Nonlinear regression least-squares method for determining relaxation time spectra for processes with first-order kinetics

Abstract

A simple method is shown for calculating the relaxation time spectrum which controls the rate at which a process following simple first-order kinetics takes place. The method involves unfolding a Fredholm equation of the first kind using least-squares and then using a modified nonlinear regression rather than a linear least-squares technique, thereby avoiding the highly oscillatory solutions which tend to occur with the latter with reduced mesh spacing or an increased number of bins. The validity and accuracy of the method for analyzing experimental data to reproduce various known input spectra are assessed and found to be excellent for data with no experimental error. For data with simulated experimental error with standard deviations up to σ=0.05 the method provides acceptable approximate solutions even though no exact solution is expected. Increasing the magnitude of the experimental error for a single lognormal input spectrum appears to have an increasing but nonsystematic effect upon the uncertainty of the approximate solution. Effects due to increasing the number of bins in the interval over which the spectrum is calculated are assessed and shown not to appreciably change the results, even for up to 60 bins. The methods is shown to be applicable to a wide variety of input spectra including single and double lognormal and box distributions. Importantly, in each of the cases studied the approximate solution appeared both to be unique and to converge toward the known input spectrum. Based upon this validation it is concluded that this method has applicability to a wide range of problems in which simple exponential decay is occurring with a spectrum of time constants. It may also be useful for problems with different kernals than that for first order kinetics; however, careful validation will be required.