ANALYTICAL CHEMISTRY
268 ing check of the peaks to be identified against a number of mass spectra. ACKNOW LEDG.M E Y T
The constructive suggestions offered by H. W. Washburn and A. P. Gifford were most helpful and are much appreciated. LITERATURE CITED
(1) Smerican Petroleum Institute Research Project 44, Natl. Bur.
Standards, “Catalog of Mass Spectral Data.” Sets of spectra may be obtained from: D. V. Stroop, American Petroleum Institute, 50 West 50th St., New York 20, N. Y. (2) Beeck, O., Otvos, J. W., Stevenson, D. P., and Wagner, C. D., J . Chem. Phus., 16, No. 3, 255 (1948). (3) Bloom, E. G., Mohler, F. L., Lengel, J. H., and Wise, E. C., J . Research Natl. Bur. Standards. 40, 437 (1948). (4) Ibid., 41, 129 (1948). (5) Bloom, E. G., Mohler, F. L., Wise, E. C., and Wells, E. J., Ibid., 43, 65 (1949).
(6) Brown, R. A., Taylor, R. C., Melpolder, F. W., and Young, W. S., ANAL.CHEM.,20, 5 (1948). (7) Gifford, A. P., Rock, S. M., and Comaford. D. J.. Ibid., 21, 1026 (1949).
(8) Haagen-Smit, A. J., Redemann, C. T., and Miroo, N. T., J . Am. Chem. Soc., 69, 2014 (1947). (9) Honig, R. E., private communication. (10) Langer, Alois, and Fox, R. E., ANAL.CHEM., 21, 9 (1949). (11) Seaborg, G. T., and Perlman, I., Rev. Modern Phys., 20, 585 (1948). (12) Shepherd, M., ANAL.CHEM., 19, 635 (1947). (13) Starr, C. E., Jr., and Lane, Trent, Ibid., 21, 572 (1949). (14) Taylor, R. C., Brown, R. A,, Young, W. S.,and Headington, C. E., Ibid., 20, 396 (1948). (15) Thomas, B. W., and Seyfried, W. D., Ibid., 21, 9 (1949). (16) Washburn, H. W., Wiley, H. F., Rock, S. M., and Berry, C. E., IND.ENG.CHEM., ANAL.ED.,17, 74 (1945). (17) Wiley, H. F., and Berry, C. E., “Mass Spectrometry” in “Mod-
ern Instrumental Analysis,” D. F. Boltz, ed., Ann Arbor, Mich., Edwards Brothers, 1949. RECEIVED March 7, 1950. Presented in part before the Divisionof Analytical and Micro Chemistry at t h e 112th Meeting of the . ~ X E R I C A N CHEMICAL S O C I E T Y , New York, N. Y .
Effect of Finite Slit Width on Infrared Absorption Measurements A. R. PHILPOTTS, WILLIAM THAIN, AND P. G. SMITH The Distillers Co., Ltd., Great Burgh, Epsom, Surrey, England Extinction coefficients in the infrared vary with the resolving powelci.e., they depend on the optical arrangement of a spectrometer and on the slit width used. This work attempts to find practically and (within the limits of the assumptions made) theoretically the extent of this variation. Plots of absorption against concentration for solutions of three hydrocarbons were made at different slit settings, using a Perkin-Elmer Model 12B spectrometer. The variations in slope and the deviations from linearity were found to fit (formally at least) the theoretical explanation given. The magnitude of the effect shows that spectroscopists must be very careful when using extinction coefficients determined under optical conditions not identical with those of the analysis. A method of correlating data taken at different slit settings on the same spectrometer in the same state of optical adjustment is given. It is hoped that the correction of extinction coefficients to “infinite resolving power” suggested will enable extinction coefficient measurements to be used on all spectrometers.
E
STIMATIOXS by absorption spectrophotometry are greatly simplified when deviations from the Beer-Lambert law are smaller than experimental error, especially when multicomponent mixtures can be analyzed by the method of solving linear simultaneous equations. The dependence of extinction coefficients on instrumental conditions (a great difficulty of infrared spectroscopy) is also bound up with the applicability of the law. Much work has therefore been devoted t o the investigation of deviations. Failures of the law due t o the inherent properties-e.g., intermolecular forces, etc.-of the sample in question are t o some extent unavoidable, but it should be possible t o correct for or a t least calculate the magnitude of instrumental limitations. These chiefly concern scattered radiation in the monochromator and the effect of finite slit width. Methods of dealing with the first have been reported (4). While attention (6, 10) has recently been drawn t o the second, and the general principles (3,7) have been laid down, there has been no discussion of the magnitude of the errors caused when conventional infrared spectrometers are used.
The case of the area of absorption bands has been considered (8, 11). While this manuscript was in the course of preparation, applications to ultraviolet problems were published b y Eberhardt ( 8 ) . Whether single- or double-beam spectroscopy is employed, the observed density is always the logarithm of the ratio of incident t o transmitted intensity, each intensity being integrated over the pass band of the monochromator. The error is introduced by assuming that the density obtained in this way is identical with the true density at the central wave length. The magnitude of the error at the maximum of an absorption band depends on the relation of the pass band width t o the true shape of the band. It is very difficult t o obtain the shape of a band at infinite resolving power and the pass band of a monochromator depends on such indefinite quantities as image aberration and line-up, as well.as the (possibly) calculable geometric slit width, diffraction effect, and image curvature. The best method of procedure seems t o be: Calculation of errors in terms of an idealized absorption band and spectrometer response
,
V O L U M E 23, NO. 2, F E B R U A R Y 1 9 5 1
269
Evaluation of parameters in this mathematical analysis by comparison with observed deviation from Beer’s law Assessment of the value of the analysis by comparison of the parameters with observable quantities
If lois constant over the range XI * R, the first term cm be integrated and the expression reduces to
MATHEMATICAL ANALYSIS OF IDEAL CASE
The Beer-Lambert law is usually stated in the form
z = Io 10-d where IO = intensitv of incident radiation, I = intensity of transmitted radiation, and d = optical density. (The term “optical density” is used rather than “absorbance” because both logarithmic bases are employed.) This statement is used in the discussion of experimental data in the next section, but for ease of manipulation let us use here the natural logarithm, so that Z
=
Ii e-*‘
where 0.4313 d’ = d. The transmission function of a monochromator depends on a number of calculable and incalculable factors. I t is therefore proposed in this exploratory survey to use the simplest form (5)-namely, that this function is constant between two limiting wave lengths (determined by the wave length and slit settings of the spectrometer) and is zero outside these.
This integral can be evaluated only if some relationship between d ’ and is known. I t has been shown (9) that in solution single absorption bands approximate to a simple error function in shape even a t high resolving power. Let us therefore assume that at infmitr resolving power d ’ = d‘,,,e-
’ 5 (4 ?,-A$
Here d’,G is the optical density a t the madmum and 1 is the quantity which determines the “peakedness” of the band. In fact, 21 is the width of the band when d’ = 0.6064 dfm-i.e., the distance between the points of infleetioil-and is used here as a measure of band width rather than the more conventional width when d’ = 0.5 d’, (the “half band width”).
To simplify,
let z =
x - xo 1
~
s
and
let
=1=
equivalent slit width band width
0.20
Expanding the exponentials again, multiplying out, and collecting, d’m 2 x2
exp
+
dlm(dln2- 3d’m 48
+ 1)
58.
..
*
..
I
Integrating Equation 1 and using the first five terms in the expansion dlobsd.= d’, - log d’,(d’,2
lo
0 2 1
Figure 1.
tNSTRUMENTAL DENSITY 0.3 0.3 0..4 0;5 0..6
+ ds n2 + d’,(d’, 4o - 1)n4 + ’m
+ 1)nB
+
d’,(dfm3 - Gd’,’ 2688
drobsd. = d’m
0;7
-
Graphs of Calculated Error [0.4343 log
(a)] against Instrumental Density (0.4343 log (z)] } for Values of n Shown
- 3d’, 336
1
[dm
Suppose the monochromator is set a t h~ and passes the wave band between XO - s and ho s where 2s is defined as the “equivalent slit width.” If the monochromator is used to measure the o tical density of a material with a maximum absorption a t XO, t f e observed value, dlobsd.,is then given by
+
Let the true,density (at infinite resolving power) a t any wave length, h, be d
- log ( 2 )
r ‘
7d’m - 1) (2)
From Equation 2 the true density can be calculated for any observed value if the ratio, n, is known. The convergence of the series depends on the magnitudes of both n and dim. It was found that with ?z > 1.55 the series did not converge rapidly enough for densities of practical interest. The best method of handling the rather formidable expression proved t o be the plotting of the calculated error, 0.4343 log (z), against the corresponding instrumental density, dobsd. = 0.4343 [d‘, - log ( z ) ] , for values of n. Some examples are shown in Figure 1, though in practice a large scale graph with more n values was used. It is believed that even with modern infrared spectrometers n is rarely less than 1 and often much greater. Table I shows examples of the calculated magnitude of the effect at typical values of n.
It can be seen that a Beer’s law curve determined when n = 1 will give a nearly straight plot with slope about 15% less than the true calibration coefficient, while one determined with n = 1.414 will have appreciable curvature.
ANALYTICAL CHEMISTRY -___
Table 1.
Calculated Density
0.217
Instrumental Density n=l n = 1.414 0.185 0.160
0.304 0.434 0.565 0.651 0.869
0.258
0.222
0.368 0.478 0.550 0.730
0.315 0.408 0.468
True Density
~~
~
0.617
/
Scattered Radiation. If the scattered radiation at XO i- n n t absorbed a t its true wave lengths h y the test substance, thr method of Hogness et al. gives an accurate correction. I t IC necessary therefore t o worh with the strongest band in the spcctrum of the test substance. Chemical Effects. T o prevent deviations due to interniolccular effects, the test substance must, be non-polar and inert solvents must be used. Solvents. It is preferable to work in solution, so that the opbical density can bc, varied by varying the concentration. This variation means an alteration in the amount of solvent in the radiation path which \vi11 influence the shape of the densityconcentration curve unless the qolutions are very dilute or the solvent is completely transparent. Shape of Band. The absorption hand used must have no obvious asymmetry, in order to give the theory a reasonable test.
/
-0.8
/ / /
0.15 0.31
.-.-. D--8-0
These conditions (togother with availability of materials) h i i t the possible bands to 6 C-II frrquencies iri hydrocarbons and the solvents to cyclohexane and earlmi tetrachloride. Cyclohexane is particularly suitah]?, bcmusc' it, is completely transparent at long wave lengths. 1Iwsurcments w r e made using the 913 c'm. --I band of n-1-decenr, t hr 829 cin. hand of p-diisopropylberizenr, and the 720 em.-' band of ct~tarie. Some experiments w r e carried out with the 728 c i n - * tolur~nr~ band, but it proved to be too narrow (with the rvsolviiig po\vc'r available) t o give niariagc7:il)lt: values of n.
/
(mm)
SLIT WIDTH
/
/ /
07
(md
SLIT WIDTH b c - 0 0.17 .-.-e
16
lo
,I
.4
,3
,2
.5
C-0-0
/
/ /
0.26 033
/
/ /
.7
,6
Figure 2. Experimental Calibration Lines for 913 C m . - ' Band of n-1-Decene a t Different Slit Widths Broken line, corrected calibration line /
The limiting cases when n is small and when the density is small are of interest. When n is small. d'obed.
E d'm
_ -d '6m
nz
=
df, (1 -
%) $)
When d', is small. 1
+ d', (f
n4 n6 - 40 +336
n8 .. 2688' __
.)1
Multiplying by 0.4343 and dividing by the concentration : Slope of tangent at origin to observed curve =
(
true calibrationcoefficient X 1
- - + - - - + --
Fi
$;
p-DIISOPROPYLBENZENE "/v ,I
Multiplying by 0.1343 and dividing b y the concentration : Observed calibration coefficient = true calibration coefficient X (1 (3)
d'obsd. = dfVz- log
1
2688"
'
EXPERlM ENTA L VERIFICATION
I n order t o reduce deviations from the Beer-Lambert law due t o effects other than the one under discussion, careful choice of instrumental conditions and of test substances was required.
,2
p
Figure 3. Experimental Calibration Lines for 829 Cm.-' Band of p-Diisopropyl Benzene at Different Slit Widths Broken line. corrected calibration line
Experimental Procedure. .4 Perkin-Elmer 12B infrared spect,rometer was used. The attenuator system y a s found extremely useful t o fulfill the double condition of constancy of slit setting and constancy of IO. The same rock salt cell (approximately 0.1 mm. thick) was used for all experiments. -1series of solutions was made up in each case and short sections of the spectra were run alternately with solution and solvent in the cell. Zeros were taken at the beginning and end of each section nith a lithium fluoride shutter. The scattered radiation correction was found by observing the fraction of incident radiation passed by the shutter but not by a strong enough solution to give an ap reci ab1 flat-topped band. This condition could not be fuffilled wit1 cetane, so the correction was a:.isumed to he the same a t 720 cm.-' as a t 728 em.-', where it TTas determined with tolutwr. B - 4 The density was calculated as log -\There B is the disT - S tance between trace and zero a t the wave length required with thr solvent in the cell, T the distance with solution in the cell, ant1 S the scattered radiation correction. A mean value for B was obtained from readings hefore and after the nieawrrmrnt of 7'.
V O L U M E 23, NO. 2, F E B R U A R Y 1 9 5 1
p"
27 1
origin of the curves in question with the corrected calibration coefficient (see Equation 4). A summary of the results is given as Table 11. In order to show the consistency of the results and also to demonstrate the curvature of the observed plots, the value of observed density divided by concentration is given for all determinat,ions on cetane in Figure 5, The variation of observed calibration coefficient with concentration (and therefore wibh density) and with slit width is apparent. The top part of Figure 5 shows how the calibration coefficient aftrr correction is constant m-ithin experimental error. The values of t,he standard deviation in Table I1 are worthy of note. The high values for the first set of points in the case of p diisopropylbenzene and of cetane are due to the unsteadiness of the recorded spectra at maximum amplifier gain. The last set for t,liese two substances also shov s high deviation, since the applied correction is high which emphasizes errors in determination--a high r:ilue receiving a high correction and B low value a low one.
SLIT WlDTH(mm) x-x-x 0.34
.OS
Q-D--Q
.-.-.
080 1.30
0.5
a-0-0
200
Y
CETANE '1" 2p 25
'1'0
5
0
IO
1.s
3.0
; 3 X
Figlire 1.. Experimental (Lilibration Lines for 720 C m - 1 Band of Cetane at Different Slit Widths Broken line, corrected ralibration line
The e~perinientalBeer's law plots a t various slit widths are shown in Figures 2 to 4. flaving obtained a series of observed densities for one slit sc.ttirig, it was possible t o find n by applying corrections such as tliost' i n Figure 1. Corrections from curves for different values of n \$vre applied in turn until, a t a particular value of n,the quandobad. 0.4343 log ( 2 ) was constant within experimental tity C t'ri or for all concentrations-Le., until the corrected density gave a IIIW:II Beer's law plot. I t was found that the Etraight lines obtained by correcting the (wrves obtained a t different slit widths for the same substance h:id the same gradient, in accordance with the theory. This "corrected calibration line" is indicated on Figures 2 t o 4 as a t)rokeIi line. When n is small the experimental plot is linear and the value of n can be obtained only if the corrected calibration c-ocffic.ienthas been found using wider slits. In that case the ratio
poi9
r
I X
+
of c,\pc,riniental to corrected calibration coefficient is
- to (1 - 3
1 1)y Equation 3. This method was, in fact, used to find the Ion value of n for cetane, and the figure is only approximate because the ratio is nearly unity. The high values of n were determined iippro\imately by comparing the slopes of the tangents at the
0 Table 11. Slit Width,
Substance n-1-Decene, cm. -1
llm.
913
p-Diisopropyl benzerie, 829 cin. - 1 Cetane, 720ctn.-1
0.11 0.13 0.31 0.50 0.17 0.26 0.33 0.265 0.80 1.30 2.00
Summary of Results ?io. of Obserrations
3 16 17 18 16 13 16
12 15 12
12
rt
Ca. 1 . 0 7 1.15 1.41 Ca. 1 . 8 3 1.23 1.50 Ca. 1 . 8 Ca. 0 3 4 0.85 1 .20 1. 5 3
0.'1'107 0.1109
0.0016 0.0014
0.2286 0.2296
0,0079 0.0041
0.02034 0.02031 0.02032 0.02032
0.00061 0.00024 0.00026 0.00035
...
'I,
IS 20 CETANE "1"
25
so
35
Calibration Coefficient
Constant calibration coefficient obtained by correcting observed density before evaluating coefficient Lomer. Variation of calibration coefficient (observed density 1 concentration) with slit width and concentration for 720 om. - 1 band of cetane t'pper.
MEANING OF OBSERVED VALUES OF
F o r n-1-decene and p-diisopropyl benzene, 1 means 1 ml. per 100 ml. of solution, and for cetane 1 gram per 100 r i d . of s o l i ~ t i o n . a
IO
Figure 5.
Corrected Calibration Coefficient Mean Standard (density deviation per 1%)"
._.
5
n
Having found that the experimental points agree forriially with Equation 1, the physical significance of the parameters found (d, and n) must be considered. Ideally, the bands should now be remeasured, using a very high resolving power iiistrument---t..g., a grating spectrometer-and d, and the band width 21 found directly. As neither the instrument nor the data are avail:tt)lc, :L less direct approach must be used.
ANALYTICAL CHEMISTRY
272 -
Table 111.
Substance n-1-Decene, 913 cm-1
p-Diisopropyl benzene 827 c m . - J i Cetane, 720 c m - 1
Slit Width, Mm. 0.11 0.15 0.31 0.50 0.17 0.26 0.33 0.265 0.80 1.30 2.00
Slit Width Band Width n
Ca. 1.07 1.15 1.41 Ca. 1.85 1.25 1.50 Ca. 1.8 Ca. 0.34 0.85 1.20 1.53
~m.-i 6.2
2.8 9.5
Equivalent Slit Width,
Cn-1 6.6 7.1 8.7 11.5
Effective Slit Width
3.5
Cm.-i 2.3 2.8 4.8 7.5 2.3
4.2
3.1
5.0
3.8 1.7
3.2 8.1 11.4 14.6
4.1
6.4 9.7
The area under a density curve for a single band is approximately independent of slit width ( 7 ) and the area per unit concentration is approximately independent of concentration (8). The area per unit concentration using a fairly narrow slit therefore gives a measure of the area per unit concentration a t infinite resolving power (IO). The area under the curve is X height X I where 1 is half the distance between the points of inflectioni.e., half the band width a t 0.6064 of maximum height. It follows density X band width is constant and approxithat the quantity concentration mately independent of resolving power. As we already know density for the ideal case, the value of the constant from Concentration observed data will enable us t o find 21 a t infinite resolving power. Four widely distributed points on each of the observed densityconcentration curves were taken and the actual band widths a t 0.6064 dobad.were measured. Both the 0.15- and 0.30-mm. slit width curves for n-1-decene gave 0.69 as the mean of four values for the constant (in density times wave numbers per 1%). This implies 6.2 cm.-’ for the true band width, 21. Similarly, the two runs a t 0 17-mm. and 0.26-mm. slit width for p-diisopropyl benzene gave 2.8 and 2.9 em.-’ band width, respectively. The lower value from the narrower slit width data is preferred. Values of 9.6, 11.1, and 12.7 em.-‘ werp given for the band width of cetane by the results at 0.265,0.80, and 1 30 mm. Extrapolation to zero slit width gives 9.5 em.-’ Multiplying these values of the band width by the values of n gives the “equivalent slit width” of the spectrometer (by definition). A comparison of these values with the “effective slit width” (calculated as the sum of the geometrical slit width and diffraction function from data supplied by the manufacturers of the spectrometer) is given in Table 111. Although no simple relationship was expected, it can be seen that the equivalent slit width is roughly the effective slit width plus a constant. It should be remembered that the low value in cetane is very approximate and also that the cetane band shows some signs of asymmetry a t high resolving power. It is clear, however, that the differences in Table I11 are not real physical quantities, inasmuch as they do not vary systematically with wave
6
Difference Cm-1 4.3
4.3 3 9 4 0
1.2 1.1 1.2 1.5 4.0 5.0
4.8
length or band width. In addition, they are too large to be actual aberration of the final image in the spectrometer. The instrument has resolving power up t o the maker’s specifications, so that an aberration of 0.4 mm. ( = 4 em.-’) a t 913 cm.-i is scarcely feasible. In view of the approximations involved in the original premises, it was unlikely that the parameters deduced would have a real physical significance. Nevertheless if an experimental relationship is established between effective and equivalent slit widths (or more practically, between n and slit width in millimeters), results a t different slit widths can be correlated for a particular absorption band. CONCLUSION
The effect of finite slit width even with instruments of reasonably high resolving power is t o cause large deviations from Beer’s law. Barnes ( 1 ) states that organic molecules have absorption bands of the same order of magnitude as the spectral slit widths (his value of 10 t o 15 cm -1 is probably large). The condition that they are equal implies an error of 15% from the true value of the calibration coefficient, and (more important) a variation of 3% in calibration coefficient for 10% change in slit width. Because so many infrared spectrometers use the slit width as a variable in obtaining an incident intensity independent of wave length, care must be taken to ensure that thc slit width used for a series of measurements is constant. ACKNOWLEDGMENT
The authors wish to thank Miss J. D. Massam for help with the experimental work, the Anglo-Iranian Oil Co. for providing a sample of pure cetane, and the directors of the Distillers Co., Ltd., for permission t o publish this work. LITERATURE CITED
( 1 ) Barnes, R. B., Gore, R. C., Liddell, U., and Williams, V. Z.,
“Infrared Spectroscopy,” Sew York, Reinhold Publishing Corp., 1944. (2) Eberhardt, K. H., J . Optical SOC.Am., 40, 172 (1950). (3) Hardv. A. C.. and Young. F. &I..I b d . . 39. 265 119491. i4j Hogn”eks, T. ~ R . ,Zscheile, F. P:, and Sidwel1,’A. E., J. Phys. Chem., 41,379 (1937). (5) Ladenberg, R.. and Reiche, F., Ann. Physik, 42, 181 (1913). (6) Lothian, G. F., “Absorption Spectrophotometry,” London, Adam Hilger, 1949. (7) Nielson, J. R., Thorton, V., and Dale, E. B., Rev. Modern Phya., 16. 307 (1944) (and biblioeraahv). (8) Ramaey, D. A,, quoted in “
