30 Determining the Half-Life of an Isotope
Purpose
Measure the radioactive decay of Barium-137, calculate the decay constant and half- life of the isotope.
Background
A nuclear reaction in which an isotope of an element emits radiation spontaneously is called radioactive decay. A concept which originated from the study of radioactive decay is half-life or t½. Half-life is the time it takes for a radioactive material to loose half of its radiological activity. Half lives can be as short as a small fraction of a second to as long as billions of years depending on the isotope.
In this activity you will use an Isotope Generator to produce a small quantity of the short lived Ba-137m isotope. You will use a G-M tube to measure the decay of the Barium- 137 and you will then be able to calculate the decay constant and half-life of the isotope. The following equation describes the process mathematically.
R(t)=R0e-lt Eq 1
Where:
R(t) = the rate of decay at time t
t = the time elapsed
R0 = the rate of decay
l = the decay constant, measured in sec-1
The half-life t of a substance can be determined by
Eq 2
Where l is the decay constant of the substance.
Materials
• PASPORT Xplorer GLX / • Three finger clamp
• PASPORT Digital adapter / • Support stand
• G-M Tube
Consumables
• Isotope generator kit, (Barium- 137) / • Styrofoam cup
Safety Precautions
• Remember, follow the directions for using the equipment
• Wear safety glasses and follow all standard laboratory safety procedures
Equipment Setup
1) Cut a styrofoam cup so that the sides are not more that 1 cm high
2) Clamp the G-M tube to the support stand with the end of the tube is facing downward toward the bench top and about 2 cm above the surface.
GLX Setup
1) Plug the PASPORT digital adapter into one of the sensor ports on the top of the GLX.
2) Plug the G-M tube into the digital adapter.
3) Select General Counting ().
4) Place the beta radiation source on the lab bench or desk top with the source facing up.
5) Clamp the G-M tube to a support stand with the end of the tube pointing toward the beta source and is about 2 cm above it. It is essential that the radiation source and G-M tube remain a fixed distance apart during the experiment.
6) From the Home screen, press to open the Graph.
Procedure
1) Follow the instructions for the Isotope generator to prepare the Ba-137 isotope and quickly transfer the solution to the shallow cup. Place the cup under the G-M tube.
2) Press to begin data recording.
3) Record data for thirty minutes and then press again to end data recording.
4) Dispose of the barium solution as directed.
Analyze
Record calculations in your lab notebook as you complete your analysis.
1) After taking data for thirty minutes, the count rate will have dropped to a nearly constant rate. You can correct the data for background radiation and for any long-lived Barium-137 in the test solution by using the average rate from data at the end of the run.
• Select the last 5 minutes of data.
• Press to open the Tools menu and select Statistics. Read the Average from the statistics window and record the value in your data table as average background counts.
2) Using algebra, Eq 1 can be rearranged to represent a linear function as follows:
• Take the natural log of both sides of the equation.
ln(R(t)) = ln (R0e-lt)
• Now rearrange the equation:
ln(R(t)) = ln (R0) + ln(e-lt)
Where:
ln (R0) = kconstant
ln(R(t)) = y
y = lt + k Eq 3
Data Analysis
1) Using Eq 3, set up the GLX to solve for l (lambda).
2) From the home screen, (), press to open the calculator screen.
3) On the calculator screen, press and then to turn off Num Lock.
4) Enter the following equation:
normalized data = ln[(pulse count)(pulses)]
press .
5) Return to the home screen () and press to open the graph.
6) Press to highlight the y-axis data source then press again and select “normalized data”.
7) Press , then to open a linear fit calculation. Select about the first 5 minutes of data.
8) Fill in the data table below with the linear fit values.
Data Table
Average Background countsFit parameters for y = -lt + k
l(sec-1)
k
R0(calculated value)
R0(actual value)
l (min.-1)
t (min)
Analysis Questions
1) Determine the relationship between half-life measured in minutes and decay constant (l measured in min.-1).
2) From the fit parameters, determine the decay constant, l, and then the half-life t.
3) Is your value of consistent with the accepted value of 2.552 minutes for the half-life of 137Ba.
4) What fraction of the initial activity of you barium sample would remain after 25 minutes? Was it a good assumption that the counts in the last five minutes would be due entirely to non-barium sources?