Single exponential

Lifetimes were fitted to a single exponential (dotted curves) with decay times of 2.56 ns y = 1.05) in (C) and 3.20... 

Figure C1.5.12.(A) Fluorescence decay of a single molecule of cresyl violet on an indium tin oxide (ITO) surface measured by time-correlated single photon counting. The solid line is tire fitted decay, a single exponential of 480 5 ps convolved witli tire instmment response function of 160 ps fwiim. The decay, which is considerably faster tlian tire natural fluorescence lifetime of cresyl violet, is due to electron transfer from tire excited cresyl violet (D ) to tire conduction band or energetically accessible surface electronic states of ITO. (B) Distribution of lifetimes for 40 different single molecules showing a broad distribution of electron transfer rates. Reprinted witli pennission from Lu andXie [1381. Copyright 1997 American Chemical Society.

Even cursory inspection of typical (v,[) data shows tliat tire evolution does not follow tire single exponential approach to saturation implied by, for example, (equation C2.14.22) witli initial concentrations Xq Such data are sometimes described as biphasic , and one encounters attempts to fit and inteiyDret tliem witli two exponentials, even tliough tliere does not seem to be any tlieoretical justification for doing so. The basic kinetics of adsorjDtion are described by ... 

The copolymer with the structure shown in Figure 16-15 displays a behavior similar to that of Ooct-OPV5-CN. In solution, one finds a fast double-exponential decay, while in a polysulfonc matrix single-exponential decay with a time constant of l. 7 ns is observed. We attribute this behavior to the same conformational phenomena. 

In order to demonstrate the possibilities and limitations of this method, we will quote some recent results (Lowdin and R dei 1958) obtained by using a wave function of the type w2(l + ar12), where u is expressed either as a single exponential or as the sum of two exponentials ... 

From this mathematical treatment we properly conclude that ki + k- is the natural rate constant. The data workup does not give the sum because of an artful algebraic manipulation. Instead, the sum k-i + k is simply the inverse of the intrinsic time constant for the single exponential that defines the approach to equilibrium. 

When this holds, the kinetic equations reduce to single exponentials. Chipperfield6 demonstrates that approximate adherence to Eq. (4-25) suffices to fit 20 absorbancetime pairs spaced at equal times over the first 75 percent of the reaction with correlation coefficients better than 0.999. 

Case E, where the three rate constants are comparable, does not feature A2 s> A3. That is, the progress curve is not a single exponential. None of the approximations is very good, although fcjmp is the best. Case F, characterized by a value of k 1 that is much smaller than kss or 2, is similar. The fit for k is quite poor that for ktmp is the best, but it is hardly adequate. 

ZnO instead of T1O2 because ZnO provides a 220 times higher mobility for photoinjected electrons, which would allow reduction of the exciting laser intensity. The slow PMC decay of TiOrbased nanostructured sensitization solar cells (the Ru complex as sensitizer), which cannot be matched by a single exponential curve and is influenced by a bias illumination, is strongly affected by the concentration of iodide in the electrolyte (Fig. 38). On the basis of PMC transients and their dependence on the iodide concentration, a kinetic mechanism for the reaction of photoinjected electrons could be elaborated.40... 

Values for degassed solutions. "With one atmosphere cyclohexane. N,N-dlmethyl-l-naphthylamlne. Trans-1,6-dlphenyl-l, 3,5-hexatrlene. Average of 13 literature values (see Ref. 50). Values from many authors agree for a single exponential, but there appear to be two exponentials that show wavelength dependence. 2,5-Dlphenyloxazole. "p-Bls(2-phenylozazolyl)benzene. Decay time dependent on Cl or I" concentration. 

Time Resolved Fluorescence Depolarization. In Equation 3, it is assumed that the polarization decays to zero as a single exponential function, which is equivalent to assuming that the molecular shape is spherical with isotropic rotational motion. Multiexponential decays arise from anisotropic rotational motion, which might indicate a nonspherical molecule, a molecule rotating in a nonuniform environment, a fluorophore bound to tbe molecule in a manner that binders its motion, or a mixture of fluorophores with different rotational rates. 

In principle, pulsed excitation measurements can provide direct observation of time-resolved polarization decays and permit the single-exponential or multiexponential nature of the decay curves to be measured. In practice, however, accurate quantification of a multiexponential curve often requires that the emission decay be measured down to low intensity values, where obtaining a satisfactory signal -to-noise ratio can be a time-consuming process. In addition, the accuracy of rotational rate measurements close to a nanosecond or less are severely limited by tbe pulse width of the flash lamps. As a result, pulsed-excitation polarization measurements are not commonly used for short rotational periods or for careful measurements of rotational anisotropy. 

For single exponential fluorescence decay, as is expected for a sample containing just one fluorophore, either the phase shift or the demodulation can be used to calculate the fluorescence lifetime t. When the excitation light is modulated at an angular frequency (o = 2itv, the phase angle f, by which the emission modulation is shifted from the excitation modulation, is related to the fluorescence lifetime by ... 

If the signal decay is a single-exponential curve, equations 16 and 17 result in values for X that are in agreement with each other. Dissimilar values indicate multiexponential decay, which usually means that the sample contains more than one fluorophore. Multiexponential decay can be resolved by using a phase fluorometer with phase sensitive detection. A time-independent, direct-current signal is produced that is proportional to the cosine of the difference between the phase angle of the detector ( D) and the phase angle of the fluorescence ( ) ... 

The model [39] was developed using three assumptions the conformers are in thermodynamic equilibrium, the peak intensities of the T-shaped and linear features are proportional to the populations of the T-shaped and linear ground-state conformers, and the internal energy of the complexes is adequately represented by the monomer rotational temperature. By using these assumptions, the temperature dependence of the ratio of the intensities of the features were equated to the ratio of the quantum mechanical partition functions for the T-shaped and linear conformers (Eq. (7) of Ref. [39]). The ratio of the He l Cl T-shaped linear intensity ratios were observed to decay single exponentially. Fits of the decays yielded an approximate ground-state binding... 

For simple outer-sphere electron transfer reactions, the effective frequency co is determined by the properties of the slow polarization of the medium. For a liquid like water, where the temporal relaxation of the slow polarization as a response to the external field is single exponential, tfie effective frequency is equal to... 

Figure 5.11 shotvs the temporal profile of the intensity change in the SFG signal at the peak of the Vco mode (2055 cm ) at OmV induced by visible pump pulse irradiation. The solid line is the least-squares fit using a convolution of a Gaussian function for the laser profile (FWFJ M = 20 ps) and a single exponential function for the recovery profile. The SFG signal fell to a minimum within about 100 ps and recovered... 

As mentioned in the introductory part of this section, quantum dots exhibit quite complex non-radiative relaxation dynamics. The non-radiative decay is not reproduced by a single exponential function, in contrast to triplet states of fluorescent organic molecules that exhibit monophasic exponential decay. In order to quantitatively analyze fluorescence correlation signals of quantum dots including such complex non-radiative decay, we adopted a fluorescence autocorrelation function including the decay component of a stretched exponential as represented by Eq. (8.11). 

Figure 12.10 Typical time traces of (a) emission intensityand (b) lifetime, measured from a single DMPBI nanocrystal, (c) Photon correlation histogram obtained from the time trace of the emission intensity (a). The lifetimes were obtained by fitting a single exponential function to the decay curves constructed for every 2000...

FIG. 3 Solvation dynamics dependence of coumarin 314 probe molecule orientation at the air-water interface. Signals are generated with a 420 nm pump photon and probed by surface second harmonic signal with 840 nm (SH at 420), x Sx element. The normalized change in SH field is plotted vs. pump delay, r is derived from a single exponential fit to the data, (a) Pump polarization S (inplane), (b) Pump polarization P (out-of-plane). (Reprinted from Ref 24 with permission from the American Chemical Society.)... 

Figure 3.18. Time dependence of the peak position of the 1570 cm Raman band of Sj trans-stilbene in chloroform solution (filled triangle). The time dependence of the anti-Stokes/Stokes intensity ratio is also shown with open circles. The best fit of the peak position change with a single-exponential function is shown with a solid curve, while the best fit of the anti-Stokes/Stokes intensity ratio is shown with a dotted curve. The obtained lifetime for both single-exponential decay functions was 12ps. (Reprinted with permission from reference [78]. Copyright (1997) American Chemical Society.)...

The resnlts from the TR experiments presented in Fignres 3.19-3.21 above were used to determine single exponential rate constants of 5.1 X 10 s for the decay of the ketyl radical, 1.6 X 10 s for the formation of the flnoranil anion and 8.4 X 10" s for the decay of the flnoranil radical anion. These rate constants snggest that the flnoranil radical anion is forming from the ketyl radical. A reaction mechanism for the intermo-lecular abstraction reaction can be described as follows ... 

Figure 4.5. Kinetic traces observed at (a) 1650cm and (b) 2096cm following 266nm photolysis (10ns, 0.4ml) of N-labeled 24 (3.1 mM) in argon-satnrated Freon-113. The dotted curves are experimental data the solid curves are the calculated best fit to a single exponential function. Reprinted with permission from Y. Wang, T. Yuzawa, H. Hamaguchi, and J. P. Toscano, J. Am. Chem. Soc., 1999,121, 2875. Copyright 1999, American Chemical Society.

Final resolution of these problems, particularly the complications from multiple matrix sites, came from investigations using spectroscopic methods with higher time resolution, viz. laser flash photolysis. Short laser pulse irradiation of diazofluorene (36) in cold organic glasses produced the corresponding fluorenylidene (37), which could be detected by UV/VIS spectroscopy. Now, in contrast to the results from EPR spectroscopy, single exponential decays of the carbene could be observed in matrices... 

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