Pre-Lab 13 Assignment: Radioactive Decay

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? 



4. Based on the definitions of the half-life (T 1/2 ) and the mean life (τ) describe how you 

would use the graph in Activity 1.1 to determine both of these quantities for the 14 6C 

isotope? 

5. What equipment will you use to determine the half-life of the unknown sample? 

PHYS-204:Physics II Laboratory 

i

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 are 

connected to a battery and a switch the capacitor will charge and discharge following an exponential 

behavior. During the charging process the voltage in the capacitor is given by the 

following equation: 

V = V 0 (1−e −t/τ ), (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 = V 0 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 = V 0 e −τ/τ = V 0 e −1 = 0.37V 0 ; 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. 

PHYS-204:Physics II Laboratory 1

Radioactive Decay v 0.1 

10 

9 

8 

7 

6 

Voltage (V) 

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 



sample is proportional to the number of nuclei in the sample 

∆N 

∆t 

= −λN, (3) 

the number of nuclei remaining in the sample will follow an exponential law: 

N = N 0 e −λt (4) 

where λ is the decay constant and has unit of inverse time, and N 0 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 = N 0 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 N 0 to decay. By solving Eq. (4) 

for T 1/2 we obtain 

T 1/2 = ln2 

λ = 0.693 

λ . (1.6) 

Question 1.1 Solve Eq. (4) with the condition that at t = T 1/2 , N = N 0 /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 14 6C 

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 

∆t = ∆t 

) 

e −λt , (1.7) 

0 

where (∆N/∆t) 0 is the initial activity of the sample. In the axes below we show the activity 

of a sample of 14 6C as a function of time. 



4 

3 

Activity (×10 10 Decays/sec) 

2 

1 

0 

0 5 10 15 20 

Time (1000 years) 

Question 1.2 Based on the definition of the half-life (T 1/2 ) and the mean life (τ) describe how 

you would use the graph above to determine both of these quantities. 

Question 1.3 Determine T 1/2 , λ and τ using the graph above. Show your work. 



Question 1.4 Compare the value you obtained in Question (1.3) with the accepted value of the 

half-life of 14 6C 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. 


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. 



Step 6: In the axes below draw the graph you obtained in Step 5. Make sure to indicate the 

scale in the axes. 

Activity (counts) 

Time (s) 

Step 7: Using the software curve fit tools try a few different fits, for example inverse (A/t), 

inverse square (A/t 2 ), 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 → ∞? 



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 (T 1/2 ): . 

Mean Life (τ): . 

Nowthatyoudeterminedthehalf-lifebydoingabestfittothegraphobtainedusingthesoftware 

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. 



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 

56Ba. 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. 


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 



• 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. 

Step 5: In the axes below draw the graph you obtained in Step 4. Make sure to indicate the 

scale in the axes. 

Activity (counts) 

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.) 



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?

Pre-Lab 13 Assignment: Radioactive Decay

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