Quantitative Analysis with Infrared Spectrophotometers . D i f e r e n,t iczl A nuly s is DAYID Z. ROBIl-SOS Baircl Associates, Znc., 33 University Road, Cambridge 38, Mass. r .
1his m-orli is a continuation of a prograni of evaluating infrared spectrophotometers for quantitative analysis. It discusses the optimum conditions for quantitatil e analysis of two-component solutions if both components absorb at the wave length at which the determination is made. There are two conditions which must be fulfilled for highest accuracy: The solution as a whole should transmit about 40qc of the incident energy; when this condition is ful-
filled the interfering component should absorb as little as possible In differential analysis with double-beam spectrophotomcters the solution is placed in one beam of the spectrophotometer, and a known similar mixture is placed in the other. Increasing the accuracy by changing the solutions in the beams is described. Constant slit widths are very important. Differential analvsis is more sensit i v e than the ordinary single-beam methods.
THIS
paper continues the examination of the theoretical problems of infrared quantitative analysis (6). It deals u ith differential analysis-the analysis of a mixture by comparison of a knoim and an unknown sample simultaneously in a double-benm spectrophotometer. This method is particularly applicable n hen tn o or more constituents absorb a t the same wave length Differential analysis is not new) and the technique has been used in a number of laboratories ( 2 , ?). I n addition, differential analypis in ordinary colorimetry has been treated by Hiskey ( 3 ) , here the conditions are somewhat different. This paper discusses some of the optimum conditions for these analyses and describes precautions that should be observed.
In analyses of two-component, systems both components may absorb a t the wave lengths where the determination is made. I t is desirable to know what cell thickness to use, so that the error in such a determination will be minimized. In differential analysis. one of the absorbing components is placed in the reference I,enni. The double-beam spectrophotometer then draws a curve which ideally, a t least, gives the ratio of the radiant powers in the two beams. This ratio is thus the curve that one works with, and the optimum (cell thickness should be obtained on the basis of this curve if possible. In deriving the results, an assumption must be made as to the source of the errors, I n this derivation the noise is considered as arising in the detector, This noise, AT*, is a definite fraction of the radiant power that would go through the sample and reference benms if there were no absorbing mat.erials in the beams. Kecorded noise, however, increases when absorbing materials are placed in both beams. In general, it has the value A T * / T ' , where T'is the t,ransniit,fanceof the material in the reference beam. Beer's law will be expected to hold for both components, so that the derivations can be made. I n order to' simplify the vocabulary, the substance to be determined is called the solute, the other absorbing component is called the solvent, and the mixture is called the solution. Actually, the results derived will hold for any two-component system. The follon-ing symbols are used:
-In t
++ +
albcl u2bc? alb (a2 - al)bc:! = -In T' (a2 - al)bcl = ( a , - al)bcr
= =
(11
Solving for c?, the concentration of the solute, n-e obtain:
To compute the error, it is necessary to realize that the error in t , At, is not independent of T'. I n computing the error n-e must substitute A T * / T ' for At, as it is AT* which is assumed constant.
cy
dt
tion 3 is obtained:
This is only the error in reading the transmittance ratio t. The errors in zero and 100% line should be included ( 6 ) ,but their omission does not materially affect the conclusions reached. This error term differs from the term where solvent absorption is not considered by the inclusion of a different term in the denominator. To obtain the optimum transmittance it is necessary to make the dE error a function of cell thickness b and set - equal to zero. ZJ When this is done, the condition for optimum transmittance is given by Equation 4: In T = -1
(4)
This condition is identical to that derived for the case of singlebeam operation, The curve of the solution alone should have a transmittance of 37yG. The differential curve will usually have higher transmittance. The optimum condition for an analysis is independent of the absorptivity of the solvent, and so cannot be obtained from examination of the differential curve alone. The error under the optimum condition is given by Equation 5:
T = transmittance of solution T' = transmittance of solvent a l = absorptivity of solvent at = absorptivity of solute h = cell length c, = concentration of solvent = 1 - c? =
-In T
The fractional error in concentration E, due to an error in reading 1 des t , is - - Af. Upon differentiating and substituting AT*, Equa-
OPTIXIUJl TRASSRIITTAKCE FOR TWO-COMPONENT DIFFERESTIAL ANALYSIS
t
If Beer's l a x holds, the following equations can be v ritten:
Emin.
=
1
-2.72 In T'
+
"*
This equation shows how the error is increased as T', the transmittance of the solvent alone, is decreased. For example, if the solvent alone transmits 50% of the radiant power when the solu-
TIT'
It, is 1 that is recorded on the chart in a differential analysis.
619
'
ANALYTICAL CHEMISTRY
620
’
tion transmits 37%, then the error is about three times as great as when the solvent does not absorb a t all. The two rules to follow for best accuracy can be written: The solution alone, uncompensated for solvent, should transmit 37% of the radiant power. 2. The wave length to be selected should be the one where the solvent absorbance will be a minimum when the sample solution transmits 37%. 1.
The error will not be greatly changed in practice when the solution transmittance runs from 20 to 60%. These conditions are discussed when the sample analysis is described. These rules apply to any analysiq of t1v-o-component mixtures, Ivhether with single-beam or double-beam instruments. In the case of single-beam instruments, the conclusions are similar to those described in the literature, but the derivation of the extent of the error made when the solvent absorbs is believed to be new.
thickness of reference cell. The transmittance of the sample cell containing the solution is given by Equation 6: -In T = alh
-In T’ = alh’
1
SOLUTE BAND
A
CASE 2
SAMPLE BEAM
SOLVENT CELL B
7
CELL A
,
SOLVENl CELL A
(8)
-In TI’
-In 2’1 alh‘
=
=
-In ti = al(b
(9)
alb
+ ( a 2 - al)cnb’ tl,
(10)
and
- (a, - al)ctb’
- b’)
(11)
This is the curve in case 4. If we now take the difference in absorbance, we will obtain -(In t - I n 11)
= (a.
- al)cl(h
+ b’)
(12)
REFERENCE BEAM
CELL A
SOLVENT
+ (an - al)ctb
-In t = al(h - b‘)
This is the curve in case 1. It is impossible to tell which of the terms in the equation causes the difference. If we put, the solvent in the sample cell and the solution in the reference cell, we obtain:
We can solve for c:: directly.
SOLUTION
CELL B
+
I
(7)
The instrument det,ermines the ratio of the transmittances t and
The instrument records the ratio,
Actually cells of exactly the same thickness are almost impossible to obtain and inequalities may cause an error when the above procedure is used. It is possible to eliminate the error due to the different cell thicknesses and also increase the accuracy by using the following procedure.
(6)
The transmittance of the reference cell containing solvent is given by Equation 7 .
EFFECT O F DIFFERENT CELLS
SOLVENT BAND
+ ( a z - al)cnb
SOLUTION CELL A
SOLVENT CELL B
+
The term ( b h’) in the denominator leads to essentially twice the deflection for t’he same concentration and thus the accuracy is greatly increased. Equation 13 holds even when the solute concentration is so high that solvent bands will shon- up. The direction of the shift between case 4 and case 1 rvill be different when the solution and solvent are interchanged, in this,case, sinc,e anis now less than al. The solvent will not be compensated for exactly, and the differences bet’ween the two solutions can be measured directly.
SOLUTION CELL
EFFECT OF CHANGIYG SLIT WIDTHS ON DIFFERENTI4L ANALYSIS
B
Because cell thickness differences cannot ordinarily be differentiated from sample differences by means of one run, another run must be made. The normal procedure is to run a curve of pure solvent us. pure solvent and note where it differs from a curve of solution us. solvent. Instead of doing this, increased accuracy can be obtained by putting the solution in the reference cell and the solvent in the sample cell. The cells are kept in the same beams in which they were originally used. Differences due to cell thickness will cause pen deflections in the same directions, while differences due to composition will cause the pen to move in opposite directions. The various methods of running differential curves are shown schematically in Figure 1. The sample cell, A , is considered a little thicker than the reference cell, B. The top curve shows the result of a single differential curve. If we simply switch the cells in the two beams, we obiain the curve shown in case 2. The difference between the two curves is twice as great as the difference between the curves and the background level, but it is not possible to distinguish between the solvent and solute bands. If the solvent is placed in cell A and a curve of pure solvent vs. solvent is obtained, the result will be as shown in case 3. In case 4 the solvent is placed in A and the solution is placed in B. The difference a t the solute band between case 4 and case l is about twice as great as the difference between case 3 and case 1. When this method of comparison is used, the solute concentration can be obtained if we use the notation above and let b‘ equal
The assumption is made in the derivations above that Beer’s law holds. It is well known (4-6) that if the absorption coefficient varies widely in the wave-length inferval admitted by the slits, erroneous readings of the true absorbance at the band maximum will be obtained. Very often a narrow side band due to an impurit,y is found on the side of a larger band due to solvent absorption. This case is shown in Figure 2 . The main band has a n-idth a t half height of FREOVENCY 100-
90
pa 4!
-19
-e
-6
-?
-2
0
2
4
6
8
0
-
50-
70-
SLITS USED I-
-.I
s 50-
Pit
+,
-+I
I C
I C I C I +
I C
40-
Figure 2. Absorption Bands Used to Compute Apparent Transmittance of Side Band as Function of Slit Width
V O L U M E 2 4 , NO. 4, A P R I L 1 9 5 2
621
2 units, while the side band has a maximum occurring one frequency unit away and a width of 1 unit. The spectrum due to both bands combined is the lower curve in the figure. In the single-beam method of analysis, one would superimpose the solution curve and the solvent curve and use the ratio of the curves a t the point where the side band maximum occurs. As can be seen from the lower curves, this procedure is difficult, because the slopes of the curves are steep a t that point. A differential curve would give only the side band and so mould make a determination easy. This is one of the chief advantages of differential analysis.
crease the radiant power, then the apparent transmittance will decrease and the Beer’s law deviations will be greater. GENERAL VALUE OF DIFFEREVTIAL 4YALYSIS
I t is difficult to analyze the increased accuracy of a differential analysis theoretically, because so many factors affect the accuracy of any particular analysis. However, it is possible to compare the accuracy of a double-beam differential analysis in a certain instrument to an analysis made with the same instrument used to record the energy going through one beam only. This is the type of comparison that leaves all conditions the same. In the most accurate single-beam analysis, one would run first the solvent curve, then the solution curve, and obtain thr ratio of the transmittances a t the correct wave length. Consider the noise level in each of these curves as-a constant AT value. Then there would be an error of about 4 2 A T in determining the ratio even if there were no error in resetting slits or the zero line. If the procedure described in cases 4 and 1 is used on a differential curve, it is still necessary to run two curves, and the error in determining their ratio is also 4 / 2 A 7 ’ . However, the difference betwern the two curves is twice as great as it is in the single-beam analysis and the error made in the concentration i i half the error of a single-tvam analysis
d
0:s
1.6
2.4
3.2
4.0
4.8
56
SLIT WIDTH
WAVE NUMBERS IN CM-I 50 IO0
Figure 3. Apparent Transmittance of Side Band as Function of Slit Width €Ion7does slit width affect the differential curve shown in Figure 2Y The effect can be calculated theoretically if some simplifying assumptions are made. If we assume that the ratio of the radiant power coming through two beams is determined, and that the effect of a finite slit is such that the transmittance over the slit width is averaged, it is possible to calculate directly the effect of different slit widths on the ratio. This v a s done as follow. The areas under the main band and under the composite band were determined for each slit q-idth shown in Figure 2. These areas were assumed to represent the radiant power transmitted through the slits by the two beams involved. The ratio of these areas is, then, the relative transmittance, t . The results are shown in Figure 3, where the per cent transmittance is plotted as a function of slit width. The per cent transmittance is what a double-beam instrument would measure if the solvent band were completely compensated for. A change in slit width will cause a change in apparent peak transmittance because of the effect on the resolution However, if the main curve and the composite curve were run separately on a single-beam instrument, a small change in slit width or in amplifier gain between the runs n-ould affect the radiant power reading directly, and would lead to large errors in concentration. Such first-order errors do not occur in a double-beam differential run, because 2’ and T’ are measured almost instantaneously. Nevertheless it is still important that the slit widths be reproducible in double-beam differential recording to minimize second-order errors of the type illustrated in Figure 3. REGlONS OF NO RESPONSE
The servosystem in a double-beam spectrophotometer requires a certain amount of energy difference in the two beams before it will respond. If both beams absorb completely, or nearly so, the instrument nil1 not give a good differential curve. For example, if carbon tetrachloride is used as a solvent, it is impossible to obtain bands in the 12- to 14-micron region, because the absorption by carbon tetrachloride takes out all the energy. This problem does not arise so markedly in quantitative analysis, as the optimum transmittance is around 50% and most modern instruments have good response even with half the energy removed from each beam, If the slits are widened to in-
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