ALUND Lab13 Radioactive Decay

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Feb 20, 2024

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PHY 112 L Lab 13: Radioactive Decay The objective of this lab is to explore the statistical nature of the radioactive decay process through a safe simulation. 1. Testable Questions: A. How is the number of remaining radioactive nuclei in a sample related to the amount of time that has passed from the beginning of the measurements cycle? B. How is the rate of decay of a sample of radioactive nuclei related to the amount of time that has passed from the beginning of the measurement cycle? 2. Hypotheses: A. If the time is increased, then the number of radioactive atoms will decrease because more time allows for more radioactive atoms to decay and become more stable. B. If the time is increased, then the rate of decay will decrease because there will be less atoms present with each decay. 3. Variables: Control(s): # of stable and unstable atoms (S o , N o ), initial rate of decay (R o ) Independent: time (t) Dependent (for A.): # of surviving radioactive atoms (N) Dependent (for B.): rate of decay (R) 4. Experimental Design: R o i t (sec) N o N R (Hz) 1 – 21 t 1-21 N o 1-21 N 1-21 R 1-21 5. Materials:  Red and white beads  containers 6. Procedure:

PHY 112 L 1. Let white extrusions/beads symbolize stable isotopes, and red extrusions/beads symbolize a radioactive isotope. 2. Count out an initial number of each (minimum 100 of each, minimum 300 combined). There is no requirement that you have equal red and white extrusions/beads. 3. Return counted extrusions/beads to a bin and mix them thoroughly. 4. With eyes closed pick one extrusion/bead from the box. Each pick represents a fixed time interval. For simplicity we are choosing the time represented by each pick to be 1 second. 5. If the picked extrusion/bead is red, it has decayed. Remove it from the sample and replace it with a white extrusion/bead. Keep track of them. 6. If the picked extrusion/bead is white, no decay occurred, and the white extrusion/bead is returned to the bin. 7. After every 10 picks, record the following: Number of red extrusions/beads remaining (N), and number of red extrusions/beads removed during the last 10 picks (–  N). Mix them thoroughly. 8. Continue until you have 21 data points (200 seconds of data). 9. The equations being investigated are as followed: Rate of decay: R = − ∆N ∆t Decay constant: λ = 1 S o + N o Half Life: T ½ = ln2 λ 7. Data: AVERAG E i t (s) white, S red, N Δ N R (Hz) 1 0 100 200 0 0 2 10 107 194 6.5 0.650 3 20 113 187 12.8 0.638 4 30 118 182 18.3 0.608

PHY 112 L 5 40 125 176 24.5 0.613 6 50 131 169 30.8 0.615 7 60 137 164 36.5 0.608 8 70 144 156 44.3 0.632 9 80 150 150 49.8 0.622 10 90 155 146 54.5 0.606 11 100 160 141 59.5 0.595 12 110 163 137 63.3 0.575 13 120 168 132 67.8 0.565 14 130 173 127 72.8 0.560 15 140 177 123 76.8 0.548 16 150 180 120 80.3 0.535 17 160 184 116 83.8 0.523 18 170 187 113 87.3 0.513 19 180 191 109 91.0 0.506 20 190 196 104 95.8 0.504 21 200 200 100 99.8 0.499 8. Analysis: Part A: 0 50 100 150 200 250 0 50 100 150 200 250 f(x) = 199.91 exp( − 0 x ) R² = 1 N vs t N vs t Exponential (N vs t) t (s) N TN o = 200 MN o = 200 % error = 0% Part B:

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PHY 112 L 0 50 100 150 200 250 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 f(x) = 0.67 exp( − 0 x ) R² = 0.92 R vs t R vs t Exponential (R vs t) t (s) R (Hz) Tλ,= 0.00333 Mλ = 0.00100 + 0.00300 2 Mλ = 0.00200 % error = | 0.00333 − 0.002000 | 0.00333 × 100 % error = | 0.00133 | 0.00333 × 100 % error = 0.399 × 100 % error = 39.9% TR o = 0.650 MR o = 0.665 % error = 2.31% TT ½ , = 208 MT ½ = 347 % error = 66.8% 9. Conclusion: A. The number of the remaining radioactive nucleotides is indirectly related to the elapsed time according to: N = 200e -(0.003s)t B. The rate of decay is indirectly related to the time elapsed according to: R = 0.665e -(0.001Hz/s)t

PHY 112 L 10. Evaluation: The hypotheses from both part A and B of this lab were supported that as the time increased, the number of radioactive nuclei remaining would decrease as well as the rate of decay would decrease exponentially. The precision of the results in part A were outstanding based on the R 2 value of 0.999 while the precision for part B was seen as terrible based on the R 2 value of 0.914. The precision of part B was based on the decay rate which is never constant. There are many factors that change this rate to either increase or decrease it overall. These external factors are examples of possible random errors within nature. In this experiment, the person pulling the beads out started pulling from top but would then pull from the bottom of the container. This changed the rate in which red and white beads were pulled.

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