Lab 3
Radioactive Decay and Half-Life
Overview
Radioactive decay follows first order kinetics. This means that the rate of decay is
proportional only to the amount of unstable isotope present. We can write:
dy/dt = ky
(where k < 0)
where y = the number of atoms of the unstable isotope at time t. As we saw in class, we
can solve this differential equation and get
y = y0ekt.
The decay constant k (“relative rate of decay”) depends on the particular isotope and
nothing else.
A representation for the decay of a nucleus is:
parent nucleus → daughter nucleus + radiation particle
In the lab, you will be looking at the radioactive isotope 137mBa. It is produced in a
container called an isogenerator, or “nuclear cow”, by the decay of 137 Cs. The decay
process looks like
137
Cs →
137m
Ba + β‫→ ־‬
137
Ba + γ
The cesium decays to metastable barium with the emission of a beta particle β‫( ־‬an
electron). The half-life for that reaction is about 30 years. The metastable barium has a
short half-life; it loses excess energy as low-energy gamma radiation (γ) to form
stable 137Ba.
Both types of barium, stable and unstable, are produced inside the cow from the decaying
cesium. They are separated from the cesium and flushed out by “milking the cow”—that
is, by passing a dilute solution of hydrochloric acid and NaCl through the cow. Barium
reacts with the acid to form soluble barium chloride and the solution is drained out. The
cesium does not react, being held by an ion-exchange medium inside the cow.
You will be using a Geiger counter, located next to a small beaker with the barium
solution, to measure the “activity rate” R (number of clicks in each time interval). Each
γ from a decaying nucleus picked up by the Geiger counter produces a click—
representing the decay of one barium nucleus. So the number of clicks we record/minute
has something to do with the number of nuclei that have decayed/minute.
To be more specific,
i) If the Geiger counter picked up every γ emitted by decay, then we’d
record a number of clicks each minute equal to the number of barium nuclei that
decayed during that minute. But the radiation γ goes out in all directions, while
the Geiger counter only picks up those γ’s in a certain “viewing window.”
Therefore the rate of γ’s counted (clicks/minute) from decaying nuclei =
M dy/dt—that is, it is proportional to the rate of decay of nuclei (nuclei/minute).
(If the Geiger counter picked up 10% of the γ’s emitted by decays, then we’d have
M = 0.1)
ii) The Geiger counter also generates clicks for background
radiation it picks up from cosmic rays. These clicks are “noise” that has
nothing to do with our decaying barium. We assume that this radiation rate
is approximately constant, say B clicks/min..
If we let R(t) denote the activity rate of the Geiger counter (clicks/min), then we have
R(t) = (clicks/minute related to barium decay) + (clicks/minute from background
radiation)
=
M dy/dt
+
B
Since y = y0ekt , we know that dy/dt = ky0ekt so that
R(t)= M(ky0ekt ) + B = Aekt+ B, where A=Mky0 , B, and k are constants
(k<0)
You will collect and plot your data about the activity rate R(t) and see how well it fits
such a model. You’ll use the model to estimate the relative decay rate and half-life for
137m
Ba .
Materials
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Computer with Lab Pro interface
Geiger counter plugged into the Dig/Sonic 1 port of the Lab Pro
Small beaker
One isogenerator (nuclear cow)
Small plastic syringe
Bottle of eluting solution: dilute hydrochloric acid and NaCl solution
Pair of rubber gloves and goggles
Note: the radioactive material we’ll be working with is harmless. The acid solution
is very dilute and not a danger. However, common sense indicates that you should
be careful; please wear gloves and goggles when working with these materials.
Procedure
Prepare the hardware and software for data collection
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The Geiger counter should be connected to Dig/Sonic 1 of the Lab Pro.
Turn the switch on the counter to Audio. You will hear random clicks as it
picks up background radiation. Then move the switch to On. (The counter will
continue working but with the clicks muted.)
Notice that the window through which radiation enters is on the bottom of the
counter. Set the counter on its side so this window is unobscured and secure it
in position. You might use two books (or wooden blocks, if available) to hold
it in this position, but be sure you don’t obscure the window.
Click on the Open button. In the Experiments folder window that appears,
double-click on lab3_halflife. A dialog box should appear indicating that you
have opened the experiment file for the correct lab. Click OK.
“Milking the cow.” Both partners should put on goggles and rubber gloves before
drawing acid from the bottle.
Decide which partner will be the “milker.”
Instructions for the “milker”:
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Take the top and bottom plastic covers off of the cow and set them on a
piece of paper toweling.
Position the cow on top of your beaker so the outlet port is in the middle
of the beaker.
Withdraw about 1 ml. (no more ) of dilute hydrochloric acid from the
bottle, using the plastic syringe.
Place the tip of the syringe into the top part of the cow and slowly inject
the solution into the cow. Some solution may overflow the top. If so, try to
inject a little slower. If quite a bit of solution overflows, pull it up into the
syringe and re-inject it.
This can be a bit tricky. When putting through the 1 ml. of HCl, it tends to
back up and overflow, even when you go slowly. Suck up the overflow at
least once and put it through again, then blow in 1 ml. of air. Even so,
you might get a sample with a low initial count.
When the syringe is empty, remove it from the port, take in about 1 ml. of
air and blow this through the cow.
When the liquid has drained through, put the stoppers in the cow, wipe up
any spills, and return the cow immediately to the designated storage
space—away from the Geiger counter.
The other partner should now place the beaker directly in front of the Geiger
counter window, as close to the window as you can, and click Collect. The
beaker should not be moved at all during data collection. (The amount of
radiation the counter picks up decreases with the square of the distance to
the beaker.) You need to collect data for 25 minutes.
While the milker is putting the cow away, the other partner should keep an
eye on the data. Answer Lab Report, 1. If the initial count is less than 150,
or the curve starts to look flat before about 10 minutes have passed, then you
will have to repeat the experiment.
(Possible sources of the problem: The beaker may have been too far away
from the counter, or not enough solution was passed through the cow.)
If you have to repeat the experiment, the same cow should not be milked
again for 30 min.
While you are waiting for the data to collect, do Lab Report, 2.
Print the graph and the table of data. (Be sure the printed table is complete.) Turn off
the counter and dispose of the solution as instructed by the lab coordinator.
Answer the questions in Lab Report, 3.
For Lab Report, 4, calculate the average value of the random background radiation.
On the graph of Radiation vs. Time, you should see that after 15 minutes or so,
the data points seem to be scattered about a horizontal line. Those data points
reflect the (random) background radiation counts.
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With the graph window active, click at Time = 20 minutes and drag over to
Time = 25 minutes. The table window should show the last fifth of the data as
selected.
Click on the Statistics button (or use the toolbar Analyze, Statistics). A
floating box will appear with the average (mean) Radiation count for the time
period (20, 25). This is the number you will use for the level of background
radiation.
We want to look at whether our data fits the model in the opening discussion—according
to which we should be able to describe the activity rate R (counts vs. time) with an
exponential model R = Aekt + B ( k < 0 ).
During the early minutes of the data collection, most of the “counts” registered are from
the barium decay rather than background radiation—that is, B is a relatively small part of
R. Therefore for these times, we’d expect a curve like R = Aekt to fit our early data fairly
well.
In a model R = Aekt , we have
ln R = ln (Aekt) = ln A + ln (ekt) = ln A + kt,
(k<0)
so that ln R would be a linear function of t.
Therefore, we’ll plot the logarithm of our R data vs. time and see what it looks like.
Open a new graph window.
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From the Window menu, select New Wide Window -> Graph. A new
graph window will appear beneath the original graph window.
Use View, Graph Options, The title of the graph will be highlighted in
the Graph Options window that appears. Type in the new title “Natural
Log of Radiation vs. Time”, then click Apply.
Click on the Axis Options tab.
Deselect the variable Radiation in the y-axis section.
The graph should now be empty, with the new title.
Create a new column containing the log of the radiation.
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Select New Column -> Formula from the Data menu. The New Column
window will appear.
Enter the following information in the Options section:
Long Name: ln(radiation)
Short Name: ln(rad)
Units: #
Click on the Definition tab. Enter ln(“Radiation”) in the Equation field
(You can select ln() from the Functions list and select Radiation from
the variables list.) Click Try New Column, then click OK.
Look at the graph of ln(Radiation). You probably need to rescale to see
what’s happened: use View, Autoscale (with the ln(radiation) graph
window active). (The last part of the graph on the right may look a little
ragged; that’s because as time passes, the background radiation is
represented more and more in the counts, so that the model R = Aekt
becomes less plausible.)
Print the graph. (Lab Report, 5a)
Fit a regression line to the ln(radiation) data over the first 15 minutes.
 Select the first fifteen minutes of the data on the graph.
 Click the Linear Fit button. The least-squares regression line will appear
as well as a floating box with the slope and y-intercept of the line.
Do Lab Report 5 b,c
Use an exponential curve to model the original radiation data over the first 15 minutes.
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Click on the Radiation vs. Time window to make it active.
Select the first fifteen minutes of data.
Click the Automatic Curve Fit button. The Curve Fit Properties window will
appear.
Select A*exp(-C*x)+B Natural Exponent from the scrolling list below the y =
box.
Enter the value of the background radiation (from Lab Report, 4) in the B Value
box.
Select Radiation from the scrolling list below Perform Fit On:
Click Try Fit, then OK
The exponential model will appear in the graph window. The decay constant and
the other parameters of the curve appear in the floating box.
Print the graph and the floating box. Do Lab Report, 6
Apply an exponential model letting LoggerPro make its own estimate of the value for B,
the background radiation.
Close the box that reports the parameters from the first exponential fit, and repeat
the exponential fitting, this time not entering a B value.
Print the graph and the floating box. Do Lab Report, 7
Using the data table (not the parameters from the exponential models), make a reasonable
estimate of the half-life of 137Ba. Lab Report, 8a
The “mean square error” (MSE) reported for each of the models is a measure of how well
the model actually fits the actual data: smaller MSE indicates a better fit. (For each time
value t in our table of data, we can imagine substituting t into the exponential equation
and getting a “predicted” value of R. We could then compute, for each t value, the
squared error: (actual R value – predicted R value)2 . The MSE is roughly the average
of the squared errors. The precise definition has a slight twist that is discussed in
statistics courses.)
Using the estimated relative decay rate, k, from the better of your two exponential
models, make a calculation of the half-life of 137Ba. Lab Report 8b