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The Journal of Physical Chemistry, Vol. 82, No. 24, 1978
(29) C. Naccache, P. Mariaudeau, M. Che, and A. J. Tench, Trans. Farahy Soc., 67,506 (1971). (30) M. Che, C. Naccache, and B. Imelik, J. Chim. Phys., 65,1301 (1968). (31) P. Meriaudeau, C. Naccache, and A. J. Tench, J. Catal., 21, 208 (1971). (32) Y. Mizokawa and S. Nakamura, Oyobutsuri, 46,580 (1977). (33) E. R. S.Winter, J . Chem. SOC. A , 2889 (1968). (34) N. N. Greenwood, "Ionic Crystals, Lattice Defects and Nonstoichiometry", Butterworth, London, 1968.
C. A. Brignoli and
H. DeVoe
(35) A. J. Tench, T. Lawson, and J. F. Kibblewhite, Trans. Faraday Soc., 68, 1169 (1972). (36) M. Misono, E. Ochiai, Y. Saito, and Y. Yoneda, J. Inorg. Nucl. Chern., 29, 2685 (1967). (37) N. B. Wong and J. H. Lunsford, Chem. Phys. Lett., 19,348 (1973). (38) T. Seiyama, N. Yamazoe, and M.Egashira, Proc. Int. Congr. Catal., 5th, 2,997 (1973). (39) A. I. Gelbstein, S. S. Stroeva, Yu. M. Kakshi, and Yu. A. Mischenko, Proc. Inf. Congr. Catal., 4th, No. 22 (1968).
Least-Squares Computer Analysis of Dye Absorption Spectra. Acridine Orange Dimerization Equilibrium Carol A. Brignoli' and Howard DeVoe" Department of Chemistry, University of Maryland, College Park, Maryland 20742 (Received February 22, 1977; Revlsed Manuscript Received June 23, 1978)
A weighted least-squares computer procedure is described for analyzing absorbance spectra of dye solutions to obtain "best" values and standard deviations of the thermodynamic dimerization constant K and of the molar absorption coefficients of the monomer and dimer species. Spectra of acridine orange hydrochloride solutions, of concentrations (1-4) X M in 0.32 mM HC1, gave K = (6.8 f 0.8) X lo3 M-l at 25.0 "C. The derived acridine orange dimer spectrum has two peaks symmetrically split in energy about the position of the monomer peak. The calculated value of K is affected by the choice of the weighting function used in the least-squares procedure, and the appearance of the derived dimer spectrum changes in a systematic way with the value of K. The uncertainty in the value of K is significantly reduced by smoothing the absorbance spectra and by using absorbance values from many solutions and many wavelengths.
Introduction Acridine orange (AO) is a much-studied member of a class of cationic dyes whose planar molecules aggregate in aqueous solution to, cause concentration-dependent spectral changes (metachromasy). Direct evidence for the aggregation of A 0 has been obtained from proton chemical shiftsa2The concentration dependence of changes in the visible absorption spectrum of dilute A 0 solutions is consistent with a dimerization e q ~ i l i b r i u m The . ~ ~ simplest such equilibrium for the A 0 monomer, D', is 2Dt = DZ2+ (1) Another possible dimer species, in which a counterion such as C1- is incorporated stoichiometrically, has been found by potentiornetricelo and spectraleB8studies to be negligible at low concentrations of the counterion and thus may be safely ignored under these conditions. Of particular interest is quantitative information on the dimerization constant for equilibrium 1, and on the absorption spectrum of the presumed dimer species, Dz2+, Values of the dimerization constant at 22-25 "C and low ionic strength have been r e p ~ r t e d in ~ - the ~ range (9-11) X lo3 M-l. Three types of dimer visible absorption spectra of differing qualitative appearance have been obtained: (1) a blue-shifted peak (relative to the monomer peak) and a second less intense red-shifted peak;6 (2) a single, blue-shifted peak with a red-shifted s h ~ u l d e rand ; ~ ~(3) ~ a symmetric blue-shifted peak with no ~ h o u l d e r .This ~ uncertainty in the shape of the dimer spectrum makes it difficult to apply theorie~ll-~~ for interpreting the spectrum in terms of electronic interactions between the two monomer units in the dimer species. This paper describes a weighted least-squares computer procedure for analyzing the absorption spectra of solutions of a dye having a monomer-dimer equilibrium, and reports 0022-3654/78/2082-2570$01 .OO/O
quantitative results obtained by this method for the thermodynamic dimerization constant and the monomer and dimer spectra of AO. The procedure is an elaboration of a method described by Bergmann and O'Konski14 and, in an improved form adapted to a computer, by Lamm and NevilleS4 The important improvements in the present procedure are that it weights the absorbance values in the least-squares minimization, it allows the concentrations to be adjusted, it takes the ionic strength into account, and it provides estimates of the uncertainties in the calculated quanti ties I
Experimental Section Dye Purification. A solution of a crude A 0 salt (Chroma) in water was filtered and titrated with NaOH to precipitate the non-ionized base of AO. A methanol solution of the base was passed through an alumina column, and the eluate containing the main red-brown band was collected and concentrated. An equal volume of water was added to form a suspension of the base, which was extracted with benzene. The benzene was evaporated and the base was recrystallized from methanol and dried under vacuum at 100 "C. The melting point (uncorrected) was 186.4-187.1 "C (Zanker3 found 180-181 OC), Anal, Found: C, 76.95; H, 7.05; N, 15.31. Calcd for C17HleN3: C, 76.95; H, 7.22; N, 15.84. Dye Solutions. A 1.3 X M stock solution of acridine orange hydrochloride in 0.32 mM HC1 was prepared by swirling 1.3 x mol of the purified base with 4,5 mL of 0.10 M HC1 until the solid was completely dissolved, and adding water to a total volume of 1 L. More dilute solutions for the absorbance measurements were prepared by combining weighed portions of the stock solution with 0.32 mM HC1 in the cleaned optical absorption cell and stirring with a miniature Teflon-encased magnetic stirring 0 1978 American
Chemical Society
Acridine Orange Dimerization Equilibrium
bar. The dilute solutions were prepared and measured in a random order of concentration. All water was deionized and distilled in glass. Solutions were not allowed to contact ground surfaces in the flask and the cell, in order to avoid losses of dye by adsorption. The measured pH of the solutions was equal to the theoretical value of 3.5. This acidic value was chosen so as to allow hydrogen ions to compete with the A 0 cations for adsorption sites. (At pH 3.5 the fraction of A 0 existing in the doubly protonated form is only 0.0008, based on a pK, value3 of 0.4.) Absorbance Measurements. Absorbances were measured with a Cary Model 14 spectrophotometer, using a stoppered l-cm pathlength silica absorption cell thermostatted a t (25.00 f 0.02) “C. The reference beam was left, empty, except that it was necessary to place a neutral density screen (Cary) in this beam for measurements of the most concentrated A 0 solution at wavelengths near the absorbance maximum. The absorbance values of the A 0 solutions were collected automatically at a wavelength interval of 1nm with a Datex SDS-1 data recording system (analog-to-digital conversion) and recorded to 0.001 absorbance unit in punched card form. Computer Calculations Absorbance Spectra. The punched absorbance values were used as input data for a computer program. These values, and derived spectral data, were stored within the computer in order of decreasing wavelength; each such one-dimensional array is referred to below as a spectrum. In most of the calculations each absorbance spectrum was first smoothed by adjusting each absorbance value to a cubic least-squares function fitted through the value and the seven preceding and seven following values in the ~pectrum.’~This smoothing procedure removed obvious noise from the spectrum without distorting the band shapes and heights. The spectrum of the solvent (0.32 mM HC1) was subtracted from the solution spectrum. In some spectra a small constant absorbance correction (50.002) was made so as to adjust the absorbance to zero at 579 nm (where A 0 is transparent). In the spectrum of the most concentrated solution, an appropriate correction was made for the presence of the neutral density filter in the reference beam. Matrix rank analyseP of various sets of absorbance spectra were made to determine the number of independent absorbing components which were present in the solutions. The analyses required estimates of the limits of error of the absorbance values, For these estimates, we tried two different functions of the absorbance to be described in a later section. Fundamental Relations. A particular solution in which the total concentration of dye monomer units is c is assumed to have an absorbance A at a given wavelength which is the sum of contributions from the monomer and dimer species according to the Beer-Lambert law: A = [QCM + (1- a ) t D ] C 1 (2) Here a is the fraction of dye monomer units present in the monomer form, 1 is the pathlength, and E M and eD are the molar absorption coefficients of a dye monomer unit in the monomer and dimer forms, respectively. The dimerization constant for equilibrium 1,in terms of concentration, is defined by K, = (1- a)/2a2c. Solving for cy, one obtains LY = [(l+ ~K,c)’/’ - 1 ] / 4 K , ~ (3) By approximating the activity coefficients of D+ and DZ2+ with the Debye-Huckel limiting law, one can relate K , to
The Journal of Physical Chemistry, Vol. 82, No. 24, 7978 2571
the thermodynamic dimerization constant (in terms of activities), K, which (unlike K,) is independent of the ionic strength: log K , = log K 2ADHp1/2 (4)
+
Here A D H is a physical constant in the Debye-Huckel theory, equal to 0.512 M-1/2(for water at 25 “C), and p is the ionic strength of the solution. Least-Squares Procedure. The experimental data for a least-squares analysis consisted of M X N values of the absorbances A,, and M values of the concentrations car where M is the number of solution spectra, N is the number of wavelengths in each spectrum, and the subscripts s and w refer to the solution and the wavelength, respectively. These values were subject to experimental errors. The adjusted or “correct” values are symbolized by A,; and c i . An equation analogous to eq 2 gives the functional relationship between the adjusted absorbances and constants to be determined: (5) As/ = [ ~ S C M W + (1- ~s)CDwlc,’& The constants K, EM,, t D w , and c,’ were treated as adjustable parameters whose “best” values were determined by the familiar least-squares criterion, according to which the “best” values are those which minimize the errorsquare sum function, S, defined by M
N
S = C C ~,,(-4,, s=l w=l
- A,,))2
Here w,, is a relative weight which is a measure of the inherent accuracy of the value of A,. In order for the least-squares criterion to be rigorously valid, the random errors affecting the values of A,, should be normally distributed and there should, of course, be no systematic err0r.l’ The selection of appropriate values of the weights, w,, was guided by the principle that they should each be inversely proportional to the variance uSw2, where us, is the standard deviation of the error distribution in a hypothetical infinite population of measurements of AB,:
w, = u2/u,2
(7)
Here, u is a constant for all solutions and wavelengths and represents the standard deviation of an absorbance measurement of unit weight. The values of us, were assumed to be proportional to the limits of error of A,, as described in a later section. The error-square sum S as defined by eq 6 is a function of the adjustable parameters through eq 3-5. The computer program minimized this function by the following sequence of steps: 1. An initial value was assumed for K. 2. A value of a, was calculated for each solution from eq 3 and 4, using the current value of K and an approximate value of p. 3. The ionic strength of each solution was calculated more exactly from p = c,* + (3 - as)c,/2 (83 (where cs* is the concentration of HCl in the solution) and used to recalculate as;this step was repeated once more to give consistent values of p, and as. 4. At each wavelength, the values of tMW and tDw which minimized S for the current values of cy, were calculated by the standard algebraic method for the fitting of a linear function (eq 5) by weighted least squares. 5. In this step, omitted in some of the analyses, the concentration values of all but one of the solutions were
2572
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The Journal of Physical Chemistry, Vol. 82,No. 24, 1978
adjusted so as to minimize S. (The largest concentration value was left unadjusted because it can be deduced, from the fact that each adjustable parameter contains the dimension of concentration or its reciprocal, that at least one of the concentrations must be assumed to be known exactly if unique “best” values of the other parameters are to be obtained). Three cycles of adjustment were performed by the Gaussian approximation formula
A/nm 550
450
500
1 /
L
I
\
1
c,’(new) = c,’(old) - (aS/a~,’)/(a~S/ac,’~) where c,’(new) and c,’(old) are the adjusted and previous values. The partial derivatives appearing in this formula were evaluated from algebraic expressions written in terms of the adjustable parameters and obtained from eq 3-6 and eq 8. At the end of each cycle, new “best” values of a,, EM,, and €Dw were calculated by the methods described in steps 2-4, using c,’ in place of c,. 6. The value assumed for K was systematically varied and steps 2-5 were repeated so as to continuously decrease the value of S. It was found convenient to have the computer vary the value of log K, first in steps of 1 and then by bisection of intervals in successively smaller steps until the three lowest values of S at an interval of T7in the values of log K were found. These three lowest values were fitted to the quadratic function
S = So + b(log K
-
log
(9)
from which the minimum value of the error-square sum, So,and the “best” value of the dimerization constant, KO, were evaluated. The final “best” values of the other parameters were calculated from KOby repeating steps 2-5. An estimate of the standard deviation, u, of an absorbance measurement of unit weight was calculated from u = [ S o / ( M N- n)]1/2where n is the number of different parameters which were adjusted in the minimization process. This value of u was then used to estimate the standard deviation (standard error), uK, of the “best” value of K, a measure of the uncertainty as judged from the consistency of the experimental data. According to the “D boundary” concept of Sill611,~~the value of UK is equal to the largest value which the quantity IK - Kol can attain when the function S is equal to So + 2. That is, if values of S calculated for various values of K, and minimized with respect to the other parameters, are plotted as a function of K, uK is equal in magnitude to the larger of the two deviations of K from KOa t the points where the curve intersects S = So u2. When the quadratic function of eq 9 adequately represents the dependence of S on K, UK may be evaluated by SillBn’s method from the relation UK = 2.303Ko~/b1/2.The computer program calculated a tentative value of uK from this relation, which we later checked by the graphical method. In most cases both methods gave the same value for UK. Standard deviations were also calculated for the final “best” values of EM, and 6Dw in the course of the leastsquares evaluations of these parameters. These standard deviations did not take into account any covariance with the values of K and c,’ and so are smaller than the true values. Absorbance Errors. The noise in the measuring system of a spectrophotometer increases with increasing absorbance. The Cary Model 14 is shot-noise limited. At a constant setting of the dynode voltage and slit-control circuit, the root-mean-square noise in the measured absorbance is independent of wavelength and theoretically proportional to18 (1 l/7‘)1/z, where T is the transmittance, or to (1 1OA)li2, where A is the absorbance. We measured the average noise on the pen chart of our in-
+
+
+
2 c
0
I
20
18
22
24
5 / 103crn”
Flgure 1. Smoothed spectra of the molar absorption coefficient, E = A / c l , as a function of wavenumber, 5,and wavelength, A, for acridine orange hydrochloride in 0.32 mM HCI, 25.0 OC. The concentrations, ~ in order of decreasing 6 values at 492 nm, are as follows ( c / ~ O -M): 1.4, 1.8, 2.3, 2.7, 3.4, 4.1, 4.3. I n this and the following two figures, the curves were produced by a Houston Instrument Model DP-5 plotter, and are made up of short line segments connecting the data points at a 1-nm wavelength interval.
strument at absorbances between 0 and 2, and confirmed that it is quite accurately proportional to (1 10A)li2. We considered two possibilities for the errors of the absorbance values after smoothing. It is likely that the smoothing procedure still leaves noise as the main source of error in the absorbance values. In this case, the limit of error X(A,,) in an absorbance value may be written
+
X(A,,) = X(O)[(l
+ 10Asw)/2]1’z
(10)
where X(0) is the limit of error of an absorbance of zero. From this expression comes a weighting function for the least-squares procedure, as defined in eq 7 :
w,, = 2/(1
+ 10ABW)
(11)
An alternative possibility is that the smoothing procedure removes all errors due to noise and leaves slidewire nonlinearity as the main source of absorbance error. In this case the limit of error should be constant NA,,) = X(0) (12) with resulting weights
w,, = 1
(13)
Both eq 11 and eq 13 assign a weight of 1 to an absorbance of zero. Thus u, the standard deviation of an absorbance of unit weight, becomes the standard deviation of an absorbance of zero. Results Absorbance Spectra. The absorbance spectra of ten solutions of A 0 in the concentration range (1.4-4.3) X lo-’ M were used for the analyses. Representative spectra are shown in Figure 1. The absorbance maximum appears at 492 nm. The well-known blue-shift and intensity reduction (hypochromism) of the absorption band are evident as the concentration is increased. The appearance of well-defined isosbestic points, at 520 and 474 nm, suggests but does not prove that there is an equilibrium between only two different absorbing species. These isosbestic points disappeared at higher concentrations. Matrix rank analyses were made on sets of smoothed spectra, using absorbance values at 5-nm wavelength
The Journal of Physical Chemistry, Vol. 82, No. 24, 1978 2573
Acridine Orange Dimerization Equilibrium
h/nm
a/nm
550
450
1
18
eo
,
I
22
3/ io3
--24
I
Figure 2. Monomer and dimer spectra derived from K = 6.75 X lo3 M-I. The error bars are equal in length to 6 times the calculated standard deviations of t.
intervals from 555 to 420 nm. The error function of eq 10 (shot noise-limited error) was used treating U O ) , the limit of error of an absorbance of zero, as an adjustable parameter. With the set of all ten spectra, it was found that the conditions X(0) I0.0014 allowed the number of independent absorbing components in the solutions to be limited to two as required by equilibrium 1. Quantitatively similar results were obtained with sets of from nine to six spectra, in which the spectra of the most concentrated solutions were progressively omitted. When equal limits of error for all absorbances were used (eq 12, slidewirelimited error), a larger value of the minimum limit of error, 0.006, was required for two independent components. This value is three times the manufacturer’s specificationls of the maximum error of the slidewire, and is thus inconsistent with the assumption of slidewire-limited error. Dimerization Constant. The smoothed spectra of the series of ten solutions were used to evaluate the “best” value of the thermodynamic dimerization constant, K , for equilibrium 1. Each spectrum consisted of 139 absorbance values in the wavelength range 555-417 nm. The weighting function of eq 11 was assumed, and the concentrations were adjusted. The resulting “best” value was K = 6.75 x lo3 M-l with a standard deviation uK = 0.28 X lo3 M-l. The maximum percentage adjustment of a concentration was only 0.370,which is approximately the uncertainty of the concentration of the stock solution but greater than the imprecision of the stock solution dilutions. An absorbance of zero had an estimated standard deviation a = 0.0006, and an estimated uncertainty (3u limits) of 0.002. No obvious systematic trends could be seen in the values of the absorbance deviation A,, - A,,‘, such as a correlation with the wavelength or a consistent trend with increasing concentration at one wavelength. Monomer and Dimer Spectra. Figure 2 shows the monomer and dimer spectra derived from the “best” value of K. The monomer has a maximum molar absorption coefficient eM = (6.37 f 0.04) X lo4 M-l cm-l occurring at 493 nm; the dimer has maxima at 513 and 468 nm, and a minimum a t 503 nm. The error bars indicated in Figure 2 were obtained from the calculated standard deviations; these error bars indicate the uncertainty due only to random errors in the measured absorbances. The appearance of the derived dimer spectrum changes in a systematic way with the value assumed for K , as illustrated by Figure 3. Curves b and c are derived from values of K which differ from the “best” value by the
I8
1
’
500 I
I
1
450 1
1
1
PO
’
22
24
?/i0Jcm-l
Figure 3. Dimer spectra derived from different values of K : (a) K = 5.00x i o 3 M-I; (b) K = 5.91 x i o 3 m-I; (c) K = 7.59 x io3 M-I;. (d) K = 10.0 X IO3 M-’. 800
’
n ’
X,,l
IO 4
Figure 4. Distribution of weighted absorbance deviations: (solid lines) lo4 times the fraction of the total number of values of x, having values in an interval of as a function of the value of xsw;(dashed curve) normal distribution function given by eq 14, with (T = (S0/MN)”* = 5.53 X The areas under both the solid lines and the curve are unity.
approximate uncertainty in this value (3aK = 0.84 x io3 M-l), and therefore a comparison of these curves indicates the uncertainty in the dimer spectrum which arises from the uncertainty in the value of K. Most of the variation is seen to occur in the wavelength region of the minimum, around 500 nm. Figure 3 also illustrates two extreme types of possible dimer spectra: two separated and almost completely resolved peaks (curve a), and one peak with no minimum (curve d). The value of K cannot be smaller than 5 X lo3 M-l (curve a) without leading to some negative values of the molar absorption coefficient, which would be physically impossible. Distribution of Absorbance Deviations. The leastsquares procedure is based on assumptions that the random errors in the measured absorbances are normally distributed, and that appropriate weights, w,,,have been chosen to properly relate the standard deviations of these distributions to one another according to eq 7. In order to examine the validity of these assumptions for our calculations, we have calculated values of the quantity x,, = ( W ~ , ) ~ ’ ~ (-AAs:) , , which is a kind of weighted absorbance deviation for solution s at wavelength w. If the assumptions stated above are correct, in an infinite population of measurements the values of xsw would be normally distributed with a variance of a2 and a distribution function given by P ( x ) = [ 1 / ~ ( 2 n ) ~exp(-x2/2a2) /~]
(14)
P(x) dx is the probability that the value of x,, lies between x and x + dx.
2574
C.A. Brignoli and H. DeVoe
The Journal of Physical Chemistry, Vol. 82, No. 24, 1978
TABLE I: Least-Squares Results (Acridine Orange Hydrochloride in 0.32 mM H a , 25.0 C) ~ / 1 0 3 uK/103 modifications“ u/10-4 M-1 M-1
unsmoothed spectra no concn adjustment equal weights (eq 1 3 ) 9 spectra 8 spectra 7 spectra 6 spectra 1 wavelength (492 nm), no concn adjustment
6.2 10.7 6.2 13.2 5.8 5.7 5.9 5.7 3.3
6.75 6.76 6.98 8.20 7.67 5.98 6.64 8.50 8.8
0.28 0.48
0.27 0.30 0.34
0.32 0.41 0.60 1.1
“ Except for the modifications noted, the calculations were carried out using ten smoothed absorbance spectra at 139 wavelengths, adjusted concentrations, and the weighting function of eq 11. The actual distribution of the finite number of x,, values obtained from the evaluation of K is shown in Figure 4 (solid lines). The distribution is seen to agree reasonably well with the normal distribution (dashed curve) calculated from eq 14 with u replaced by the root-mean-square deviation of an absorbance of unit weight. This agreement is consistent with the assumptions stated above, and increases our confidence in the validity of applying the least-squares procedure to our data. Effects of Modifications. The results of making various kinds of modifications in the least-squares analyses are shown in Table I and discussed below. 1. Effect of Using Unsrnoothed Spectra. When the absorbance spectra were not smoothed, practically the same “best” value of K was obtained and the monomer and dimer spectra were quantitatively indistinquishable from those derived from smoothed spectra, except for the presence of random noise. The disadvantage of using unsmoothed spectra shows up as an almost twofold increase in the standard deviation uK of the dimerization constant. 2. Effect of Not Adjusting Concmtrations. The main effect was a 3% increase in the “best” value of K , The change was slightly less than the standard deviation, OK. There was practically no change in the monomer and dimer spectra. 3. Effect of Weighting Function. The choice of weighting function for the absorbances had a large effect on the calculated “best” value of K. The weighting function of eq 11 (based on shot noise-limited error) allowed a better fit of the data (smaller value of a) than did an assumption of equal weights (slidewire-limited error). The uncertainty of an absorbance of zero (3u = 0.004) calculated from the latter assumption was larger than the instrument specification18of the maximum slidewire error (0.002), and was thus inconsistent with the assumption. 4. Effect of the Number of Spectra. As the spectra of the more concentrated A 0 solutions were progressively omitted from the set of spectra used in the calculation, there was an erratic variation in the “best” value of K and a progressive increase in the standard deviation of these values. The “best” value obtained with all ten spectra has the least uncertainty, showing the advantage of using a large number of spectra, and this value of K agrees with the values obtained from fewer spectra within the uncertainty ( 3 4 of those values. 5. Effect of Wavelength. The calculations of K described above were made with absorbances measured a t 139 wavelengths. In principle any number may be used. Calculations made with absorbances at only one wavelength (and, necessarily, without concentration adjustment)
gave “best” values of K which varied widely with the wavelength choosen, but at most wavelengths the value of K was within two standard deviations of the value obtained using 139 wavelengths (K = 6.75 x lo3 M-l). In other words, large deviations of the “best” value of K from 6.75 X lo9M-l were accompanied by large standard deviations. The smallest standard deviations were obtained a t wavelengths close to the absorbance maximum at 492 nm. The results obtained a t 492 nm are shown in Table I; the value of uK is seen to be four times greater than for the calculation based on 139 wavelengths, illustrating the substantial reduction in the uncertainty of the “best” value of K which can be achieved by using many wavelengths.
Discussion For the least-squares analyses we assumed that, up to the highest A 0 concentration used (4.3 X lo-&M), the solutions contained only two independent absorbing species (D’ and D 2 T This assumption is justified by the presence of good isosbestic points (Figure l),and by the finding that neither the results of the matrix rank analyses nor the fit of the absorbance spectra to equilibrium 1 (as measured by 5) were significantly affected by omitting the more concentrated solutions, which would be expected to contain a higher fraction of higher aggregates if they had been present. Absorbance errors in the smoothed spectra were treated more satisfactorily by assuming they were shot-noise limited, increasing with increasing absorbance, than by assuming they were slidewire-limited and independent of absorbance. This conclusion is based on the results of the matrix rank analyses and the effect on 5 of changing the weighting function. Our “best” value of the thermodynamic dimerization constant a t 25 O C calculated on the basis of shot noiselimited absorbance errors, K = (6.8 f 0.8) X lo3 M-I, is lower than previously reported values of K,: K,/103 M-l = 10.5 (at 25 10.6 (at 25 O C ) ? and 9.2 (at 22 OCh6 In each case, the solutions contained acridine orange hydrochloride at such low ionic strengths that there should be little difference between the values of K and Kc. Our lower value partly resulted from the weighting function used, which assigned relatively less weight to the higher absorbance values. Our derived value for the molar absorption coefficient of the monomer peak, eM = (6.37 f 0.04) X lo4 M-’ cm-” at 493 nm, is within 6 % of values obtained by othersa4+ The “best” derived dimer spectrum has two peaks, symmetrically split in energy about the position of the monomer peak (Figure 2) as predicted theoretica1ly.’l-l3 Kurucsev and Strauss6 obtained a similar dimer spectrum, while others derived dimer spectra having either a single peak and a shoulder4v5or a single symmetric peak? These variations can be explained by the different values of the dimerization constant used to derive the dimer spectra from absorbance spectra; the larger is the assumed value of K , the less prominent i s the red-shifted peak in qualitative agreement with our calculations shown in Figure 3.
Table I shows the advantage, in reducing the uncertainty of the calculated value of K , of smoothing the absorbance spectra and of using as many spectra and wavelengths as possible. Adjusting the experimental concentration values turned out not to affect the results much, apparently because the errors in these values were small.
Acknowledgment. This research was supported in part by the General Research Board of the TJniversity of Maryland, and by Research Corporation. The computer
The Journal of Physical Chemistry, Vol. 82, No. 24, 1978 2575
Photoluminescence Spectroscopy in Low Temperature Matrices
time was supported in full through the facilities of the Computer Science Center of the University of Maryland.
(7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)
References and Notes (1) Taken in part from the F’h.D. Dissertation of Carol A. Brignoli, University of Maryland, 1973. (2) D. J. Blears and S. S. Danyluk, J . Am. Chem. Soc., 89, 21 (1967). (3) V. Zanker, Z. Phys. Chem. (Leipzig), 199, 225 (1952). (4) M. E. Lamm and D. M. Neville, Jr., J. Phys. Chem., 69, 3872 (1965). (5) Y. Kubota and M. Miura, J . Sci. Hiroshima Univ., 30, 49 (1966). (6) T. Kurucsev and U. P. Strauss, J . Phys. Chem., 74, 3081 (1970).
R. E. Ballard and C. H. Park, J . Chem. SOC. A , 1340 (1970). R. Larsson and 8. Nordbn, Acta Chem. Scand., 24, 2563 (1970). F. Watanabe, J . Phys. Chem., 80, 339 (1976). F. Watanabe, Bull. Chem. SOC.Jpn., 49, 1465 (1976). E. G. McRae, Aust. J . Chem., 14, 344 (1961). H. DeVoe, J . Chem. Phys., 41, 393 (1964). H. Ito, T. Eri, and Y. J. I’Haya, Chem. Phys., 8, 68 (1975). K. Bergmann and C. T. O’Konski, J. Phys. Chem., 67, 2169 (1963). A. Savitzky and M. J. E. Golay, Anal. Chem., 36, 1627 (1964). R. M. Wallace and S.M. Katz, J. Phys. Chem., 68, 3890 (1964). L. G. SillBn, Acta Chem. Scand., 16, 159 (1962). Cary Instruments Application Report AR 14-2, ”Optimum Spectrophotometer Parameters”, Sept 1964.
Electronic and Vibrational Relaxation Studied by Photoluminescence Spectroscopy in Low Temperature Matrices. 2. i 1 A 2 State of Formaldehydes (H,CO, HDCO, and D2CO)‘ Luisa T. Molina, Kenneth Y. Tang,+ John R. Sodeau, and Edward K. C. Lee* Department of Chemistry, University of California, Irvine, California 927 17 (Received April 28, 1978; Revlsed Manuscript Received August 4, 1978) Publicatlon costs assisted by the National Science Foundation
The photophysics of electronically excited formaldehyde molecules in low temperature matrices have been studied by photoluminescence spectroscopy and lifttime measurement. Laser-induced emission spectra show only fluorescence from the zero-point level of the AIAz state. A rapid vibrational relaxation from the predissociated upper vibrational levels is also observed. The lifetimes of the A state of H,CO, HDCO, and DzCO in a Xe matrix at -20 K are 54, 105, and 508 ns, respectively, and the matrix-induced electronic quenching rates at 20 K are in the range of 2-6 X lo6 s-l, The lifetime of DzCO is reduced by ca. tenfold from the gas phase value. It is suggested that the nonradiative behavior of formaldehyde is more like a “small” molecule than a “large” molecule.
Introduction Gas phase studies of photophysics and photochemistry of formaldehyde from its single vibronic levels in the first excited singlet state (A1A2)have drawn much attention r e ~ e n t l y . ~The - ~ mechanisms of radiationless transitions and the dynamics of photodissociation processes are of both theoretical and experimental interest. The present study of low temperature matrix photoluminescence spectroscopy of H2C0, HDCO, and DzCO was initiated to complement our on-going gas-phase s t u d i e ~ The . ~ present experimental work was carried on in a manner similar to our earlier study of the vibrational and electronic relaxation of the l(n,x*) states of cyclic ketones in low temperature matrices.&In the meantime, a recent study of low temperature matrix photochemistry of formaldehyde in our laboratory has shown that photoexcited formaldehyde regranges to a transient radical species, hydroxymethylene (HCOH), and undergoes chemical reactions with a variety of matrix reactantsa6In view of this new development, we have studied the fluorescence decay behavior in the dilute matrix as well as in pure formaldehyde solid. We have been able to establish the following in the low temperature matrix (20 K): (1)vibrational relaxation is sufficiently rapid to give “fluorescence stabilization” even at the exciqation wavelength of 280 nm where the lifetime of the SI (A1A2)state in the gas phase has been estimated to be 10-10-10-12s due to molecular predisso~iation;~~~’ (2) all of the photoluminescence originates from the ground Maxwell Laboratories, 9244 Balboa Ave., San Diego, Calif. 92123. 0022-3654/78/2082-2575$0 1.OO/O
vibrational level of the AIAz state; (3) no emission attributable to the phosphorescence from the Z3A2state has been seen, even from D2C0 for which laser-induced phosphorescence emission has been seen in the gas phase,* although the observation of the phosphorescence emission in the low temperature solid was previously reported;O (4) the observed values of the fluorescence decay time ( T ~ ) indicate that the matrix-induced radiationless decay rate is relatively slow, i06-i07 s-l.
Experimental Section Formaldehydes were prepared by heating the polymer in the range of 100-140 “C, according to the method of Spence and Wild.lo (CHzO)xwas obtained from Aldrich Chemical and (CHDO)x and (CD20)xwere obtained from Merck Sharp and Dohme of Canada. Research grade Ar and Nzand chemically-pure grade Xe and SF6were used directly as the matrix gas. Formaldehyde was premixed with a matrix gas in a molar ratio of 1:lOO for many runs. Matrix samples were prepared by spraying the mixture for deposition at 20 K at a rate of -1 mmol/h for a period of 1-2 h as described previously el~ewhere.~ Pure formaldehyde matrices were prepared by the pulsed deposition method.’l Three different types of experiments were carried out: a frequency-resolved emission; a time-resolved emission; an excitation spectrum (viewing total emission). The emission from formaldehyde was excited at 325 nm with a continuous wave He-Cd laser (Spectra Physics Model 185, 15 mW). The exciting beam illuminated the cold 0 1978 American
Chemical Society
