Name:
Lab Partners:
Date:
Pre-Lab 13 Assignment:
Radioactive Decay
(Due at the beginning of lab)
Directions: Read over the lab handout and then answer the following questions about the
procedures.
1. What is your answer to Question 0.1?
2. What is your answer to Question 0.2?
3. What is your answer to Question 0.3?
4. Based on the definitions of the half-life (T1/2 ) and the mean life (τ ) describe how you
would use the graph in Activity 1.1 to determine both of these quantities for the 146 C
isotope?
5. What equipment will you use to determine the half-life of the unknown sample?
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Name:
Lab Partners:
Date:
Lab 13:
Radioactive Decay
Objectives
• To understand phenomena that have an exponential behavior.
• To determine the relationship between the decay constant and the half-life.
• To determine the half-life of an unknown radioactive sample.
Overview
In Lab 7 (Capacitors and RC Circuits) we learned that when a resistor and a capacitor
connected to a battery and a switch the capacitor will charge and discharge following an
ponential behavior. During the charging process the voltage in the capacitor is given by
following equation:
V = V0 (1 − e−t/τ ),
are
exthe
(1)
where τ = RC, and R is the resistance of the resistor and C is the capacitance of the capacitor.
The discharge is given by
V = V0 e−t/τ .
(2)
The constant τ called the characteristic time (or ”time constant”) tells how fast or slowly
the capacitor will be charged. The larger the value of τ the more time it will take to charge
the capacitor and this will depend on the values of R and C. The constant τ also tells how
fast or slowly the capacitor will be discharged. For example, from Eq. (2), at time t = τ ,
V = V0 e−τ /τ = V0 e−1 = 0.37V0 ; that is, after τ seconds, the voltage is reduced to 37% of the
original value.
Question 0.1 In the axes below an exponential decay of an RC circuit with a 100Ω resistor and
a capacitor of unknown capacitance is shown. Determine the time constant τ for the circuit.
Show all your work.
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9
8
7
Voltage (V)
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Question 0.2 Using the result from Question 0.1 determine the capacitance of the capacitor.
show your work
Question 0.3 Using the graph above determine at what time the voltage dropped to about 5%
of its original value. This value corresponds to approximately how many time constants (τ )?
Many other phenomena have exponential behavior such as population growth and decay and
radioactive decay. In this lab we will examine the exponential decay of a radioactive sample
with a short characteristic time. Radioactivity is a result of the disintegration or decay of
unstable nuclei. Because of this instability the nucleus will decay with the emission of some
kind of radiation which can be a particle or electromagnetic radiation. The rate of decay of a
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sample is proportional to the number of nuclei in the sample
∆N
= −λN,
∆t
(3)
the number of nuclei remaining in the sample will follow an exponential law:
N = N0 e−λt
(4)
where λ is the decay constant and has unit of inverse time, and N0 is the number of nuclei
present in the sample before any decays occur. In this lab we will investigate the decay of
radioactive samples and the relationship between the decay constant and the half-life and mean
life of a radioactive material.
Investigation 1:
Half-Life and Mean Life of an Isotope
Eq. (4) describes the exponential decay in terms of the decay constant λ, however one can
rewrite it in a way very similar to the decay in the RC circuit described by Eq. (2). The mean
life (τ ) of a certain unstable nuclei is defined as τ = 1/λ. And Eq. (4) becomes
N = N0 e−t/τ .
(1.5)
Just like the RC circuit τ is the time constant that tells us how fast or how slow the sample will
decay. Its meaning is that one expects, on average, that a particular nuclei will decay within
this characteristic time. But, of course, some nuclei will decay faster and others will decay
slower than τ . τ is determined exactly the same way as we did for the RC circuit, that is we
determine how long it takes the sample to have 37 % of the original number of nuclei. Another
quantity of interest is the half-life of the isotope, which is defined as the time it takes for the
sample to have 50 % of the original number of parent nuclei N0 to decay. By solving Eq. (4)
for T1/2 we obtain
0.693
ln2
=
.
(1.6)
T1/2 =
λ
λ
Question 1.1 Solve Eq. (4) with the condition that at t = T1/2 , N = N0 /2 and prove Eq. (1.6).
Show all your work
One of the common uses of radioactive decays is carbon dating. Carbon dating is done by
measuring the rate of decay (or ”activity”, the number of decays per unit time) of the 146 C
isotope in a sample of an artifact made of organic material. It can be shown that the activity
follows the same exponential law as the number of parent nuclei in the sample, i.e.
∆N
∆N
e−λt ,
(1.7)
=
∆t
∆t 0
where (∆N/∆t)0 is the initial activity of the sample. In the axes below we show the activity
of a sample of 146 C as a function of time.
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Activity (×1010 Decays/sec)
3
2
1
0
0
5
10
15
20
Time (1000 years)
Question 1.2 Based on the definition of the half-life (T1/2 ) and the mean life (τ ) describe how
you would use the graph above to determine both of these quantities.
Question 1.3 Determine T1/2 , λ and τ using the graph above. Show your work.
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Question 1.4 Compare the value you obtained in Question (1.3) with the accepted value of the
half-life of 146 C of 5730yr. Do they agree?
Now that you know how to determine the half-life, the mean life, and the decay constant of a
given isotope by looking at the activity of the radioactive sample as a function of time you are
ready to determine the half-life of an unknown isotope by measuring its activity over a period
of time.
Investigation 2:
Determining the Half-Life of a Radioactive Sample
In this section we will determine the half-life, the mean life and the decay constant of a radioactive sample of an unknown isotope.
For the activities in this investigation you will need:
• computer-based laboratory system
• experiment configuration files
• radioactivity probe
• small beaker
• radioactive sample (handled by the lab technician)
Precaution The level of radioactivity in the samples is very low, and will not cause any harm,
provided they are not ingested or allowed to come into contact with the skin. However, care
should always be exercised when handling any radioactive substance. If a spill should occur, it
should be wiped up immediately and the paper towels discarded in a container provided. DO
NOT EAT OR DRINK while handling radioactive material.
Activity 2.1: Measuring the Activity of the Sample
Step 1: Open the Lab file Lab13 A 1
Step 2: Connect the radioactivity probe to the socket named Dig.1 in the Vernier adapter.
Step 3: Place the measuring window of the probe directly in front of the clean beaker.
Step 4: When you are ready to start your experiment request that the lab technician put the
sample in your beaker.
Step 5: Immediately after your sample is poured in the beaker click the collect button in the
Logger Pro software. Be patient as the total collection time is 20 minutes. Once the
collect button appears again, you are ready to analyze your results. The probe will be
collecting raw counts of the activity of the sample. These are the number of decays over
a period of 5 seconds.
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Activity (counts)
Step 6: In the axes below draw the graph you obtained in Step 5. Make sure to indicate the
scale in the axes.
Time (s)
Step 7: Using the software curve fit tools try a few different fits, for example inverse (A/t),
inverse square (A/t2 ), and natural exponent (Aexp(−Ct) + B).
Question 2.1 Which of the curves best fit the data? And what are the values of the constants
obtained in the fit?
If the experiment went according to plan your best fit should have been the natural exponent.
But, as you might have noticed that instead of the expression in Eq. (1.7) which goes to zero
as times becomes very large (t → ∞) there was an extra constant (B).
Question 2.2 In the expression y = Ae−Ct + B what is the value of y as t → ∞?
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In the experiment the values of the fitting function y corresponds to the activity ∆N/∆t and
the variable x corresponds to time t.
Question 2.3 What is the meaning of the constants B and C?
As you might have concluded the value of C corresponds to the decay constant λ and the
value of B, which is the activity as t → ∞, is the activity of the background, often referred as
the background radiation.
Question 2.4 From the values obtained in question 2.1 what is the decay constant and the
background activity (note: In the plot the software gives a raw count that is taken over a period
of 5 s. To obtain the activity you have to divide C by 5.)
Decay Constant λ from curve fit:
Background Activity from curve fit:
.
.
Question 2.5 Determine the half-life and the mean life of the measured sample using the decay
constant determined in Question 2.4.
Half-Life (T1/2 ):
Mean Life (τ ):
.
.
Now that you determined the half-life by doing a best fit to the graph obtained using the software
you will determine the half-life directly from the graph just like you did in Investigation 1. In
the axes above draw a curve that best fits your data points and from that extract the half-life
of your sample. Compare your result with the one obtained in Question 2.5. Do they agree?
Why or why not?
Repeat steps 1-7, draw the new graph in the above axes (preferably in a different color) and
determine the half-life, decay constant and mean life of the second sample. Write the values
below and compare them with the numbers determined in the first run.
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Question 2.6 Are your values from the second run in agreement with the ones obtained in the
first run? If not determine the possible source of errors.
Question 2.7 The sample you used is of 137
56 Ba. The accepted value of its half-life is about 156
seconds. Compare your measured results with the results obtained in the lab. Roughly estimate
the error in your estimate of the half-life. Are your results in agreement (within error bars)
with the accepted value? If not, determine the possible sources of errors.
Question 2.8 What modifications in the described procedure could lead to smaller errors in
the determination of the half-life? Explain your answer.
Investigation 3:
Background Radiation
As noted in Investigation 2 the best fit to your experimental data was one where the activity
did not go to zero as t → ∞. This is due to what is called background radiation, which is
radiation that we are exposed to everyday. The sources of this radiation are cosmic rays as
well as sources from our soil such as radon gas. This background radiation will change from
one place to another. However, if no other artificial sources are brought to a given site the
background radiation on that site should be nearly constant. In this activity we will measure
the background radiation in our lab and compare to the values we determined in investigation
2.
For the activities in this investigation you will need:
• computer-based laboratory system
• experiment configuration files
• radioactivity probe
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• small beaker
Step 1: Open the Lab file Lab13 A 1
Step 2: Connect the radioactivity probe to the socket named Dig.1 in the Vernier adapter.
Step 3: Point the measuring window to of the probe directly in front of the clean empty
beaker. The probe will be collecting raw counts of the activity of the sample. These are
the number of decays over a period of 5 seconds.
Step 4: Start collecting your data. Run your experiment for two minutes and then stop.
Activity (counts)
Step 5: In the axes below draw the graph you obtained in Step 4. Make sure to indicate the
scale in the axes.
Time (s)
Step 6: Using the analysis tools in the software determine the average activity in the lab and
write your answer below:
Background activity:
.
Question 3.1 Is the answer you obtained in agreement with the results obtained in Activity 2? If not, determine the probable causes for the difference. (note: In the plot the
software gives a raw count that is taken over a period of 5 s. To obtain the activity you
have to divide C by 5.)
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Question 3.2 Is the background activity obtained in Question 3.1 in agreement with the
number displayed in the window of the radioactivity probe?
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