Pre-Lab 13 Assignment: Radioactive Decay

Transcription

1 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 (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

2 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

3 Voltage (V) 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 PHYS-204: Physics II Laboratory 2

4 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 = 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 t = ( N 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. PHYS-204: Physics II Laboratory 3

5 4 3 Activity ( Decays/sec) 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. PHYS-204: Physics II Laboratory 4

6 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. 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. PHYS-204: Physics II Laboratory 5

7 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? PHYS-204: Physics II Laboratory 6

8 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. PHYS-204: Physics II Laboratory 7

9 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 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 PHYS-204: Physics II Laboratory 8

10 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.) PHYS-204: Physics II Laboratory 9

11 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? PHYS-204: Physics II Laboratory 10