and we will work out the exact chemometrics for measurementof molecular chlorine with analytical accuracy. It will also beimportant to investigate the influence of water (as a possiblesource of hydrogen for HOCl) and other compounds present inthe sample gas and produced during the photolysis on theoriginal chlorine concentration.24–27

ACKNOWLEDGMENT

The authors thank Ian S. Butler (McGill University, Montreal, Canada) forhis assistance in the preparation of this paper.

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Implications of Area Normalization ofMulti-Component Spectra

DEBASHIS PANDA and ANINDYADATTA*Department of Chemistry, Indian Institute of Technology

Bombay, Powai, Mumbai 400 076, India

Index Headings: Area normalization; Pseudo-isosbestic point; Pseudo-

isoemissive point.

INTRODUCTION

The existence of isosbestic points in spectra of multicom-ponent systems with varying concentrations of components is awell-established phenomenon for systems that follow theLambert–Beer law.1–3 The importance of isosbestic points liesin the quantitative and qualitative information of a sample. Thecause of nonexistence of such points is the nonlinear behaviorof absorption spectra and the fluctuation of the totalconcentration of the components in solution through theirdecomposition, specific interaction with solvents, or experi-mental error in sample preparation. In such cases, thedetermination of the number of absorbing species in solutionis erroneous. Earlier, this problem has been circumvented forinfrared bands in CDC13 and di-n-butyl ether by normalizingthe observed bands to unit area. The consequent isosbesticpoint had been termed a pseudo-isosbestic point.4 The recoveryof an isosbestic point in a two-component system ofcondensed-phase conducting polymers has been demonstratedusing Lorentz–Lorenz approximation for the local field.5 Time-resolved area normalized emission spectra (TRANES) havebeen shown to exhibit isoemissive points when two emissivespecies are present in the finite time scale.6–9 Area normalizedultraviolet (UV)-visible absorption spectra have been shown toreveal hidden isosbestic points, which are not observedotherwise.10 It can be mentioned here that the necessities ofother methods such as the rank method, direct comparison, andspectra difference are also discussed in order to interpret theabsorption spectra of wastewater.11,12 In the present Note, it isattempted to formulate this problem with a linear combinationof two Gaussian functions and, thereafter, extend the findingsto the experimental observation where both the appearance anddisappearance of a common point in a spectrum are observed.

FORMULATION OF AREA NORMALIZATIONMETHOD

The algebraic formulation of the problem is as follows:consider the composite spectrum to be represented by afunction, f (k), which is a linear combination of two functions,f1(k) and f2(k):

f ðkÞ ¼ c1 3 f1ðkÞ þ c2 3 f2ðkÞ ð1Þ

where c1 and c2 are constants for a given composite spectrum.

Received 27 July 2007; accepted 2 January 2008.* Author to whom correspondence should be sent. E-mail: anindya@chem.iitb.ac.in.

APPLIED SPECTROSCOPY 341

Consider a series of such spectra, starting from pure f1(k) (c1

¼ 1, c2 ¼ 0), through a series of composite spectra for whichboth c1 and c2 are non-zero, to pure f2(k), (c1 ¼ 0, c2 ¼ 1).

Now, consider two composite spectra, f (k) and f*(k), wherethe coefficients of f1(k) and f2(k) are c1, c2 and c�1 , c�2 ,respectively. The areas areZ

f ðkÞdk ¼Z

c1 f1ðkÞdkþZ

c2 f2ðkÞdk ð2Þ

for coefficients c1 and c2, andZf �ðkÞdk ¼

Zc�1 f1ðkÞdkþ

Zc�2 f2ðkÞdk ð3Þ

for coefficients c�1 and c�2 .Let a, b, k, and k* be the area normalization constants of

pure spectra f1(k), f2(k), and composite spectra f (k), f*(k),respectively, such that

a

Zf1ðkÞdk ¼ 1 ð4aÞ

b

Zf2ðkÞdk ¼ 1 ð4bÞ

k

Zf ðkÞdk ¼ 1 ð4cÞ

k�Z

f �ðkÞdk ¼ 1 ð4cÞ

The expressions for the area normalized spectra f (k) and f*(k)can be expanded as follows:

k

Zf ðkÞdk ¼

Zkc1 f1ðkÞdkþ

Zkc2 f2ðkÞdk ¼ 1 ð5Þ

k�Z

f �ðkÞdk ¼Z

k�c�1 f1ðkÞdkþZ

k�c�2 f2ðkÞdk ¼ 1 ð6Þ

Subtracting Eq. 6 from Eq. 5, we obtain

ðkc1 � k�c�1 ÞZ

f1ðkÞdk ¼ ðk�c�2 � kc2ÞZ

f2ðkÞdk ð7aÞ

Zf1ðkÞdk

Zf2ðkÞdk

¼ k�c�2 � kc2

kc1 � k�c�1¼ b

að7bÞ

as the area normalization constants for the pure spectra f1(k)and f2(k), which are a and b, respectively.

At the point of crossing (kc) of area normalized pure spectra,

a 3 f1ðkcÞ ¼ b 3 f2ðkcÞ ð8Þ

For area normalized composite functions at kc,

f ðkcÞ ¼ kc1 3 f1ðkcÞ þ kc2 3 f2ðkcÞ ð9Þ

Similarly, for f*(kc), where c1 and c2 are replaced by c�1 andc�2 , respectively,

f �ðkcÞ ¼ k�c�1 3 f1ðkcÞ þ k�c�2 3 f2ðkcÞ ð10Þ

Subtracting Eq. 10 from Eq. 9, we obtain

ðkc1 � k�c�1 Þ3 f1ðkcÞ þ ðkc2 � k�c�2 Þ3 f2ðkcÞ¼ f ðkcÞ � f �ðkcÞ ð11aÞ

or

ðkc1 � k�c�1 Þ3 f1ðkcÞ �b

aðkc1 � k�c�1 Þ3 f2ðkcÞ

¼ f ðkcÞ � f �ðkcÞ ð11bÞ

or,

ðkc1 � k�c�1 Þ f1ðkcÞ �b

af2ðkcÞ

� �¼ f ðkcÞ � f �ðkcÞ ð11cÞ

Since f1(kc)¼ (b/a) f2(kc) from Eq. 8, therefore, f (kc)¼ f *(kc).Thus, all the area normalized spectra pass through a point withx-xo-ordinate of kc.

RESULTS AND DISCUSSION

In order to exemplify this situation, a linear combination oftwo Gaussian functions, f1(k) and f2(k), differing in full widthat half-maxima (FWHM), peak positions, and height isconstructed. The width of f1(k) and f2(k) are 55 nm and 75nm, respectively. The peak positions of f1(k) and f2(k) are at420 nm and 449 nm, respectively. The composite functionsf (k) shown in Fig. 1a are generated by varying the amplitude(c1 and c2) of functions f1(k) and f2(k), respectively, but thesum of c1 and c2 is kept constant, thereby simulating a situationwhere the two components are in equilibrium with a constanttotal concentration. All the curves are found to pass through acommon point at 432 nm, analogous to isosbestic orisoemissive points observed experimentally in such situations.Interestingly, the area normalization of curves also yields sucha point, but this ‘‘pseudo-isosbestic point’’ is red-shifted by 7nm (Fig. 1b). Next, f (k) is generated by altering the totalconcentration of components (c1þ c2 is not constant), in orderto simulate a situation where the total concentration of thecomponents is not fixed (Fig. 1c). As expected, there is noisosbestic point for these spectra. Area normalization of thesespectra yields a pseudo-isosbestic point, as is expected from thealgebraic formulation proposed earlier in this letter (Fig. 1d).This point is found to be the same as that observed in Fig. 1b,as expected. Thus, the usefulness of area normalization as ahandy tool to identify the number of components in a set ofcomposite spectra is re-emphasized.

For components with very small spectral shifts, the numberof curve crossing points might be more than one, even for asystem having two components, which can lead to anerroneous prediction of the number of components present ina system. Such a situation is simulated with linear combina-tions of functions of the same height and FWHM as f1(k) andf2(k), but with a separation of maxima of only 5 nm (Fig. 2a).The spectral domains of interest have been magnified for thesake of clarity (Fig. 2a, inset). Two curve crossing points areobserved at 430.8 nm and 446.6 nm. Upon area normalization,the points reappear at 411.6 nm and 466.7 nm, respectively(Fig. 2b). Thus, area normalization provides a better resolutionby inducing an increase in the relative separation of points from16 nm to 55 nm, as a consequence of a 19 nm blue shift of the

342 Volume 62, Number 3, 2008

higher energy point and a 20 nm red shift of the lower energypoint, without a change in the number of isosbestic points. So,the shortcoming of the method is that the reappearing point hasno significance as do isosbestic or isoemissive points, whichare indicative of equilibrium between two components.

So far in this Note, the discussion has been restricted to onlysynthesized functions. At this stage, it is appropriate to test theapplicability of area normalization in experimental spectra.This method is tested on spectra of two different fluorophores:3-aminoquinoline (3AQ) and 2-(2 0-pyridyl)benzimidazole(2PBI). Earlier, the photophysics of 3AQ were studied.13 Thered-shifted absorption band in Fig. 3a is due to the cationicform of this fluorophore. With decrease in pH of the medium,the equilibrium is shifted towards the cationic form. Noisosbestic point is observed, even though a two-stateequilibrium is operative and the sum of the concentrations ofthe neutral and the cationic species is constant (Fig. 3a). Inorder to impose the condition of fixed total concentration of thetwo species, the composite spectra are generated as linear sumsof these spectra (the one at pH 11 denoted as f1(k) and the one

at pH 3 denoted as f2(k)), keeping the sum of theirconcentrations (c1þ c2) constant. Two isosbestic points appearat 317 nm and 347 nm in these synthesized spectra (Fig. 3b).The area normalization of the experimental spectra, on theother hand, yields a pseudo-isosbestic point at 355 nm (Fig.3c), which conforms with a two-component system. Notably,there is no isosbestic point at 302 nm. The spectra do not crosseach other at this point. Thus, area normalization of theexperimental spectra yields a realistic picture in this case.

Finally, the effect of area normalization on a three-component system is examined. Recently it has been observedthat a fluorophore, 2PBI, in protein–surfactant complexesexists in three different forms, neutral, cationic, and photo-tautomeric.14 The superposition of fluorescence spectra yieldsisoemissive points at 361 nm and 422 nm, corresponding to theequilibria between neutral-cationic and cationic-phototautomerforms of 2PBI, respectively (Fig. 3d). Area normalization ofthese spectra yields two pseudo-isoemissive points, notablywith a larger shift, due to blue and red shifts of 6 nm each, forthe higher energy and lower energy points, respectively (Fig.3e). This is reminiscent of the situation depicted in Fig. 2b.

CONCLUSION

In summary, area normalization is a good method to confirmquantitatively the number of components contributing tospectra. The method can be applied usefully in systems wheresome preliminary idea about the components is available.Moreover, it can be extended to the absorption spectra of

FIG. 1. (a) The plots of the function f (k), which are linear combinations oftwo Gaussian functions f1(k) and f2(k) varying c1 and c2, with (c1 þ c2)constant. All the curves pass through a point at 432 nm. (b) The areanormalization of the curves in (a). An isosbestic point is obtained at 439 nm. (c)The plot of the function f (k), varying c1 and c2, with variable (c1þ c2). (d) Thearea normalization of the curves in (c) yields an isosbestic point at 439 nm.

FIG. 2. (a) The plots of the function f (k), which is a linear combination of twoGaussian functions f1(k) and f2(k) with varying c1 and c2 but constant (c1þ c2).The isosbestic points are at 430.8 nm and 446.6 nm. (b) The area normalizationof the above-mentioned curves, with isosbestic points at 411.6 nm and 466.7nm.

APPLIED SPECTROSCOPY 343

naturally occurring samples. As has been demonstrated in theprevious section, the method of area normalization eliminateserroneous isosbestic points as well and thus provides a realisticpicture of the situation. An obvious limitation in the model isthe use of Gaussian components, which may be an oversim-plification. The best way to proceed would be to use the actualspectra of the components, wherever available.

ACKNOWLEDGMENTS

This work is supported by SERC, DST grant no. SR/S-1/PC-19/2005. D.P.thanks CSIR for a Senior Research Fellowship.

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FIG. 3. (a) The pH-dependent absorption spectra of 3AQ. No isosbestic pointis present. (b) The spectra generated by taking the spectrum at pH 11 as f1(k)and the spectrum at pH 3 as f2(k), keeping the sum of their concentrations (c1þc2) constant. The isosbestic points appear at 317 nm and 347 nm. (c) Areanormalized pH-dependent absorption spectra of 3AQ, showing a common pointat 302 nm and pseudo-isosbestic point at 355 nm. (d) The fluorescence spectraof 2PBI in the presence of HSA and SDS. Two isoemissive points are present at361 nm and 422 nm. (e) Area normalization of the spectra in (d) yields twopseudo-isoemissive points at 355 nm and 428 nm.

344 Volume 62, Number 3, 2008