Radioactivity

 

PURPOSE

 

The purpose of this activity is to determine the half-life of a radioactive species.

 

 

INTRODUCTION

 

It can be shown using calculus that whenever the next value of a changing quantity depends on the current value of the quantity, then the mathematical form of the relationship is an exponential function.  Examples of exponential relationships include population growth and radioactive decay.  Barring any other factors such as limited food supply, population growth is an example of an exponential increase, since the population of the next generation depends on the number of fertile adults in the current generation.  Radioactive decay is an example of exponential decrease, since the number of radioactive nuclei remaining in the next instant depends on the number that have not yet decayed.

 

Barium -137 is a radioactive nucleus that naturally decays into Cs - 137 by beta emission.  The half-life of this process is on the order of minutes.  The general form of the equation governing radioactive decay is:

 

A = A0 e -kt                                                                  (1)

 

where A is the activity, A0 is the activity at time t = 0, k is a constant and t is the time. 

 

The activity is measured in counts per second (cts/sec).  We can determine an average activity by measuring the total number of counts over a 15 second period and dividing the counts by 15 (after subtracting out the background activity).

 

It isn’t possible to graph equation (1) as written because one of the variables, time, is in the exponent.  However, we can manipulate the equation to produce the equivalent relation:

 

ln (A/A0) = -k t                                                                        (2a)

or

ln (A0 /A) = k t                                                             (2b)

 

where A0 is the average activity over the first 15 second count.  A graph of ln (A0 /A) versus time should be a straight line with the slope equal to the decay constant, k.

 

 

 

 

 

Another useful concept is half-life ( t ˝ ).  The relationship between the decay constant and half-life is:

 

k t˝ = ln(2) = 0.693                                                                (3)

 

For a decay process following equation (1), the activity of a radioactive sample will be exactly one half the activity at some initial time after one half-life of time has passed.

 

 

INSTRUCTIONS

 

Obtain the background radiation count by collecting data for two minutes away from the radioactive source.  Since counts will be collected for 15 seconds at a time, divide this number by 8 to determine the average background activity over a fifteen second time interval and record it on the data sheet.

 

Extract a radioactive barium sample by forcing eluting solution through the sample holder and collecting the solution in a plastic cup.  Place the Geiger counter directly over the solution to obtain the most accurate data.  Record the total counts for 15 seconds then wait 15 seconds before starting another count.  Be sure to clear the counter before recording the total counts for the next fifteen seconds.  Repeat this process of ‘15 seconds on, 15 seconds off’ for at least ten minutes.

 

Subtract the background count (for 15 seconds) from each raw count.  Divide that number by 15 to obtain the number of counts per second and record the activity in the next column of the data table.  The first activity recorded will serve as the value of A0 (which is the activity, A, at time t = 0).   Complete the rest of the table by dividing this value of A0 by each value of A.  Next, find the natural log of each ratio.

 

Construct a graph  and graph ln (A0 /A) versus t, where t is the initial time of the 15 second count (0, 30, 60 secs., etc.).  Find the slope to determine the decay constant.  Also calculate the half-life of barium 137. 

 

Use the adjusted 15-second count to check if the conditions of the half-life are being met.  To do this, choose a time in the data table and note the activity.  Add the calculated half-life to this time and estimate the activity of the sample at that time.  If half-life is a valid concept, then the second activity should be approximately one-half of the first.

 

 

 

 

 

 

 

 

RADIOACTIVITY – REPORT SHEET

 

 

Background reading for 2 minutes  ________________

 

Average background for 15 seconds         ________________

 

 

 

 

Elapsed time

 

Counts / 15 secs

 

Adjusted count

 

Counts / second

 

A0 / A

 

ln(A0/A)

 

0 seconds

 

 

 

 

 

A0 =

 

 

 

 

 

30

 

 

 

 

 

 

 

 

 

 

 

60

 

 

 

 

 

 

 

 

 

 

 

90

 

 

 

 

 

 

 

 

 

 

 

120

 

 

 

 

 

 

 

 

 

 

 

150

 

 

 

 

 

 

 

 

 

 

 

180

 

 

 

 

 

 

 

 

 

 

 

210

 

 

 

 

 

 

 

 

 

 

 

240

 

 

 

 

 

 

 

 

 

 

 

270

 

 

 

 

 

 

 

 

 

 

 

300

 

 

 

 

 

 

 

 

 

 

 

330

 

 

 

 

 

 

 

 

 

 

 

360

 

 

 

 

 

 

 

 

 

 

 

390

 

 

 

 

 

 

 

 

 

 

 

420

 

 

 

 

 

 

 

 

 

 

 

450

 

 

 

 

 

 

 

 

 

 

 

480