Introduction

This document provides a short background on exponential decays followed by user notes for the following Excel simulations:

transient-single-exponential.xls (50 kB)
Enter a lifetime and initial intensity to view a single exponential decay curve. Also finds the intensity at any given time for the entered lifetime. [more info]
transient-biexponential.xls (60 kB)
Enter lifetimes and initial intensities to view two exponential decay curves and their sum.
transient-single-exponential-curve-fit.xls (50 kB)
Does a nonlinear least-squares curve fit to experimental data using the lifetime, initial intensity, baseline, and start time as adjustable parameters. Also functions as a simulation by entering these parameters. [Curve fitting requires the Solver add-in.]
transient-single-exponential-integration.xls (50 kB)
Displays an exponential decay and calculates the relative area under the curve between two x positions. (No notes on this one yet.)

Background

Nonlinear decays are common in physical systems. You will encounter them in the attenuation of radiation passing through an absorbing substance, the emission of radiation (atomic emission, molecular fluorescence, and phosphorescence), and in first-order reaction kinetics. Taking the decay of a radioactive substance as an example, the half-life, T1/2, is defined as the time for one-half of the radioactive nuclei to decay. If we were to isolate 1000 radioactive nuclei at a time that we will call time = 0, we would find that at one half-life later there remains 500 of the original radioactive nuclei (plus the disintegration products). If we wait another half-life there now remains one-half of 500 or 250 of the original nuclei. The following plot shows the continuation of this series, where N is the number of radioactive nuclei.

No matter how many radioactive nuclei exist at any given time, one half-life later only one-half of those nuclei will remain. So how do we describe how the amount of a radioactive substance changes with time? Try laying a straight edge along the top of each bar in the plot. You can't do it, at least not for all of the data. The decay of a radioactive substance is not a linear function. What we find is that the decay can be modeled with a number raised to an exponent containing the variable time. We call a plot such as this one an "exponential decay".

We plotted the example above in terms of half-lives. Different radioactive elements decay at different rates, so the exponent can also be expressed as a rate constant, k, times the variable time, t. Since the amount decreases with time, the exponent is also multiplied by negative one. The plot above is modeled by the number e to raise to the exponent -kt:

where N(t) is the number of radioactive nuclei at any time t and N(0) is the number at time = 0.

An example similar to radioactive decay is the decay of a population of atoms, molecules, or ions (referred to as atoms from here on) in an excited state. It is common to express the rate constant as the inverse of the lifetime, , which can be measured directly.

A more general treatment of the lifetime is elsewhere. The following simulations are set up to describe the transient emission signal that occurs after creating an excited-state population. The expressions are in terms of emission intensity, I, which is directly proportional to the number of excited atoms, N.

Single Exponential

The mathematical expression for an exponential decay versus time, t, is:

where I is intensity at time t, Io is the initial intensity at time=0, and is the lifetime.

The spreadsheet simulation, transient-single-exponential.xls, has the following input data and resulting plot:

You can change the cells in gray to see how the plot changes.

Biexponential

The fluorescence transient from complex samples often include multiple components. The mathematical expression for a signal that results from the sum of two exponential decays is:

where I is intensity at time t, A1 and A2 are the initial amplitudes (intensities) at time=0 for the two decay contributions, and 1 and 2 are the lifetimes of the two components.

The spreadsheet simulation, transient-biexponential.xls, is similar to the single exponential simulation described above but displays two single exponential decays and their sum:

Curve Fitting

A more general mathematical expression for a single exponential decay is:

where I is intensity at time t, Io is the initial intensity at time=0, is the lifetime, b is a constant baseline, and to is the start time. For many sets of data b and to will be zero. Including them allows simulation or fitting of experimental data that has a y offset or does not start at time = 0.

The spreadsheet simulation, transient-single-exponential-curve-fit.xls will do a nonlinear least-squares curve fit to experimental data using the lifetime, initial intensity, baseline, and start time as adjustable parameters. It can also function as a simulation by entering the parameters.

The curve fitting requires the Solver add-in. In Excel'97, Solver is under the Tools menu. If it is not there, try add-ins... If Solver is not listed as a possible add-in, you will have to run setup and install it from the CD-ROM. Notes on using Solver are in the spreadsheet.

Self-study Questions

Use the single exponential simulation to answer the following questions:

• How does the lifetime relate to the number e?
• If we define an intensity as gone at 0.0001 x Io, how many lifetimes must we wait for a signal to disappear?

Use the biexponential simulation to answer the following questions:

• You are trying to measure a fluorescing analyte in a sample, but there is a lot of signal from a short-lived interference. The interference has a lifetime of 0.2 ms and the analyte has a lifetime of 3.0 ms. At what time should you begin to measure the fluorescence of the analyte if the interference amplitude is the same as the analyte? What time is best if the interference amplitude is 10 times greater than the analyte?
• Predict the initial amplitude and lifetime when you sum two decays with different amplitudes but the same lifetime. Use the simulation to check your prediction.

Using the curve fitting simulation, try to enter parameters that produce a good match to the experimental data. After you are satisfied with your "eyeball" fit, try it using Solver.

Repeat this excercise with the following set of data. This set has to and b of zero. You can highlight this data and paste it into the spreadsheet. Under the Edit menu use Paste Special... "as text", and not as HTML format. If pasting the data produces #REF errors, click the undo button and try again.

```Time	I
0.0	10.48
1.0	7.54
2.0	5.49
3.0	4.02
4.0	2.74
5.0	2.02
6.0	1.50
7.0	1.09
8.0	0.68
9.0	0.57
10.0	0.37
11.0	0.31
12.0	0.19
13.0	0.15
14.0	0.13
15.0	0.11
```