Fluorescence Decay of %Naphthol
(4) W. C. E. Higginson, Chem. SOC.,Spec. Pub/., No. 10, 95 (1957). (5) F. Bottornley, Q. Rev. Chem. Soc., 24, 617 (1970). (6) P. B. Pagsberg, I V International Congress on Radiation Research, Abstract No. 641 (1970). (7) B. H. J. Bielski and J. W. Sutherland, I V International Congress on Radiation Research, Abstract No. 88 (1970). (8) N. A. Slavinskaya, B. M. Kozlov, N. V. Petrukhin, S. A. Kamenskaya, and S. Ya. Pschezhetskii, Khim. Vys. Energ., 8 , 68 (1974).
The Journal of Physical Chemistry, Vol. 83,
No. 7, 1979 795
(9) (10) (11) (12) (13)
M. Sirnic and E. Hayon, J. Am. Cbem. Soc., 94, 42 (1972). P. B. Pagsberg and J. W. Sutherland, unpublished results. H. A. Dewhurst and M. Burton, J. Am. Chem. Soc., 77,5781 (1955). M. Lefort and M. Haissinsky, J. Cbem. Phys., 527 (1956). J. Belloni and M. Haissinsky, Int. J . Radiat. Pbys. Chem., 1, 519 (1969). (14) R. Goodrich and J. W. Sutherland, unpublished results. (15) B. H. J. Bielski and H. A. Schwarz, J. Pbys. Cbem., 72, 3836 (1968).
Analysis of Two-State Excited-State Reactions. The Fluorescence Decay of 2-Naphthol William R. Laws and Ludwlg Brand* Department of Biology and McCollum-Pratt Institute, The Jobns Hopkins University, Baltimore, Maryland 2 12 18 (Received July 11, 1977; Revised Manuscript Received October 16, 1978) Publication costs assisted by the Institute of General Medical Sciences
The fluorescence decay of 2-naphthol has been investigated as a model system for the general case of two-state excited-state reactions. Decay kinetics are presented as a function of emission wavelength and pH. Depending on the pH, the fluorescence emission of 2-naphthol may arise from one state or from two states reversibly or irreversibly approaching the excited-state equilibrium. Several errors and corrections, common to these types of reactions, must be taken into account and are described. The rate constants for the excited-state reaction have also been determined by several methods. The results are in agreement with the mechanism proposed by Weller based on steady-state fluorescence measurements. Decay measurements for the acetate-facilitated proton transfer of 2-naphthol are also presented.
Introduction Advances in instrumental methods and procedures for data analysis have made nanosecond fluorometry a powerful tool for the study of excited-state reactions. Although the kinetics of excited-state interactions can be quite complex,l numerous processes follow simple two-state reversible or irreversible mechanisms. Excimer, exciplex, and proton transfer reactions are examples of excited-state reactions that, under suitable conditions, follow two-state mechanisms. The origin of excited-state proton transfer lies in the difference between the ground-state and excited-state ionization constants, and has been recently reviewede2 2-Naphthol (pK, = 9.5 and pK,* = 2.8) is a useful compound to study in this regard since the rate of proton transfer is comparable to the rates of fluorescence and radiationless decay processes. The classical work of Weller3-6on excited-state proton transfer was based in large part on steady-state fluorescence experiments with 2-naphthol. Trieff and Sundheim7have studied the effect of solvent on this reaction, while several investigators have utilized fluorescence decay measurements to study the excited-state proton transfer of this m o l e ~ u l e . ~ In -~~ addition to some discrepancy in the values obtained for the excited-state rate constants, some doubt has also been raised regarding the basic mechanisms involved. The aim of the present paper is to reevaluate the fluorescence decay of 2-naphthol as a function of pH and emission wavelength, to describe some experimental artifacts that must be considered, and to discuss the criteria for determining whether an excited-state reaction follows a two-state reversible mechanism.
experiment. For the pH dependence studies, samples were dissolved at low pH (-2), deaerated with nitrogen, brought to the desired pH, and sealed in a Teflon-stoppered cuvet. All other work was done without purging, with the solution made to the desired pH immediately. Measurements were carried out a t 24 f 1 "C. Fluorescence decay curves were collected on a singlephoton counting fluorometer similar to the one described by Schuyler and 1 ~ e n b e r g . A l ~more detailed description of the instrument and of the method used to generate the time-resolved emission spectra has been presented by Easter et a1.16 Excitation was by an air flash lamp through a Baird-Atomic 3406-A interference filter for the pH and acetate decay studies, a Ditric 313-nm interference filter for the time-resolved emission spectra, or a Baird-Atomic 3300-A interference filter for the wavelength dependence work. Monochromator bandpasses were 6.6 nm for the pH, wavelength, and acetate decay measurements and 13.2 nm for the time-resolved work. Absorption spectra were obtained on a Cary 14 recording spectrophotometer. Fluorescence decay curves were analyzed either by the Laplace t r a n ~ f o r m , 'the ~ method of moments,18 or by nonlinear least-~quares.'~J~ "Goodness"-of-fit was judged by inspection of the reduced x2,the weighted residuals, and by the autocorrelation function of the re~idua1s.l~ Criteria for analysis of decay curves have been published elsewhere.20,2' Data analysis was accomplished using a Hewlett-Packard 2100 minicomputer. Numerical derivatives were calculated over consecutive nine point intervals by a second-order polynomial. The resulting plot of derivative/intensity vs. intensity ratio was analyzed by a linear least-squares routine.
Experimental Section Solutions of %naphthol M) in deionized water were used for the decay measurements. 2-Naphthol was purified as previously d e ~ c r i b e d . HC1 ~ and NaOH were used to adjust to pH. The pH was checked before and after each
Results (A) Criteria for a Two-State Excited-State Reaction. In this section we describe the kinetic characteristics of two-state reactions in general and give special attention to excited-state proton transfer. The reaction scheme
0022-3654/79/2083-0795$0 1.OO/O
1979 American Chemical Society
796
The Journal of Physical Chemistry, Vol. 83, No. 7, 1979
H+
H+ Figure 1. The kinetic scheme for excited-state proton transfer where A is naphthol and B is naphtholate. The rate constants are explained in the text.
applicable to proton transfer with 2-naphthol is illustrated in Figure 1. A and B refer to naphthol and naphtholate, respectively. Excited naphthol, A*, can return to the ground state by fluorescence (hFA) or quenching (hQA) which are designated by the combined rate constant kA. A* can also lose a proton to form B* as designated by kBA. The excited naphtholate, B*, can fluoresce (kFB) or undergo nonradiative conversion to B (hQB)as designated by the combined rate constant k B , or it can accept a proton to re-form A* as indicated by the bimolecular rate constant kAB.
P W
Figure 2. Lifetimes as a function of pH. The solid curves are the computer-generated functions based on eq 3 and 4 using the rate parameters determined in this study: k,, = 7.0 X lo7 s-’, k,, = 4.7 X loioM-’s- k, = 1.38 X 10’ S-’, and k, 1.06 X lo’s-’. The dashed curves are the generated fits using parameters from the lite r a t ~ r e :k~,, = 5.1 x 10’ s-’, k,, = 5.0 x IO” M-’ s- k, = 1.49 X 10’ s-I, and k, = 1.1 X 10’ s-’. The points (0)are the experimenal data taken from Table IV.
’,
’,
The differential rate equations for the decay of A* and B* are given in eq 1 and 2. Integration of eq 1 and 2 with -d[A*]/dt =
+ ~BA)[A*] - hAB[H+][B*] (1) k +~ ~AB[H+])[B*]~ B A [ A * ] (2)
(kA
-d[B*]/dt = (
-
the initial boundary conditions that only A is directly excited, that is [A*] = [A,*] and [B*] = 0 at t = 0, yields for the fluorescence decay a t wavelength A:
+ a2(X)e-t/T2
(3)
+ pz(A)e-t’T2
(4)
IA(X,t) = al(A)e-t/T1 IB(X,t) =
/?l(X)e-t/rl
The decay times and amplitudes are related to the rate constants indicated in Figure 1 as shown in eq 5-8, where y1,y2 =
71-1,72-1
~ ~ B A ~ A B [ H (+5 )] ] ~
C A ( = ~CA(U[A~*I(X - YJ~FA - /YZ) ( Y(6) ~ -
72) ( 7 )
-pl(X) = pZ(X) = P(X) = CB(X)b = CB(X)hBA[A@*lkFB/(rl - Y2) (8)
kA +
and Y = k B + hAB[H+],while C,(X) and CB(X) are the spectral emission contours, normalized to unit area, of species A and B, respectively. The important features of eq 3-8 to be emphasized are that the two decay times, T~ and r2,are the same for the decay of both species and that the two amplitudes describing the decay of B* are identical in magnitude but opposite in sign. Also, eq 5-8 have [H+] dependent terms; thus the lifetimes and amplitudes will have a known p H dependence. These features provide important diagnostics for determing whether an excited-state reaction, such as 2-naphthol, falls into the class of a two-state reversible reaction. The rate equations for this system (eq 1 and 2) are simplified under some conditions. At high hydrogen ion concentration (pH before correction
PH 2.5 3.1 3.9 5.1 a
4, -0.476 -0.480 -0.483 -0.487
Normalized such that
B, 0.524 0.520 0.517 0.513
after correction B, -0.490 -0.492 -0.493 -0.495
IO, I t IO, I =
1.
0% 0.510 0.508 0.507 0.505
Measured at
450 nm.
at pH -0. R2,cB(360)/cB(450),is the ratio of naphtholate emission at 360 nm to that at 450 nm and is obtained at pH -12. It is assumed that the spectral distribution of the fluorescence of A* and B* is pH independent. Consequently I A ( X , ~ )= (IA’(X,~) - &l~’(X,t))/(l - R1R2)
IB(A,t) = (IB’(A,t) - R ~ I A ’ O , ~ ) )-/R1R2) (~ In the results to be presented below, spectral corrections were made for each set of decay curves obtained at 360 and 450 nm. The decay curves were properly shifted in time prior to the correction. The magnitude of the effect of spectral overlap on the amplitudes obtained at 450 nm is indicated in Table I. As expected the effect is most significant at low pH where the intensity of A* emission is high compared to B* emission. Although there is still a small difference after correction, there are several errors that could be involved. The most significant error in making the spectral correction is in the determination of R1 and R2, since this involves the measurement of low emission intensities. Other errors to be considered are the assumption of spectra shape independence with pH and the ability of analysis methods to extract equal and opposite amplitudes. Time-resolved emission ~pectra~O3~~ and Laplace transforms12 can also be used to do the spectral correction. Direct excitation of B in the ground state becomes significant for the 2-naphthol system above neutral pH. Although eq 3 and 4 still hold, the boundary conditions used for eq 6-8 no longer apply. The amplitude expressions with the new boundary conditions of [A*] = [&*] and [B*] = [Bo*]at t = 0, for the mechanism presented in Figure 1, are as follows:
+ &(~)]e-t/~z (9)
Equation 9 indicates that if the emission at some wavelength, A, has its origin only in /3*, the observed amplitudes will be equal in magnitude. If, however, there is a significant contribution by emission from A*, the amplitudes will not be equal. In the experiments to be described, decay curves representing A* and B* emission were collected with emission wavelengths of 360 and 450 nm, respectively. The decay curves are readily corrected for spectral overlap. If I’refers to the observed experimental decay curve, then
The ratio [Bo*]/[Ao*] was expressed by a / ( l - u ) , where u was taken to be
IA’(A,~~ = IA(A,~)& I ~ ( h , t ) IB’(A,~) = IB(A,t)
RiIA(A,t) R1, c~(45o)/C*(360),is the ratio of naphthol emission at 450 nm to that at 360 nm and is obtained experimentally
a valid approximation under the condition of low optical densities. €*(A) and cB(A) are the extinction coefficients of A and B, respectively, and L(A)is the lamp profile over the bandpass described by AJ2. For the 2-naphthol
The Journal of Physical Chemistry, Vol. 83, No. 7, 1979
W. R. Laws and L. Brand
TABLE 11: Correction of Decay Amplitudes Based o n the Ground-State Naphtholate Effect
problems. As indicated in Table 11, p2conis now quite close to one, well within the experimental error of determining the spectral overlap factors. Ground-state naphtholate only had a noticable effect for one point in Table IV (pH 7.7). However, other two-state processes of general interest might have significant amounts of ground-state B which would need to be corrected by the above procedure. A variety of techniques have been used to analyze fluorescence decay curves. These include graphical methods,26 the method of moment^,^^^^^ the Laplace transform,17 the method of modulating functions,28Fourier transform^,^^,^^ and the method of nonlinear leastsquares.16J9 The fluorescence decay of 2-naphthol was obtained a t 360 and 450 nm a t pH 2.7. The data were corrected for timing distortions and spectral overlap and then analyzed by three different methods. The results are compared in Table I11 and indicate very similar decay constants and amplitudes. It has previously been pointed that the accuracy of exponential analysis depends more on the degree of correlation between the parameters than on the method of analysis used. It is our experience that most algorithms fail to correctly analyze decay curves containing short lifetimes and/or small amplitudes. Consequently, caution must be used when trying to fit decay data to Figure 1, or any mechanism, when parameters may be small, as indicated in certain regions of Figures 2 and 3. ( C ) Evaluation of the 2-Naphthol System. The fluorescence decay parameters obtained for 2-naphthol as a function of pH are summarized in Table IV. Naphthol was found to decay as a double exponential in the pH region 1.7-3.9, where the reaction is reversible, and as a single exponential at higher pH. The decay of naphtholate is a double exponential throughout the entire pH range 1.7-7.7. The two lifetimes, r1 and r2, obtained at the different emission wavelengths are identical within experimental error, and are shown as a function of pH in Figure 2. The amplitudes P1 and pz for the decay of naphtholate are equal in magnitude but opposite in sign within experimental error. lpsl is slightly larger than ]&/and this is attributed to the errors mentioned above. The more
798
as
PH
0,'
U
gd
0.018 0.084 0.225
0.028 0.142 0.451
a2
1.04 1.17 1.50
atcon
1.012 1.028 1.049 a Measured at 450 nm. Corrected for time and s ectral overlap effects. Normalized such that 14 I = 1. 'Experimental values used were hB* = 7 x 10' s - ' , T , = 4.6 ns-', and r 2 = 9.2 ns-'. 7.1 7.8 8.3
-1.0 -1.0 -1.0
system, the excited-state reaction is irreversible when ground-state B is present; thus y2 approaches Y and [H'] = 0. As noted in Figure 3, cy&) = 0 under these conditions, while terms are also lost from the other amplitude expressions. Equations 10-13 were generated for their pH dependence and are shown by the dashed lines in Figure 3. As indicated, deviations due to ground-state appear around pH 7. As [B] increases the amplitudes associated with A* (a1)and proton transfer (pl) tend toward zero, until by pH 10 the majority of the fluorescence is from directly excited naphtholate. That ground-state B dominates so close to the pK,, where [A] will still be significant, is due to the experimental conditions of the overlap of the lamp and absorption spectra. The decay of B* can now be expressed as
Normalization of P1
to 1 means
+ P2 =
(e)( r) 71 -
Yz
=
and where pzCorr should be equal to one. Decay curves were collected to test this correction procedure and were analyzed correcting for the time and spectral overlap TABLE 111: -
Comparison of Analysis Methodsa B*C
A*b
method of analysis Laplace method of moments nonlinear least-squares
ns 3.5 3.6 3.5
T , ,
r 2 ,ns
8.2 8.3 8.3
012
2.6 2.5 2.6
a 2-Naphthol, pH 2.7; corrected for timing and spectral distortions.
ns 3.4 3.4 3.4
T , ,
ffl
3.4 3.5 3.6
ns 8.3 8.3 8.4
0,
T?,
Measured at 360 nm.
- 1.1 - 1.2
-1.2
02
1.1 1.2 1.2
' Measured at 450 nm.
TABLE IV: Exponential Analysis of 2-Naphthol as a Function of pHa B*C
A*b
PH 1.7 2.2 2.5 2.7 3.1 3.4 3.9 4.8 5.5 6.05 6.6 7.25 7.7
r l ,ns
ns 7.30 7.67 7.70 8.29 8.24 9.45 9.60
r2,
NI
0.35 1.33 1.37 3.03 2.16 4.24 1.83 4.74 4.40 3.97 4.04 3.49 3.04
0'
1.91 6.74 2.59 2.68 0.82 0.67 0.09
r I I ns
0.85 1.7 2.68 3.25 4.02 4.33 4.60 4.77 4.83 4.89 4.70 4.85 4.54
r l I ns
Dl
7.36 -0.14 2.14 7.68 -0.57 2.48 7.93 - 0.60 - 0.95 3.48 8.38 - 0.88 4.03 8.77 - 1.16 4.35 9.19 4.65 9.44 - 0.72 - 1.36 4.85 9.46 9.44 - 1.21 4.87 - 1.10 4.81 9.30 -1.10 4.84 9.47 - 1.02 4.81 9.38 - 0.83 4.82 9.51 Measured at 360 nm. a Corrected for timing and spectral overlap distortions, but not for ground-state B. at 450 nm. 0.80
02
0.16 0.60 0.62 0.98 0.91 1.21
0.74 1.43 1.28 1.19 1.17 1.18 1.01 Measured
Fluorescence Decay of 2-Naphthol
2. 0
-
impulse response^.'^
Q,T,
>
-
c
VOI. 83. NO. 7, 1979 799
Figure 8. A three-dimensional view of the fluorescence emission of ?-naphthol as a function of time in 1 mM phosphate buffer. pH 7.15. Decay C U N ~ Swere collected at 40 wavelengths and deconvolved. Emission spectra were then generated at 60 different times from the
.- ....'.
1.0
The Journal of rnyskd Chemistry.
I
I
\
...... .. *
S
O
D
B
.
0
0
E
. o -
~
0 2 3 ~
~
O
0
.
D
W A V E L E N G T H lnml
WAVELENGTH
(NM)
Figure 5. Relative contribution of each ilfetlme, expressed by a i . fa the parameterr shown In Fgre 4. air, (B)is the cantnbuikm associated with the longer decay time T , in Flgwe 4, while a2r,is assodated with the shorter lifetime
T?
significant deviation between & and B2 a t pH 7.7 is due t o the ground-state absorption by naphtholate. Some of the experimental amplitudes as a function of pH are shown in Figure 3. The dependence of the decay parameters on the emission wauelength was studied a t pH 3.15. At this pH the reaction is reversible and double exponential decay behavior is predicted for both species; therefore, no variance in the lifetimes is expected with emission wavelength. As shown in Figure 4 this prediction was found t o hold with average lifetimes of 8.8 and 4.05 ns constant throughout the spectrum. The deviations a t 385 and 390 nm can he explained by realizing that a t a particular wavelength interval, two amplitudes will cancel one another and the decay will he monoexponential (eq 9). This occurs for 2-naphthol around 390 nm, as shown in Figure 5. The a times T plot, which is a measure of relative contribution of a particular exponential component, shows that the amplitude associated with r2 is decreasing before becoming negative. Consequently, the analysis algorithms try to compensate for the small amplitude by altering the lifetime. The overall emission characteristics of the 2-naphthol system a t pH 7.15 are illustrated in Figure 6. This three-dimensional representation shows the fluorescence
Figure 7. NapMhol contribution (time zero of Figure 6) subtracted from I(X. f) Naphtholate emissions, normalized to peak height, are shown for 4.9 ns (-), 9.8 ns (--), and 15.3 and 20.2 ns (---). Other times examined. also with no spectral changes. were 25.1 and 30 ns. Subtraction was done after normalizing the time zero spectrum to the leading edge of the naphthol band of the indicated times: variances observed are due to normalization errors. intensity as a function of emission wavelength and time. The direct decay of A* is seen at the lower wavelengths, while the initial rise from zero followed by a slower decay is observed for B* at the higher wavelengths. This nanosecond time-resolved emission spectrum is not distorted by the convolution artifact.16 Proton transfer in the excited state creates a charged molecule; consequently, solvation around B* might occur after its formation resulting in a time-dependent shift of the emission maximum.16 Although the decay kinetics do not indicate any other processes occurring (Figure 4), naphthol contribution was subtracted out of the timeresolved spectra (Figure 6). leaving the emission due t o naphtholate at the various times. As shown in Figure 7, there appears to he no additional process occurring in the excited state BS the spectral characteristics of the emission are constant. However, solvation could have already taken place prior to emission. (0)Determination of the Excited-State Rate Constants. Once the kinetic behavior of an excited-state reaction has been established, rate constants for the reaction can be obtained from fluorescence decay data. As is shown in Figure 3, the decay of A* becomes a single exponential beyond neutral pH with a decay time equal to (kA + k&.
800
The Journal of Physical Chemistry, Vol. 83, No. 7, 1979
TABLE V: Quenching of Naphthol by Hydrogen Iona
a
H' activity
lifetime, ns
0.0796 0.378 1.016
7.1 6.8 6.0
H+activity
lifetime, ns
3.949 10.75
3.9 2.2
W. R. Laws and L. Brand
TABLE VI: Effect of Acetate on the Decay of 2-Naphthol a t pH 6.9' [-OAcI,
M 0.0 0.024 0.048 0.072 0.104 0.128 0.152 0.176 0.20
Measured at 360 nm.
A*b T,
ns
4.54 3.35 2.96 2.25 1.86 1.75 1.48 1.36 1.24
B*C CY
2.09 1.41
1.10 0.61 1.07 0.84 0.32 0.94 0.58
T,,
ns
4.41 3.24 2.72 2.17 1.77 1.57 1.36 1.20 1.10
r1,
ns
9.07 8.98 9.08 8.97 8.91 9.00 8.94 8.95 8.94
p1
Pl
-0.72 -0.58 - 0.61 -0.38 -0.48 -0.35 -0.28 - 0.39 -0.35
0.75 0.60 0.62 0.40 0.49 0.36 0.29 0.40 0.34
Corrected for timing and spectral overlap distortions, Measured at 360 nm. Measured at 4 5 0 nm.
a
(71+ y2) vs. [H+] plot which yields kAB;insertion of (a1 + a2)into eq 8 for [A,*] allows h g A to be calculated, and
a
l_II
t
. ~0 ~ . 0"2 " . 0"4 " . 0" 6 " . O" B " . "10 Yl
. 1 2 . 1 4 . 16 . 18
Y2
+
Figure 8. A plot of y1 y2 vs. y1y2for the data in the reversible pH region in Table I V (0). The slope, (kA)-l,was determined to be 7.23 ns by least-squares fitting analysis.
Likewise in strong acid the decay of A* becomes monoexponential with a decay time equal to (hA)-l. Thus hgA can simply be obtained as the difference of the reciprocals of two lifetimes, if it is assumed that hA is independent of hydrogen ion concentration. However, this assumption is not valid for 2-naphthol. Table V shows the decrease in the lifetime of naphthol with increasing hydrogen ion activitya31The data give rise to a linear Stern-Volmer plot (aH+vs. ( T ~ / T ) 1) with a quenching constant h, 3X lo7 M-l s-l. Chloride ion also quenches naphthol but to a much smaller extent. Experimentally this problem cannot be overcome in the 2-naphthol system since a t hydrogen ion concentrations low enough to avoid quenching, proton transfer occurs; the decay is then quite complex. The lifetime could be estimated by extrapolating to zero quencher activity; however, a more desirable approach to the problem exists that does not require measurements under extreme acid
-
condition^.^^
use of derivative plots allow the determination of hA, hB, k B A , and k A g . For the data presented in Table IV, average values obtained were hA = 1.38 X los s-', hB = 1.06 x lo8 7.0 x 10' S-', and h A B = 4.7 X 10" M-' S-'. S-l, hgA The rate constants obtained above were used together with eq 5-8 to generate the theoretical plots of the 7'5, a's, and IPI vs. pH shown in Figures 2 and 3. The dashed curves in Figure 2 are based on rate constants obtainedg under conditions where many of the errors described here were not yet recognized. Small variations in the parameters lead to significant changes in the lifetime vs. pH profiles. The experimental decay times (Table IV) are in good agreement with the theoretical curves based on the rate constants obtained here. (E) Experiments in the Presence of Other Proton Acceptors. Proton transfer is facilitated in the presence of a proton acceptor such as acetate. The mechanism shown in Figure 1 can be modified according to
A*
The corresponding rate equations then become -d[A*]/dt =
+ hgA + k[R])[A*] -
-d[B*]/dt = ( h +~~ A B [ H + + ]h[RH])[B*] ( ~ B A+ k[Rl)[A*l (19)
l/[(x+ Y)f {(Y x) 4 h ~ ~ ~ ~ ~ where x = h A hgA + h[R], ~ B = ~A A B~[ H +~]h[RH], ~ hBAaPP kgA + h[R], and Y = k B + hm[H+] + h[RH]. The
h~
+ ~AB[H+]
elimination of hAB[H+]gives Y1
(KA
(~AB[H+ +Ih[RHl)[B*l (18)
y1,')'~=
and
+ yz = x + Y = x
(17)
Equations 18 and 19 are of the same form as eq 1 and 2 and will integrate similarly to the sums of two exponentials. The lifetimes will now be expressed by
From eq 5
71
k
+ R AB* + RH k2
+ YZ = x[1- h B / k A l + h B + ? l Y Z / h A
(l6)
A plot of y1 + yz vs. ylyzshould then yield a straight line with a slope = (hA)-l. This plot is shown in Figure 8 for 2-naphthol in the p H region where the reaction is reversible (Table IV). The slope was obtained by linear least-squares and was found to be 7.23 ns f 0.570,which means h A = 1.38 X lo8 s-'. At neutral pH the decay of A* The rate is monoexponential and T~ = ( k A + kBJ1. constant hgA can now be obtained by difference. Under irreversible conditions ( h A + hBA)-l = T~ = 4.81 ns (Table IV) and thus hgA = 7 X lo7 S-'. A number of other methods were employed to obtain the excited-state rate constant^.^ Included in these were
-
-
amplitude expressions are also changed accordingly. Thus double exponential decay behavior is still predicted but the rate of proton transfer will be enhanced by proton acceptors such as carbonate, acetate, or phosphate. Analysis of the exponential decay behavior can yield the rate constants as before. For example, since (71 + yz) = h~ h g 4-~ hi[R] + h~ + ~AB[H'] k,[RH], plots Of (71 + y2) vs. [H+], [R], or [RH] will give AB, kl,and hz, respectively. An example of an experiment showing facilitated proton transfer is given in Table VI. Decay curves of 2-naphthol at constant pH (6.9) were obtained in the presence of varying concentrations of acetate. The short decay time decreases with increasing acetate concentration reflecting the more efficient depopulation of A*. The rate param-
+
h ~ g
The Journal of Physical Chemistry, Vol. 83, No. 7, 1979 801
Fluorescence Decay of 2-Naphthol
TABLE VII: pH Effect on t h e Decay of 2-Naphthol in the Presence of 8 x
1.7 2.7 3.1 3.6 4.2 4.5 4.7 4.75 4.8 5.0 5.7 6.1 6.5 6.8
7.0 x 10-5 7.0 x 10-4 1.7 x 10-3 5.1 x 1.7 X lo-’ 2.75 X lo-’ 3.7 x 4.0 x 4.4 x 105.2 X lo-’ 7.2 x l o - > 7.7 x 10-2 7.85 x lo-’ 7.9 x 10-
7.26 3.69 3.48 3.82 3.40 3.09 2.84 2.84 2.71 2.64 2.27 2.19 2.11 2.19
8.06 7.56 7.58 7.29 17.61 22.8 27.9
a Corrected for timing and spectral overlap effects.
l o w 2M Sodium Acetatea
1.01
1.96 1.20 1.68 1.55 1.48 1.14 1.36 0.87 0.92 0.90 1.00 0.96 0.93 0.95
3.70 3.70 3.80 3.26 2.91 2.66 2.62 2.55 2.42 2.09 2.04 1.98 2.03
0.82 0.65 0.20 0.05 0.003 0.002 0.0006
Measured at 360 nm.
eters were obtained as indicated above and kl = 3.2 x lo9 M-l s-l. A Stern-Volmer plot of the decay parameters of A* vs. acetate concentration gave a quenching constant hQ = 3.1 X lo9 M-l s-’. That hl = h~ is to be expected since facilitated proton transfer can be considered as another quenching process of A*. The value of k l obtained here is in accord with the value of 2.9 X lo9 M-I s-l obtained by Weller6 and may be compared to the value of 2.4 X lo9 M-l s--l due to Trieff and S ~ n d h e i m .The ~ value for h2 could not be determined since the reaction was not reversible under these conditions. Experiments with acetate were also carried out as a function of pH. Solutions of 2-naphthol containing 8 X M sodium acetate were prepared and decay measurements were obtained between pH 1.7 and 6.8. The results are summarized in Table VII. A t low pH, where no acetate ion is present, the decay behavior is similar to that of 2-naphthol alone. At higher pH, acetate ion becomes available and influences the decay behavior as predicted. This data can now be used to obtain the pK, of acetate. It has previously been showng that rate constants can be obtained by means of instantaneous derivative-intensity plots. Equation 18 can be rewritten in terms of eq 20. (dIA/dt)/IA = - ( k ~+ ~ B + A h[RI) + (~AB[H+] + ~ [ R H I ) I B C B / I A C(20) A Plots of (dIA/dt)/IAvs. I B / I A were constructed at each pH and the intercepts on the (dIA/dt)/IA axis were determined. Points obtained from early-time data were distorted by convolutiong and were ignored. (The distortions caused by convolution as well as the noise in the data after taking a numerical derivative can be avoided, if needed, by working with the impulse responses obtained from exponential analysis and using analytical derivatives.) The intercepts, which are equal to (hA + k g A + hl[RI), are plotted as a function of pH in Figure 9. The solid curve is the best fit of the data according to the HendersonHasselbalch equation (pH = pK, + log [R]/[RH]). The pK, of acetate determined in this manner is 4.6. This represents an example of a way in which excited-state reactions can be used to obtain analytical information about ground-state species.
Discussion Reactions involving the lowest singlet excited state can be complex and emission can take place from more than one species. A two-state mechanism represents the simplest scheme for an excited-state reaction. Numerous
L
J
7.24 7.78 8.46 8.63 8.79 8.82 8.85 8.88 8.89 8.93 8.96 8.99 8.97 8.98
0.09 0.44 0.64 0.77 0.54 0.49 0.52 0.56 0.48 0.50 0.44 0.43 0.39 0.37
-0.08 -0.43 -0.64 -0.77 -0.54 -0.49 -0.52 -0.56 -0.48 -0.50 -0.44 -0.43 -0.39 -0.37
Measured at 450 nm.
l
1L
1
2
3
4
5
6
-
7
PH
Figure 9. The intercept term, k A f kBA4- k,[-OAc], from the derivative method for 2-naphthol as a function of pH in the presence of 0.08 M NaOAc (0). The solid curve is the Henderson-Hasselbalch expression (pH = p K + log ([R]/[RH])) with a pK, of 4.6.
processes such as exciplex or excimer formation or excited-state proton transfer may, under suitable conditions, fall into the two-state class. This paper has described some experimental criteria for determining whether a fluorophore is undergoing a twostate excited-state reaction. Excited-state proton transfer of 2-naphthol was used as a model system for this study. Its reaction mechanism can be changed from single state to two-state reversible or irreversible by a simple change in pH. These changes in experimental conditions have a predictable kinetic effect; while the influence of pH will be unique to proton transfer reactions, other experimental variables such as solvent composition can be used in a similar way to elucidate the excited-state mechanism of the types of reactions. Although the proton transfer of 2-naphthol is a wellstudied process, there has not been complete accord in the literature either in regard to the the qualitative reaction mechanism or the quantitative evaluation of the rate constants. The fluorescence of the 2-naphthol system was found to be in excellent accord with the proposed mechanism. Apparent deviations were shown to be due to instrumental and analysis artifacts as well as the experimental problems of spectral overlap and the presence of both species in the ground state. Once the mechanism for an excited-state reaction has been established, the decay parameters can be treated in various ways to obtain the rate constants. While various values for the rate constants have been p ~ b l i s h e d , ~ ~ ~ - ~ J ~ - l ~ the results obtained in the present study are in good
802
The Journal of Physical Chemistty, Voi. 83, No. 7, 1979
F. Doweli
agreement with those of Weller6 and Almgren,12 and also reflect the excited-state pK,. It is finally worth mentioning that many excited-state reactions will be influenced by the nature of ground-state reaction partners. In an example presented here, the fluorescence decay kinetics of 2-naphthol were used to determine the pK, of acetate (Figure 8). Another possible perturbation of the system that could yield information would be the use of added proton donors to promote the reverse reaction. These approaches may be of value when it is not easy to obtain the desired information in a more direct manner.
(2) J. F. Ireland and P. A. H. Wyatt, Adv. Phys. Org. Chem., 12, 131 (1976). (3) A. Weiier, Prog. React. Kinet., 1, 187 (1961). (4) A. Weiier, Z. Phys. Chem. (Frankfurt am Main), 3, 238 (1955). (5) A. Weiier, Z. Phys. Chem. (Frankfurt am Main), 15, 438 (1958). (6) A. Weiier, Z. Phys. Chem. (Frankfurt am Main), 17, 224 (1958). (7) N. M. Trieff and B. R. Sundheim, J. Phys. Chem., 69, 2044 (1965). (8) J. L. Rosenberg and I.Brlnn, J . Phys. Chem., 76, 3558 (1972). (9) M. R. Loken, J. W. Hayes, J. R. Gohike, and L. Brand, Biochemistry, 11, 4779 (1972). (IO) M. Ofran and J. Feiteison, Chem. Phys. Lett., 19, 427 (1973). (11) J. Feiteison, Isr. J . Chem., 11, 509 (1973). (12) M. Aimgren, Chem. Scr., 6, 193 (1974). (13) M. Hauser and G. Heidt, Rev. Sci. Insfrum., 46, 470 (1975). (14) T. Kishi, J. Tanaka, and T. Kouyama, Chem. Phys. Lett., 41, 497 (1976). (15) R. Schuyier and I. Isenberg, Rev. Sci. Instrum., 42, 813 (1971). (16) J. H. Easter, R. P. DeToma, and L. Brand, Biophys. J., 16, 571 (1976). (17) A. Gafni, R. L. Modlin, and L. Brand, Biophys. J., 15, 263 (1975). (18) I.Isenberg and R. D. Dyson, Biophys. J., 9, 1337 (1969). (19) A. Grinvaid and I. Z. Steinberg, Anal. Biochem., 59, 583 (1974). (20) A. Gafni, R. L. Modiin, and L. Brand, J . Phys. Chem., 80, 898 (1976). (21) A. Gafni and L. Brand, Biochemistry, 15, 3165 (1976). (22) J. Yguerabide, Methods Enzyrnoi., 26, 499 (1972). (23) M. Shinitzky, J. Chem. Phys., 56, 5979 (1972). (24) P. Wahi, J. C. Auchet, and B. Donzei, Rev. Sci. Instrum., 45, 28 (1974). (25) R. P. DeToma and L. Brand, Chem. Phys. Lett., 47, 231 (1977). (26) L. A. Shaver and L. J. C. Love, Appl. Spectrosc., 29, 485 (1975). (27) I. Isenberg, J . Chern. Phys., 59, 5708 (1973). (28) B. Vaieur and J. Moirez, J . Chim. Phys., 70, 500 (1973). (29) R. E. Jones, Ph.D. Dissertation, Stanford University, 1970. (30) S. W. Provencher, Biophys. J., 16, 27 (1976). (31) G. N. Lewis and M. Randall, "Thermodynamics",2nd ed,McGraw-Hiii, New York, 1961, p 645.
+
Acknowledgment. The plot of y1 y2 vs. y1y2was proposed by R. P. DeToma of this laboratory. We thank A. Nason for the use of the Cary spectrophotometer and R. P. DeToma for helpful discussion. L.B. was supported by NIH Grant No. GM 11632 and NIH career award development Grant No. GM 10245. W.R.L. was supported by NIH training Grant No. GM-57. A preliminary account of this work was presented a t the second annual meeting of the American Society for Photobiology, Vancouver, Canada, July, 1974, Abstract No. FAM-C4. Publication No. 988 from the McCollum-Pratt Institute.
References and Notes (1) J. B. Birks, "Photophysics of Aromatic Molecules", Wiiey, New York, 1970.
Reduction Parameters in a Phenomenological Three-Parameter Corresponding States Theory for n-Alkanes F. Dowellt Polymer Science and Standards Division, Center for Materials Science, National Bureau of Standards, Washington, D.C. 20234 (Received October 6, 1978) Publication costs assisted by the National Bureau of Standards
Reduction (scaling) parameters in a phenomenological three-parameter corresponding states theory for n-alkanes are determined at the gas-liquid critical point and compared with values previously determined away from the critical region. The relative reduction parameters for volume and temperature remain virtually constant, but the relative reduction parameters for energy and entropy change; the trend in the relative entropy reduction parameter as a function of the number of carbon atoms changes. From these results, certain implications are observed regarding the parameters for corresponding states models for chain molecules.
Introduction When the phase diagrams or thermodynamic properties of two substances have been appropriately scaied or reduced, such that invariant points (triple point, critical point, etc.) and lines of transition in the phase diagrams of the two substances coincide, the substances are said to be in corresponding states and may be described by a scaled or reduced equation of state. In addition to their theoretical interest, corresponding states theories are of practical importance since the behavior of a given compound can be predicted from the behavior of a reference compound in the same situation and the knowledge of a few parameters of the two compounds. +Addresscorrespondence t o the author at the Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, TN.
A phenomenological three-parameter corresponding states theory for chain molecules has been rather successfully applied to the liquid state of pure n-alkanes and of their The phenomenological formulation has been used to test corresponding states theories based on specific molecular models (including those of Flory, Orwoll, and co-workers; of Simha and co-workers; etc.) in which explicit theoretical equations of state have been d e r i ~ e d Patterson .~ and Bardin3 have given very detailed analyses of the artefacts and errors encountered in these molecular models (especially the Flory model) when compared with experimental data over a range of temperature and pressure away from the gas-liquid critical region. The three scaling or reduction parameters of the phenomenological theory are not explicitly tied to any molecular model (or, thus, explicit theoretical molecular
This article not subject to US. Copyright. Published 1979 by the American Chemical Society
