In Chapters 2 and 3, exponential decay of a signal was used extensively to illustrate various aspects of nonlinear regression analysis. Analysis of this type of data is quite important for processes such as luminescence decay, radioactive decay, and irreversible first-order chemical reactions. The general reaction is

Recall that the model for a single exponential decay of a signal y as a function of independent variable x (such as time) can be expressed as (Chapter 3)

The parameters are bx, the pre-exponential factor, b2 = kx the rate constant for the decay, and b3, a constant offset background. The decay lifetime is r = 1/k-i, or l/b2.

Often, data containing two or more time constants are encountered. For mixtures of species that are all undergoing first-order decay with lifetimes Th the appropriate model composed of sums of exponentials is given in Table 13.1. This model requires a judgment concerning the correct number of exponentials that best fit the data. The problem is complicated by the

Table 13.1 Model for Multiple Exponential Decay Kinetics

Assumptions: Background and extraneous instrumental signals unrelated to the decay have been removed from the data Regression equation:

Special instructions: Test for goodness of fit by comparing residual plots and by using the extra sum of squares F test (see Section 3.C.1); use proper weighting, such as for a counting detector Wj = 1 !yt in eq. (2.9) (cf. Section 2.B.5)

fact that each successive higher order model contains two more parameters than its predecessor. That is, the single exponential contains two parameters, and a system with two decaying species contains four parameters. Thus, the best model can be chosen relying on comparisons of deviation plots and the extra sum of squares F test. As discussed in Section 3.C.1, the extra sum of squares F test provides a statistical probability of best fit that corrects for different numbers of parameters in two models.

Resolution of multiple exponentials becomes difficult when three or more exponentials overlap. Success in separating these contributions is a complex function of the relative values of y, and 77. Degradation in the precision of these parameters for components with equal y, values was found [1] when ratios of lifetimes were decreased from 5 to 2.3. Studies with simulated data showed similar results, although the difference between single, double, and triple exponentials could be ascertained [2] from noisy theoretical data for ratios of t2Itx of 2.3 and t3/t2 of 4.3. However, for the triple exponential, the errors were 36% in r1; 23% in r2, and 7% in r3 [2]. As an approximate rule of thumb, nonlinear regression is not expected to give reliable results for t values with the model in Table 10.1 for more than three overlapped exponentials (p > 3) or when the ratio of successive lifetimes falls below about 2.5.

These statements should be taken only as guidelines and need to be evaluated within the context of the problem to be solved. The success of the data analysis will also depend on the quality and the resolution of the data. For example, if an experiment produced a signal to noise (S/N) ratio of >1000, a data set containing >5000 individual points, and if all ratios of Tk+]/rk were >5, it might be possible to fit these data with four exponentials. On the other hand, for data with (S/N) < 50, tk+lhk < 3, and 30 individual data points, it might be difficult to get acceptable results even for a two or three exponential fit.

Assumptions: Background and extraneous instrumental signals unrelated to the decay have been removed from the data Regression equation:

Special instructions: Test for goodness of fit by comparing residual plots and by using the extra sum of squares F test (see Section 3.C.1); use proper weighting, such as for a counting detector Wj = 1 !yt in eq. (2.9) (cf. Section 2.B.5)

Regression parameters: y, and r, for z = 1 to p

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