Errors in Ultraviolet and Visible Spectrophotometric Measurements Caused by Multiple Reflections in the Cell Robert W. Burnett Clinical Chemistry Laboratory, Hartford Hospital, Hartford, Conn. 06115 IN ORDER TO ACHIEVE high accuracy in spectrophotometric measurements, it is necessary to eliminate or correct for all sources of significant systematic bias in the measurement procedure. Multiple reflections within the cell constitute one such source of bias which is present in virtually all absorbance measurements. Although the existence of this effect was pointed out several years ago ( I ) , only a rough approximation of its magnitude was given. Since, as will be shown, the relative absorbance error due to multiple reflections may approach three parts per thousand, it is felt that a more rigorous discussion of this effect is warranted. When light passes from one medium to another, a fraction of the light is reflected at the interface. If the angle of incidence is zero, the Fresnel expression for the fraction reflected reduces to
In a spectrophotometric measurement, a fraction of the radiation reaching the detector has undergone an even number of reflections within the cell and has thus passed through the absorbing solution three or more times. It will be shown that this radiation gives rise to a systematic bias in the measured transmittance or absorbance which is not compensated by a reference measurement. Mention is made at this point of two phenomena in physical optics which have been ignored in the discussion to follow. First, to be strictly correct, the refractive index of a solution in the vicinity of an absorption band should be written as the complex refractive index n* = n - ik, where k = 2.3 EMXJ~T and E is the molar absorptivity, M is the molarity, and X is the wavelength in the absorbing medium. However, substituting typical values for e, M , and X shows that in the ultraviolet and visible regions, k is of the order of 10-5 or smaller, and it is thus permissible to neglect the imaginary term in the refractive index expression. Second, it may be noted that the parallel faces of an absorption cell constitute the essential elements of a Fabry-Perot interferometer. The spacing of the interference fringes in wavelength units is given by AX = X2/2nb, where b is the separation between the faces. Again, for a l-cm path length and wavelengths in the ultraviolet and visible, substitution shows that the fringe spacing ranges roughly from 0.002 nm to 0.02 nm, too close to be resolved by a conventional spectrophotometer. Interference effects may, therefore, be ignored in a discussion dealing only with ultraviolet-visible spectrophotometry; the topic is much more important in the infrared due to the longer wavelengths and much shorter pathlengths employed (2, 3). Figure 1 shows the primary ray and the six rays which undergo just two internal reflections. By tracing the rays ( I ) L. S . Goldring, R. C . Hawes, G. H. Hare, A. 0. Beckman, and M. E. Stickney, ANAL.CHEM., 25, 869 (1953). (2) J. U. White and W. M. Ward, ibid., 37, 268 (1965). (3) T. Fujiyama, J. Herrin, and B. L. Crawford, Jr., A p p / . Specrrosc., 24,
9 (1970).
further, it may be easily shown that there are 31 rays which undergo four internal reflections and 157 which undergo six internal reflections. Also given in Figure 1 are the transmittance factors for each of the six rays shown. These factors are the fractions of the primary ray intensity which are transmitted to the detector by the paths indicated. The factors which appear in Figure 1 are obtained simply by accounting for the fraction of light reflected at each interface traversed by the ray. Additionally, each time a ray passes through a solution of transmittance T , its intensity is reduced to T times the incident intensity (from the definition of transmittance). Using the primary ray in Figure 1 as an example, the incident intensity, Po,is reduced to (1 - fi)Poas a result of crossing the air-cell interface. Since the ray emerging from the cell has crossed the air-cell interface twice, the cell-solution interface twice, and the solution itself once, the emergent intensity is (1 - fi)"l - f2)'TP0. It is also important to recognize that the six reflected rays shown represent all possible reflection patterns, and higher order reflections simply involve some combination of these six paths. It follows that the intensity of any multiply reflected ray is given by a product of some number of the above six factors times the primary ray intensity. It is useful to define a function F(T) as the sum of the transmittance factors for each reflected ray under consideration. Thus if only the six reflected rays shown in Figure 1 are to be considered, then F(T)
=
2hfi
+ 2fif2(1 - f2)'T2+ h 2 T 2+ f i Y 1 - .bJ4T2 ( 2 )
Writting additional terms in F(T) to take into account higher orders of reflections is straightforward. Clearly the total light intensity reaching the detector is given by the sum of the primary ray intensity and the infinite series whose terms are the intensities of the individual multiply reflected rays. In order to determine the value to which this series converges, the partial sums have been calculated, first considering only rays undergoing two reflections, and then adding terms to take into account the rays which undergo four internal reflections. As will be seen, it is not necessary to add additional terms because of the rapid convergence of the error function. The relative absorbance error due to multiple reflections within the cell is defined as
6 =
Aspparent
-A
A
where A is the solution absorbance, and an expression for 6 is derived below. The photon flux emergent from a cell filled with an absorbing solution may be written
Similarly the photon flux emergent from a reference cell filled with pure solvent is Psolvent =
since T
=
Po(1
- fi>'(1 - JJ2[1
+ F(1)1
1 in this cell.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973
383
Table I. Relative Absorbance Error (parts per thousand) Due to Multiple Reflections in Aqueous Solutions Absorbance
220
243
257
308
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
3.72 3.33 3 .oo 2.72 2.47 2.25 2.06 1.90 1.75 1.62 1.51 1.41 1.32 1.24 1.17 1.10 1.04 0.99 0.94 0.89 0.85 0.82 0.78 0.75 0.72 0.69 0.67 0.64 0.62 0.60 0.58 0.56 0.55 0.53 0.52 0.50 0.49 0.47 0.46 0.45
3.40 3.05 2.75 2.48 2.26 2.06 1.89 1.74 1.60 1.49 1.38 1.29 1.21 1.13 1.07 1.01 0.95 0.90 0.86 0.82 0.78 0.75 0.71 0.68 0.66 0.63 0.61 0.59 0.57 0.55 0.53 0.52 0.50 0.49 0.47 0.46 0.45 0.43 0.42 0.41
3.22 2.89 2.60 2.35 2.14 1.95 1.79 1.64 1.52 1.41 1.31 1.22 1.14 1.07 1.01 0.95 0.90 0.85 0.81 0.77 0.74 0.71 0.68 0.65 0.62 0.60 0.58 0.56 0.54 0.52 0.50 0.49 0.47 0.46 0.45 0.43 0.42 0.41 0.40 0.39
2.86 2.57 2.31 2.09 1.90 1.73 1.59 1.46 1.35 1.25 1.16 1.09 1.02 0.95 0.90 0.85 0.80 0.76 0.72 0.69 0.66 0.63 0.60 0.58 0.55 0.53 0.51 0.50 0.48 0.46 0.45 0.43 0.42 0.41 0.40 0.39 0.38 0.37 0.36 0.35
0.80
0.85 0.90 0.95 1 .oo 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00
Wavelength (nm) 340 397
468
546
656
768
2.76 2.47 2.23 2.02 1.83 1.67 1.53 1.41 1.30 1.21 1.12 1.05 0.98 0.92 0.86 0.82 0.77 0.73 0.70 0.66 0.63 0.60 0.58 0.56 0.53 0.51 0.49 0.48 0.46 0.45 0.43 0.42 0.41 0.39 0.38 0.37 0.36 0.35 0.34 0.33
2.49 2.24 2.01 1.82 1.66 1.51 1.39 1.27 1.18 1.09 1.01 0.95 0.88 0.83 0.78 0.74 0.70 0.66 0.63 0.60 0.57 0.55 0.52 0.50 0.48 0.46 0.45 0.43 0.42 0.40 0.39 0.38 0.37 0.36 0.35 0.34 0.33 0.32 0.31 0.30
2.43 2.18 1.96 1.78 1.61 1.47 1.35 1.24 1.15 1.06 0.99 0.92 0.86 0.81 0.76 0.72 0.68 0.65 0.61 0.58 0.56 0.53 0.51 0.49 0.47 0.45 0.44 0.42 0.41 0.39 0.38 0.37 0.36 0.35 0.34 0.33 0.32 0.31 0.30 0.29
2.36 2.12 1.91 1.73 1.57 1.43 1.31 1.21 1.11 1.03 0.96 0.90 0.84 0.79 0.74 0.70 0.66 0.63 0.60 0.57 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.41 0.40 0.38 0.37 0.36 0.35 0.34 0.33 0.32 0.31 0.30 0.29 0.29
2.33 2.09 1.89 1.71 1.55 1.42 1.30 1.19 1.10 1.02 0.95 0.89 0.83 0.78 0.73 0.69 0.65 0.62 0.59 0.56 0.54 0.51 0.49 0.47 0.45 0.44 0.42 0.40 0.39 0.38 0.37 0.35 0.34 0.33 0.32 0.32 0.31
2.62 2.35 2.12 1.91 1.74 1.59 1.45 1.34 1.23 1.14 1.06 0.99 0.93 0.87 0.82 0.77 0.73 0.69 0.66 0.63 0.60 0.57 0.55 0.53 0.51 0.49 0.47 0.45 0.44 0.42 0.41 0.40 a.38 0.37 0.36 0.35 0.34 0.33 0.33 0.32
Figure 1. Schematic diagram showing the primary ray and the six doubly-reflected rays formed when a light beam enters a spectrophotometer cell. See text for explanation of transmittance factors
solution 384
ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973
air
0.30
0.29 0.28
The apparent transmittance of the solution is
1 1
- - -Peolution =T(-) Tapparent Psolvent
AT T 1 , Equation 3 may be rewritten as
Since Aapparent- A
(Tspyent
+ F(T) + F(1)
)
AA
=
=
AA
-0.434
=
(3)
=
-0.434
0.434 (F!%,”f’)
and
( : =a
0434 F(1) - F(T) A F(1) 1
+
)
(4)
The calculation of the relative absorbance error has been carried out for aqueous solutions at 25 “ C in quartz cells and the results are tabulated in Table I. It has been assumed for the calculation that the refractive index of the absorbing solution does not differ appreciably from that of pure water.
Values given in Table I were calculated with F(T) including terms for the thirty-one rays which undergo four reflections. However, if only the six doubly-reflected rays are considered, the calculated 6’s differ from those given in Table I by less than 0.5%. It is apparent then, that F(T) converges quite rapidly, and no practical advantage is gained by including more than the first six terms. In summary, all conventional absorbance measurements include a systematic bias due to multiple reflections within the cell. The magnitude of this bias is easily and accurately calculated from the refractive indices of the solution and the cell at the wavelength of interest. The appropriate refractive indices are used in Equation 1 to calculate reflection coefficients, which are substituted in Equation 2 along with the solution transmittance. Equation 4 may then be evaluated to find the relative absorbance error. The error commonly amounts to 1 to 2 parts per thousand for aqueous solutions and will be slightly smaller for most organic solvents. RECEIVED for review June 12, 1972. Accepted September 15, 1972.
Mercuric Iodate as an Analytical Reagent. Spectrophotometric Determination of Certain Anions by an Amplification Procedure Employing the Linear Starch Iodine System Willie L. Hinze and Ray E. Humphrey Department of Chemistry, Sam Houston State University, Huntsville, Texas 77340 SEVERAL RELATIVELY INSOLUBLE metal iodates have been used for the determination of anions which form a less soluble or undissociated compound with the metal ion involved. The anion being measured is “traded” for the iodate ion which is released into the solution. Iodide ion and acid are added to yield triiodide ion. These reactions are shown in the equations below for the simplest case.
+ X- MX + + 81- + 6H+ 313- + 3H10
MI03 103-
+
103-
(1)
+
The triiodide formed is then either titrated with standard sodium thiosulfate solution, the absorption determined, or the starch-iodine complex measured spectrophotometrically. The procedure has been called an “amplification” reaction when a titration is involved as 6 iodine atoms are produced for each monovalent anion exchanged ( I ) . Chloride has been determined titrimetrically in this way using either mercuric or mercurous iodate ( I ) , or silver iodate ( I , 2 ) , fluoride has been determined with calcium iodate (3) while barium iodate has been applied to the measurement of sulfate (4, 5). Apparently chloride is the only anion which has been determined Belcher and R. Goulden, Mikrochim. Acta, 1953,290. (2) J. Sendroy. Jr., J . Biol. Cliem., 120,405 (1937). (3) W. I. Awad, S. S. M. Hassan, and M. B. Elayes, Mikrochim. Acta, 1969,688. (4) D. A Webb, J . Exp. Biol., 16,438 (1939). (5) J. L. Lambert and D. J. Manzo, Anal. Cl7im. Acta, 54, 530 (1) R.
(1971).
spectrophotometrically employing a metal iodate. Silver iodate was used and the absorption of the triiodide (6, 7) or the starch iodine complex measured (7, 8). In this work, the absorption of the starch-iodine complex was employed for the spectrophotometric determination of bromide, chloride, cyanide, iodide, sulfide, sulfite? thiocyanate, and thiosulfate. These methods are about as sensitive as any which have been reported for most of these anions and have higher sensitivity for some. EXPERIMENTAL
Apparatus. Absorption measurements were made with Beckman ACTA 111, DB-G, and DK-2A spectrophotometers. An International clinical centrifuge model CL and International Model SBV Centrifuge were used to separate the mercuric iodate from the solutions. Reagents. Mercuric iodate was obtained from City Chemical Corp., New York, N.Y. A mortar and pestle was used to reduce the chemical to a fine powder. Compounds used to provide the anions studied were Baker and Adarnson or Baker analyzed reagent grade materials. The starch iodide reagent was prepared using Baker analyzed cadmium iodide and Fisher Scientific Co. potato starch. Stock solutions of the various anions were in the range of 0.01 to 0.001M using 1 :1 ethanol-water as solvent. Sulfite solutions contained 5 glycerol to retard oxidation. (6) F. L. Rodkey and J. Sendroy. Jr., Clii7. Clienz., 9, 668 (1963). (7) J. Sendroy, Jr., J . Biol. Chem., 120,419 (1937). 27,444 (1955). (8) J. L. Lambert and S. Y . Yasuda, ANAL.CHEM.,
ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973
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