Radioactive Decay Part 2

Radioactive Decay

Part 2: Rapid Decay

In order to determine a half-life directly from obervational data, the radioactive substance must decay rapidly enough that one can actually measure a change. Not surprisingly, the earliest radioactive substances studied were ones that decayed fairly rapidly. The data in the following table are from a paper published by Meyer and von Schweidler in 1905. Their numbers are not concentrations but rather measurements of "relative activity." Since the rate of radioactive emission is proportional to the amount present, one can measure amounts (relative to the starting amount) by observing instead the emission rates (relative to the starting rate).

Time (days)	Relative Activity
0.2	35.0
2.2	25.0
4.0	22.1
5.0	17.9
6.0	16.8
8.0	13.7
11.0	12.4
12.0	10.3
15.0	7.5
18.0	4.9
26.0	4.0
33.0	2.4
39.0	1.4
45.0	1.1

Source: S. Meyer and E. von Schweidler, Sitzungberichte der Akademie der Wissenschaften zu Wien, Mathematisch-Naturwissenschaftliche Classe, p. 1202 (Table 5), 1905, as reported in J. Tukey, Exploratory Data Analysis, Addison-Wesley, 1977.

Enter the data points in your worksheet, and plot them. Does the decay pattern look roughly exponential -- that is, of the form

y = y₀b^t,
with b < 1? If so, can you determine either y₀ or b directly from either the graph or the data table?

Use your helper application to display the data on a semilog plot. Does this plot confirm or deny an exponential form for the data? Why?

Calculate the coefficients m and c for a line of the form

Y = mx + c
that appears to fit the data in the semilog plot. Explain how you go about this.

Note that there is a problem in plotting your line from Step 3 directly on the semilog data plot, because the line has an ordinary Cartesian equation. Plot your line by itself, and compare its appearance with the semilog plot in Step 2. If it's way off, rework Step 3. (We will have a better check in the next step, but this check will catch major errors.)

From your coefficients m and c, find a function of the form

y = y₀b^t
that should fit the data points well. Plot this function together with the data, first in a semilog plot and then in an ordinary (Cartesian) plot. If you are not satisfied with the fit, try to adjust the constants to make it better.

In a later module we will explore how to fit a model function to data for a slowly decaying radioactive substance.

Send comments to the authors <modules at math.duke.edu>

Last modified: September 16, 1997