufficiently large number of standards must be processed to fix the curve throughout the entire range of values apt to be encountered in the analysis. The accuracy of such a procedure is limited by the analyst’s skill in tracing a curve to fit the experimental points. Furthermore, it becomes necessary to repeat all the standard determinations a t intervals to test the fidelity of the plot, especially when new batches of reagents are employed.
The need for a practical solution to this problem in the clinical laboratory has long been apparent, particularly in dealing with yellow-colored solutions. Two such systems are considered: alkaline potassium chromate, commonly employed as a spectrophotometric standard (3); and alkaline solutions of p-nitrophenol, an indicator whose liberation by the enzymic hydrolysis of colorless ester conjugates (as p-nitrophenyl sulfate and p-nitrophenyl phosphate) is used as a direct colorimetric index of enzyme activity. EXPERIMENTAL
p-Nitro hen01 (Eastman grade) was recrystallized twice from water a n 8 dried over sulfuric acid in vacuo. Stock aqueous solutions were prepared containing 0.025 to 0.5 micromole per 2 ml. The colored (or blank) solutions consisted of 2-ml. portions of the standard p-nitrophenol solutions (or water), 5 ml. of tungstic acid ( 7 ) , and 0.2 ml. of saturated potassium carbonate solution to bring the final pH in the neighborhood of 10. The potassium chromate standards, prepared directly from the C.P. grade without further purification, contained 0.5 to 10 mg. er 100 ml. of solution. Potassium hydroxide, 0.05N, serve! both as the solvent and color blank. Absorption measurements were made in a Model DU Beckman spectrophotometer a t 400 mp using 1-cm. Corex cells and a tungsten lamp; in a Model 6R Coleman junior spectrophotometer a t 400 to 440 mp with matched Kahn tubes (12 X 75 mm.) selected from the serology stock supply; and in several KlettSummerson photoelectric colorimeters with No. 42 filters using flat-bottomed (micro) test tubes. The Klett readings were converted to absorbances by multiplying by 0.002 (5). RESULTS
Absorption studies of both colored systems, conducted a t 400 to 440 mp, disclosed in each case a curvilinear response by the Klett-Summerson colorimeter as well as by the Coleman junior spectrophotometer, in contrast to the conipletely linear response of the Beckman spectrophotometer (Figures 1 and 2). These findings are consistent with the steep absorption spectra of the solutions. Attempts to rectify the aberrant Beer’s law curves by plotting the data on semilog and log-log graph paper were unsuccessful, although in the latter case linearity was obtained over several experimental points in the lower absorbance ranges. Subsequent treatment of the data, described below, has been more successful in eliciting a simple and appropriate choice of axes.
691
V O L U M E 26, NO. 4, A P R I L 1 9 5 4
as
c
i
CHROMATE
w
0 a6 2
a
m
a
$
0.4
I
m
a
0
2
0
6
4
CONCN.,
e
10
M G K 2 C r 0 4 PER IOOML
Figure 1. Conventional A us. C Plots for Alkaline Potassium Chromate Solutions
The nonlinear relationship between chromogen concentration,
C, and absorbance, A , may be represented with any desired degree of accuracy by assigning a sufficient number of terms to the general series
C
=
aA
+ 6.42 +
+ d-4‘ + e S 5 + . . .
cd3
(1)
relationships, the chromate absorbance and concentration data used to construct the curves in Figure 1 have been replotted in the appropriate forms (Figure 4). The curves are entirely linear. TTith the A/C cs. A plot, the Beckman results fall on a line parallel to the A axis, since the A/C values are constant according to Beer’s law: A / C = k , and the slope, -m, of Equation 6 is 0. On the other hand, because of the lag in increase of A with increasing C, the N e t t and Coleman data yield straight lines which regress with A . An example of the type of graph corresponding to Equation 7 is given in the inset to Figure 4. The nature of the reciprocal plot is such as to spread out the low absorbance-concentration points and cluster the high points along the line in the direction of the axes. Equation 7 therefore offers greatest sensitivity in the lower, rather than the higher, conrentration-absorbance ranges, but does not permit interpolations down to zero absorbance. This is possible with Equation 6, which has the further advant,gc of giving a mole uniform point spread and sensitivity over the entire range of absorbances. On this basis Equation 6 is to be preferred to Equation 7 in practice. Equation 6 has also been found to be applicable to the Klett p-nitrophpnolate data (Figure 5 ) . The results obtained with the Coleman spectrophotometer are more involved and have been studied in greater detail in the range 400 to 440 mp. With this instrunierit there is initial adherence to Beer’s law at 400, 420, and 430 m& as indicated by the horizontal lines in Figure 5 .
where the coefficients a,b,c,d,e,. . . .are constants. When Beer’s law isobeyed, 6 = c = d = e = . . . = 0, and C = aA. As a first approximation to the nonlinear plots we may retain the first and second power il terms and tentatively drop the others, yielding
.
C = aA
+ b d 2 = A ( a + bA)
(2)
+ bA
(3)
and
C/A = a
W
whence CIA would expectedly be a linear function of A Actually, of all the data tested in this way, a linear CIA us. A plot was noted only for the Klett chromate system. From this it can be deduced that the higher order terms in Equation 1 have considerable weighting value, and consequently cannot be ignored. h more satisfactory treatment is possible if additional terms are adopted in Equation I beyond the second and their ratios are assumed to be relatively constant and equal to the ratio of the first two terms. Then
0
z a m [L
0
m m 4
0
01
02
05
CONCN, ).lM
Figure 2.
04
(15
P E R 7.2 M L
Conventional A z’s. C Plots for Solutions of p-Nitrophenol in Carbonate Buffer
By this procedure the properties of a geometric progression are imputed to Equation 1, which seems to find justification in that the equations derived therefrom are in general accord with the data. Setting l / a = k and b/a2 = rti, the expression
c=-
k
A
- mA
(5)
is obtained. Its form is suggestive of the Beer relation, with the term mA a correction factor on the coefficient, k . Equation 5 may be rearranged to give
A/C = k
- mA
(6)
-m
(7)
or 1/C = k / A
According to Equations 6 and 7 , straight lines should be obtained on plotting AIC us. A (Figure 3) or l/C us. 1/A. To test these
ABSORBANCE
Figure 3.
A / C us. A Plot for Rectification of Nonlinear Beer’s Law Curves Extended line intercepts the A axis at k / m
692
ANALYTICAL CHEMISTRY
The concentration errors incurred by extrapolating this plot down to zero absorbance, indicated by the broken 0 line in Figure 5 , are 0.7, 2.0, and 3.3% 0 a t ahsorhances of 0.075, 0.050, and 0.025, respectively, which are equivaz 0 leut t,o absorbance errors of $0.001, the z 0 limit of reproducibility of the instrument. The choice of the 430 mp sett,ing W would t’herefore appear to give niaxi0 mum color sensitivity consistent, with z a simplicity of graphical construction arid m 4 a calculations. 0 tn b s to the practical use of Equations m 5 and 6, if the linear A / C us. A plot is a 2 shown to hold sat,isfact.orilythrough the U desired concentration range, it, may b e used as such in interpolating the con0 rentrations of “unknowns.” C is then equal to .4 divided by the corresponding ordinate value. Or one may alterna0 01 02 0.3 0.4 05 0.6 0.7 0.8 tively asceertain 6he values of k and in ABSORBANCE from the graph (Figure 3) or by direct Figure 4. .4/C t’s. A Plots of -4lkaline Chromate Data in Figure 1 calculation from the experimental data (described below), and use Equation Inset. Reciprocal plot of Coleman response a t 400 mfi 5 to determine the concentrations of unknowns from their corresponding As the concentration is increased, two breaks appear in the plots absorbances. I n this case the calculation is somewhat expedited at 400 and 420 nip, while only one is evident at 430 mp. At by putting Equation 5 in the form 440 mp the a / C z‘s. A plot is entirely linear. I n short, aa the n-ave-length setting recedes from the absorption maximum a t 400 mp, a shift occurs which is marked by a foreshortening of the incipient horizontal segment with a transition to the “normal” where K = k / m . It is then only necessary to subtract A from A / C us. -4plot at 440 mp. K and complete the calculation on a slide rule. The constants k , V I , and K can he computed directly from 6he absorbances, 8 , and A ? , of two standard solutions of concentraI I I I I I tions Ci and C‘? with t h e formulas IO
q6
I
p-
- 1 j
Figure 5 .
A,/C
z‘s.
I
I
NITROPHENOLATE
I
I
I
I
.4 Plots of p-Nitrophenolate Data in Figure 2
Additional Coleman data included at 420 and 430 m u
Although the Coleman setting a t 440 mp affords a straight 9 / C us. A line xvhich can be established with a minimum number of points, the use of this wave length limits the sensitivity of the measurements, since the p-nitrophenolate absorbances a t 440 mp are only 56 to 5’JmO of the maximum values a t 400 nip. ilt 430 mp, however, the absorbances rise to 73 t o i57c of the maximum; further, there is an exact adherence to a single linear A / C us. A plot over a t least a sevenfold range of concentrations-i.e., from under 0.075 to over 0.5 micromole of p-nitrophenolate ions per 7.2 ml. of solution, corresponding to absorbances of 0.09 to over 0.575 under the given experimental conditions.
These relations n-ere del ived by solving the simultaneous equations resulting from the substitution of Cl, A I and CP, A? for C‘ and A in Equation 5 . DISCUSSION
Ilthough the A / C us. .4 relationship is by no means a universal linear function, it may veiy likely hold for a number of solutions \Thich absorb in regions of the spectrum other than the violet. For example, preliminary Etudies indicate its applicability to the divergent data obtained with the ferrous-phenanthroline system using a Klett colorimeter and a No. 52 filter. No information is available, however, on its significance relative to colored solutions whose deviations from Beer’s law are of chemical origin. The results of the p-nitrophenolate studies with the Klett and Coleman instruments suggest that linear A / C us. A plots may be obtained more generally with filter photometers than with wide-band-width spectrophotometers a t the wave length of maximum absorption. I n the latter case, polyphasic plots may arise when there is initial adherence to Beer’s IaR a t low chromogen levels followed by deviations a t higher conrentrations.
V O L U M E 26, NO. 4, A P R I L 1 9 5 4 Applications Outside the Field of Colorimetry. The expressions presented herewith are not limited to colorimetry. Completely analogous relations have been discerned in several protein adsorption phenomena and in the quantitative t,reatment of enzyme-substrate systems n-hich do not show strict adherence t o zero-order kinetics ( 2 ) . Indeed, ti)- appropriate approximations with suitable geometric progressions, a number of linear equations may be derived for enzyme s>.stems acting in the zone of transition kinetics ( 2 ) . ACKNOWLEDGRTEST
Readings on the Nett-Summerson colorimeters were kindly takeii by AIorris Goldberg.
693 LITERATURE CITED
(1) (2) (3)
(4)
(5) (6) (7)
Ashley, s. E. Q., IND.ENG.C H E M . , . ~ N . A L .ED., 11, 72 (1939). Goldenberg, H., manuscript in preparation. Haupt, G. W., J . Research Ll”atZ.Bur. Standards, 48, 414 (1952). Hawk, P. B., Oser, B. L., and Summerson, R. H., “Practical Physiological Chemistry,” 12th ed., Chap. 23, Philadelphia, Blakiston Co., 1947. Klett Alanufacturing Co., Kew York, “C‘linical Manual for the Klett-Summerson Photoelectric Colorimeter.” Sandell. E. B., “Colorimetric Determination of Traces of Metals.” 2nd ed., pp. 67-9, Kew York, Interscience Publishers, 1950. Tan Plyke, D. D., and Hawkins, ,J. A , . J . B i d . Chem., 79, 739 (1928).
RECEIVED for review August 26, 1953. Accepted December 29,
1923.
Spectrophotometric Determination of Cobalt with 2-N itroso-1 -naphthol-4-sul fonic Acid WARREN M.WISE and WARREN W. BRANDT Department of Chemistry, Pordue University, Lafayette,
The investigation was undertahen to ascertain whether the color of the water-soluble red complex which is formed when cobalt(I1) reacts with 2-nitroso-1-naphthol-4-sulfonic acid could be emploj ed as the basis for a spectrophotometric method for the quantitati\e determination of the metal. The sjstem was found to obe) Beer’s law between the concentration limits of 4.00 X to 1.25 X VI cobalt(I1) and remain stable with respect to time and temperature. The effects of pH, reagent concentration, and di\ erse ions-chiefly iron(IlI), copper(II), and nickel(I1)-were studied. Procedures for effectively remo\ ing nickel(l1) and pre1 enting the interferences of concentrations of iron(Il1) or copper(l1) up to 1.00 x 10-3 M were developed. The method is conkenient, sensiti\e, reproducible, accurate, and precise.
Ind. operating on a band nidth of 10 mp and equipped with t n o matched 1-cm. cells was used for all spectrophotometric measurements. -2 Beckman LLodel 8 - 2 p H meter was utilized to indicate pH values. During the investigation the precipitates hich were formed, because of the presence of diverse ions, were separated from the solutions ith the aid of a centrifuge. The reagent, 2-nitroso-1-naphthol-4-sulfonic acid, was prepared and purified from 1-naphthol-+sulfonic acid (Sational Aniline), sodium nitrite, and hydrochloric acid according to published techniques ( 8 ) . The required amount of purified 2-nitroso-lnaphthol-4-sulfonic acid was dissolved in distilled water so that the reagent concentration was 1.00 X AI. A11 other solutions were prepared from reagent grade compounds and distilled water. (‘itrir acid was always utilized as the solid when it bqas used to complex diverse ions.
Table I. [Cobalt(II)
T
HE most popular analytical procedure that exists for determining minute amounts of cobalt is based on the spectrophotometric measurement of theamount of colorwhich is produced when cobalt(I1) reacts n.ith nitroso R salt ( 4 ) . This met,hod has been applied t o man? types of materials, including soils, grasses, steels, arid carbide3 ( 7 ) . However, if reproducible results are desired, appreciahle quantities of iron( 111)and copper(11) must first be removed (5). Also. the amount of time the solution is boiled after the reagent and nitric acid are added anti the filial concentration of the acid are important factors which can aft’ect the intenrity of the color ( 3 ) . Sii1c.e 2-nitroso-1-naphthol-&sulfonic arid has a chelating s5-rtc.m which is almost identical with the one possessed by the nitroso R salt, it seemed desirable t o investigate the known color re:iction of the former Ivith cobalt(I1). The reaction was first, mentioned by Hoffman ( 2 ) . Some 50 \-ears later Sarver (6) described the color reactions of the reagent with iron(III), iron(11). cobalt(II), nickel(II), and copper(I1). These experiments were conducted to develop qualitative $pot tests only, and no effort was made t o adapt them for use as spectrophotometric procedures. Consequently, an investigation was undertaken t o determine whether the color reaction of cobalt(I1) with the reagent could be utilized as the basis for a convenient, sensitive, and quant it ative spectrophotometric method. APP 4RATU S A\ D R E 4G EYT S
A General Electric automatic recording spectrophotometer
Effect of Ammonium Ion Concentration on Absorbance of the Systeni =
5.00 X 10-6M; hlolar Concn. of (”4) +
0.00 0.01 0.10 1.00
= 6.00 X 10-4.11; Absorbance of System a t 5 2 ; .\Ip
reagent
p H = 7.0,
0,720 0.660 0.610 0.600
EXPERIMENTAL
Effect of pH on System. Cobalt(I1) will react with 2-nitroso1-naphthol-4-sulfonic acid t,o produce a water-solutile red coniples. Figure 1 shows that a t 525 mp the absorbance of an aqueous solution of cobalt(11) and reagent, when measured against a reagent blank, increases as the p H is increased from 4.5 to 6.0; it then remains constant between 6.0 and 10.0. Figure 2 eshihits the effect of p H on the spectrophotometric properties of an aqueous solution of the reagent. I n the latter caw a distilled water blank was employed. Examination of Figure‘ 1 and 2 revealed that p H 7.0 is most suitable. At this value the absortmnce at 525 mw which is due to the cobalt complex is developed to a maximum and that of the reagent ir a minimum. Dilute sodium hydroxide instead of ammonium hydroxide was chosen for increasing the pH, for an increase in the concentration of the ammonium ion a t constant p H was found to cause a decrease in the absorbance of the system a t 525 mp (Table I). This effect can be attributed to the formation of complexes of cobalt and ammonia. Effect of Reagent Concentration. -4 stud!. \vas undertaken to esamine t,he effect of the concentration of the reagent on the
