Quantitative Analysis with Infrared Spectrophotometers . D i f e r e n, t iczl A nu ly s is

DAYID Z. ROBIl-SOS

Baircl Associates, Znc., 33 University Road, Cambridge 38, Mass.

r . 1 his m-orli is a continuation of a prograni of evaluat- ing 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 condi- tions 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-

HIS paper continues the examination of the theoretical T problems of infrared quantitative analysis (6). It deals u ith differential analysis-the analysis of a mixture by compari- son 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 , ?). In addition, differential analypis in ordinary colorimetry has been treated by Hiskey ( 3 ) ,

This paper dis- cusses some of the optimum conditions for these analyses and de- scribes precautions that should be observed.

here the conditions are somewhat different.

OPTIXIUJl TRASSRIITTAKCE FOR TWO-COMPONENT DIFFERESTIAL ANALYSIS

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 analy- sis. 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, In 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. Ke- corded noise, however, increases when absorbing materials are placed in both beams. In general, i t has the value A T * / T ' , where T' is the t,ransniit,fance of the material in the reference beam.

Beer's law will be expected to hold for both components, so that the derivations can be made. In order to' simplify the vo- cabulary, 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:

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? t = TIT'

It, is 1 that is recorded on the chart in a differential analysis.

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. In- creasing the accuracy by changing the solutions in the beams is described. Constant slit widths are very important. Differential analvsis is more sensi- t i v e than the ordinary single-beam methods.

If Beer's l a x holds, the following equations can be v ritten:

-In T = albcl + u2bc?

-In t = (a , - al)bcr

( 1 1 = alb + (a2 - al)bc:! = -In T' + (a2 - al )bc l

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'. In computing the error n-e must substitute AT*/T ' for At, as i t is AT* which is assumed constant. The fractional error in concentration E, due to an error in reading

1 des t , is - - Af. Upon differentiating and substituting AT*, Equa- 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 omis- sion 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 i t is necessary to make the

error a function of cell thickness b and set - equal to zero.

When this is done, the condition for optimum transmittance is given by Equation 4:

dE ZJ

In T = -1 (4) '

This condition is identical to that derived for the case of single- beam 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 :

-2.72 1 + In T' "* E m i n . =

This equation shows how the error is increased as T' , the trans- mittance of the solvent alone, is decreased. For example, if the solvent alone transmits 50% of the radiant power when the solu-

619

620

A CASE 2

7 + ,

A N A L Y T I C A L C H E M I S T R Y

thickness of reference cell. The transmittance of the sample cell containing the solution is given by Equation 6:

-In T = alh + (az - al )cnb (6) The transmittance of the reference cell containing solvent is

given by Equation 7 .

-In T’ = alh’ ( 7 )

The instrument det,ermines the ratio of the transmittances t and

-In t = al(h - b‘) + (an - al)ctb ( 8 )

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:

-In 2’1 = a l b (9)

(10) -In TI’ = alh‘ + ( a 2 - al)cnb’

-In ti = al(b - b’) - (a, - al)ctb’

The instrument records the ratio, t l , and

(11)

This is the curve in case 4. If we now take the difference in ab-

(12)

sorbance, we will obtain

-(In t - I n 11) = (a. - a l )c l (h + b ’ )

We can solve for c:: directly.

SOLUTION SOLVENT

CELL A CELL B

SOLVENT SOLUTION

CELL B CELL A

SOLVENT

CELL A CELL B

SOLVENl SOLUTION

CELL A CELL B

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:

1. 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%.

The error will not be greatly changed in practice when the solu- tion 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.

EFFECT O F DIFFERENT CELLS

Actually cells of exactly the same thickness are almost im- possible 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.

SOLVENT SOLUTE SAMPLE REFERENCE BAND BAND 1 BEAM I BEAM

Because cell thickness differences cannot ordinarily be dif- ferentiated 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 i t 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 dif- ferential 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 i t 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 be- tween 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 concentra- tion can be obtained if we use the notation above and let b‘ equal

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 an is now less than al. The solvent will not be compensated for exactly, and the differ- ences bet’ween the two solutions can be measured directly.

EFFECT OF CHANGIYG SLIT WIDTHS ON DIFFERENTI4L ANALYSIS

The assumption is made in the derivations above that Beer’s law holds. It is well known (4-6) that if the absorption co- efficient varies widely in the wave-length inferval admitted by the slits, erroneous readings of the true absorbance at the band maxi- mum 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- -19 -e -6 -? -2 0 2 4 6 8 0

90 -

100- -19 -e -6 -? -2 0 2 4 6 8 0

90 -

p 50- a

SLITS USED !4 7 0 -

-.I I C I-

s 50-

it 40- P

+, I C -+I I C

I + I C

Figure 2. Absorption Bands Used to Compute Ap- parent 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 fre- quency 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 solu- tion 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 ac- curacy of any particular analysis. However, i t is possible to com- pare the accuracy of a double-beam differential analysis in a cer- tain 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 differen- tial 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

Figure 3. Apparent Transmittance of Side Band as Function of Slit Width

€Ion7 does slit width affect the differential curve shown in Fig- ure 2Y The effect can be calculated theoretically if some simplify- ing 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 vas 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 in- volved. The ratio of these areas is, then, the relative transmit- tance, t .

The results are shown in Figure 3, where the per cent transmit- tance is plotted as a function of slit width. The per cent trans- mittance 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 be- cause of the effect on the resolution However, if the main curve and the composite curve were run separately on a single-beam in- strument, a small change in slit width or in amplifier gain be- tween 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 i t 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 ob- tain 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 analy- sis, 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-

WAVE NUMBERS IN CM-I 50

IO0

00

< 60 2

t* 5 6

z < p: I-

n

20

5 0

2 3 4 WAVE LENGTH IN MICRONS

Figure 4. Single-Beam and Differential Curves of Benzene and Benzene with 0.025$% Cyclohexane

Added

h much more important advantage of a practical double-beam analysis occurs when i t is necessary to compensate for absorption of one of the components. If a determination must be made where the slope of the background curve is steep, the error in the transmittance obtained is due not to the random noise but to the error in reading the transmittance minimum. This case is illus- trated by the example that follows.

DETERMINATION OF CYCLOHEXANE IN BENZENE

.4s an example of the techniques used in differential analysis, the determination of small amounts of cyclohexane in benzene was at- tempted.

When the curves of the pure materials were first obtained, it was clear that there were many cyclohexane bands which were

622

well separated from benzene bands. For example, cyclohexane has a strong band a t 11 microns, whereas benzene has no bands between 10 and 12.8 microns. However, even when a very thick solution was used, it was impossible to see the effect of 0.025% cyclohexane on the benzene curve in this region. The cyclohexane band was not strong enough a t these low concentrations. By far the strongest cyclohexane band occurs a t 3.5 microns. One might ordinarily not c,onsider it as a good place to do the analysis because of the overlap from the strong 3.3-micron band of benzene. How- ever, 3.5 microns is the position where the solvent absorbs least when the solution as a whole transmits 40% and so this is the wave length a t which this particular determination should be made. This position might not he the best a t other ranges of composition.

The results of the analysis can be seen by examining Figure 4. The following runs \yere made on a Baird Associates, Inc., in- frared spectrophotometer ( I ) .

Benzene in the sample cell and benzene with 0.025% cyclo- hexane in the reference cell (case 4).

Benzene with 0.025% cyclohexane in sample cell and benzene in reference cell (case 1).

Benzene alone in sample beam. 0.025y0 cyclohexane in benzene alone in sample beam.

The bottom two curves are almost identical except for the re- gions around 3.5 microns which have been circled. Figure 5 has the same set of curves, except that 0.0125% cyclohexane in ben- zene was used.

The bottom curves are so similar that it mould be estremely dif- ficult to obtain a measure of the concentration of the cyclohexane. HoTyever, the different,ial analysis magnifies this difference graph- ically and enables accurate analyses to be made even a t these very low concentrations. The results of differential analysis are most striking when similar solutions are compared. Both instrumental operation and computation are simplified and the basic accuracy is improved.

The method of case 4 and case 1 of Figure 1 ensures that the com- position differences can be obtained even if the samples vary markedly in composition and the cells have different thicknesses.

There is no need for cell matching in these analyses.

A N A L Y T I C A L C H E M I S T R Y

SUMMARY

For the most accurate differential analyses, the slit ~ - i d t h s should be constant from run to run. A single-beam curve of the solution should transmit about 40% and the solvent should ab- sorb as little as possible under these conditions. I n doing qualita- tive differential analyses, the instrument a ill not respond through the regions of strong solvent absorption bands and read- ings around these regions are meaningless.

Differential analysis can magnify graphically the small differ- ences between two solutions. It is especially useful when the elope of the solvent or background curve is steep, for then the error is usually due not to the random noise but to the difficulty of making accurate transmittance readings.

The effect of differences in cell thickness can be eliminated by

WAVE NUMBERS IN CM-I

2 3 4 5 WAVE LENGTH IN MICRONS

Figure 5 . Single-Beam and Differential Curves of Benzene and Benzene with 0.0125% Cyclohexane

Added

running the solution in the sample cell against the solvent in the blank cell, then running the solution in the blank cell against the solvent in the saniple cell. Differences in solute composition are determined directly from the ratio of the transmittances.

ACKNOWLEDGIIENT

The author would like t,o thank Eleanor Flaherty and Arthur Dinlattia, who did the experimental work reported here.

LITERATURE CITED

(1) Baird, K. S., O'Bryan, H., Ogden, G.. and Lee, D., J . Optical

(2) Heigl, ,J., personal communication. (3) Hiskey, C. F., .%XAL. CHEY., 21, 1440 (1949). (4) h-ielsen, J. R., Thorriton, V., and Dale, E. B.. Rers. X o d e m Phvs.,

(,5) Philpotts, d. R., Thain, IT., and Smith, P. G.. d s a ~ . CHEY., 23,

(6) Robinson, D. Z., Ibid., 23, 273 (1951). (7) Wright, S., peraoiial communication.

RECEIVED for review .Ipril 18, 1951.

SOC. Am., 37, 754 (1947).

16, 307 (1944).

268 (1951).

Accepted December 17, 1951.