12: Radioactivity Simulation
- Page ID
- 516595
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. Prerequisite Math & Theory
- First-Order Decay: Radioactive decay follows the same mathematical formulation as first-order kinetics:
\[ \ln N_t = -\lambda t + \ln N_0 \]
- y-axis: \(\ln(\text{Number of Dice Remaining})\)
- x-axis: Round Number (\(t\))
- Slope: \(-\lambda\) (Decay constant)
- Half-Life Math: \(t_{1/2} = \frac{\ln 2}{\lambda}\). Knowing the decay constant \(\lambda\) allows calculation of the half-life.
2. Required Technical Skills
- Simulation logic: Recognizing that repeated trials with a small population (e.g., rolling 10 dice 10 times) statistically mirrors rolling a large population (100 dice) once.
- Graphical analysis: Linearizing decay data (\(\ln N\) vs. Round Number) in spreadsheet software to extract the decay constant.
3. Critical Safety
- Choking Hazard: Dice are small parts. Keep them secure and never ingest them.
- Slip Hazard: Contain dice within trays or boxes during rolling to prevent floor spills and tripping hazards.
- To simulate radioactive decay using a random probabilistic process (rolling dice).
- To investigate the concept of half-life (\(t_{1/2}\)) by observing the exponential decrease of a population over successive rounds.
- To determine the experimental decay constant (\(\lambda\)) and half-life via graphical and statistical analysis.
INTRODUCTION
Radioactive decay is a spontaneous and completely random process. It is impossible to predict the exact moment an individual radioactive nucleus will disintegrate; however, the probability of decay over a given time interval can be modeled using statistical and probabilistic simulations such as coin tosses or rolling dice.
Radioactive isotopes decay at widely varying rates, characterized by their half-life (\(t_{1/2}\))—the time required for exactly one-half of the radioactive nuclei in a given sample to decay. For example, Polonium-218 has a half-life of about 3 minutes, whereas Uranium-238 has a half-life exceeding 4 billion years. Regardless of initial sample size, the fraction remaining always decreases by 50% per half-life.
In this experiment, radioactive decay is simulated by rolling dice. Designating a specific outcome (e.g., landing on a '6') as "decay" models the probabilistic nature of nuclear disintegration.
| Round (Coin Toss) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Initial Coins | 100 | 50 | 25 | 12 | 6 | 3 | 1 |
| Coins Decayed | 50 | 25 | 13 | 6 | 3 | 2 | 1 |
| Coins Remaining | 50 | 25 | 12 | 6 | 3 | 1 | 0 |
Radioactive decay is mathematically identical to first-order kinetics:
- Rate Law: \(\text{Rate} = \lambda N\), where the rate of decay depends directly on the number of active nuclei (dice) present.
- Linearization: Plotting the natural logarithm of remaining dice (\(\ln N_t\)) versus time (\(t\)) yields a straight line with slope \(-\lambda\).
First-Order Decay Law:
\[ N_t = N_0 e^{-\lambda t} \quad \text{or} \quad t = -\frac{1}{\lambda}\ln\left(\frac{N_t}{N_0}\right) \]
Half-Life Relationship:
\[ t_{1/2} = \frac{\ln 2}{\lambda} \]
- 12.1: Radioactivity Simulation - Experiment
- This page provides safety guidelines for using dice in an experiment simulating radioactive decay, highlighting risks like choking and tripping. It lists necessary materials and outlines a detailed procedure for rolling the dice, recording which ones "decay," and monitoring remaining dice. The page discusses the Law of Large Numbers, emphasizing that larger sample sizes yield more accurate data and illustrating the difficulties in calculating half-lives with smaller samples.
- 12.2: Radioactivity Simulation - Pre-lab
- This page covers radioactive decay with a focus on Strontium-90 (Sr-90), detailing its half-life of 29 years and first-order kinetics. It includes methods to calculate elapsed time when 1250 out of 10,000 atoms remain, the creation of a decay curve graph, and an exploration of decay probabilities over specific time frames. Theoretical exercises using dice are introduced to illustrate decay probabilities and half-life concepts.
- 12.3: Radioactivity Simulation - Data and Report
- This page outlines an experiment analyzing radioactive decay using dice, covering data collection over 20 rounds from 100 rolls. It emphasizes recording decayed dice, graphing results in Excel, and examining curve smoothness affected by sample size. The page explains calculating the decay constant (λ) and comparing it to theoretical values, including percent error analysis.