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Investigating the Concept of Half-Life in Radioactive Decay Using Coins.
Abstract
The main aim of the two experiments is to verify that 50% of coins will decay (heads-up) in the first throw. In the first experiment, 100 coins (tails-up) will be placed in a box with a cover then shaken thoroughly. Coins with heads –up will be removed to represent the decayed atom. It will be realized that 52% of the coins will decay within the first throw. Similarly, the second experiment will test the same concept. Eight coins will be used for the second experiment. Eight coins will be thrown for each decay cycle. The total numbers of pennies decayed and total number of pennies thrown will be calculated. It will be realized that 50.25% of the pennies will decay during the first throw. The second experiment will also test the average value for three half-lives. It will be noted that the average value for three half-lives is 6. Data will be represented in form bar graphs, tables as well as plotted curves.
Introduction
Radioactivity is a spontaneous and self-disruptive process that involves emission of radiation or particles as a result of disintegration of some nuclides (Michael, 2). Half-life refers to the time taken for a substance to decay by half its original value (Jha, 4). In this experiment, the removal of the coins represents the radioactive decay of the nucleus of an atom. Serway, et al (482) states that the number N of the radioactive nucleus of the same kind that are found in a substance at time t is given by
N =No · e-λt, where No is the original number of radioactive nuclei before the time t. The equation shows that the number of particles of a given radioactive element decreases exponentially with respect to time provided no new particles are added. In this experiment coins will be used to represent the atoms. The coins are used in the experiment because the radioactive process is unpredictable. Tossing of coins to get the head and the tail is a perfect model to use because the probability is 50%. Another reason is that coins have the same shape and size so there is no bias in terms of the outcomes. Apart from that, coins are also readily available and easier to use. The two Lab experiments will be done in order to compare values when using large numbers and a few numbers of coins. The hypothesis for the two experiments is that approximately 50% of the coins will decay in the first throw. The second hypothesis for Lab 2 is that four of the eight coins will show their heads up.
Method for lab 1 To perform the experiment, hundred pennies and a box with cover were used.
All the pennies were placed with tails up in a flat box with cover. The box was then covered and shaken. After shaking, all the pennies with heads up were removed and their number recorded in a table. The number of pennies remaining was also recorded. The coins removed represents the number of atoms that had decayed in one half-life. The steps above were repeated until all the pennies were removed from the box. The number of trials required to complete the process was recorded as well. Every trial entailed shaking the box and removing the pennies with heads up. The values were then used to plot a graph of the number of pennies used versus the number of trials. A graph of the number of coins left versus number of trials was also plotted for analysis.

Method for lab 2 For lab 2 experiment, eight pennies were thrown for every decay cycle. A decay cycle was completed when only one or no penny was left. Fifty decay cycles were done in this experiment. The values for each trial was recorded by recording the number of heads that showed up on the first throw. The number of throws, which was required to have one or no pennies left, was also done. A bar graph was drawn to represent the number of pennies which decayed on the first throw. Another bar graph showing the frequency of the number of throws needed to get one or zero pennies was done for analysis.
Data presentation
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Analysis In the second experiment (lab 2), the total number of pennies that decayed on the first throw is given by adding the number that decayed the first time= (4x1)+(8x2)+(8x3)+(7x4)+(12x5)+(8x6)+(3x7)= (4+16+24+28+60+48+21)=201.
Total number of pennies thrown on the first throw is given by (50x8)= 400
The percentage of the number of pennies that decayed on the first time [pic]
Average time for three half-lives is given by the average of the number of throws for three for one or zero pennies to be left. It is equal to [pic]
Discussion
The results show that 50.25% of the coins decayed in the first throw. The value obtained is in agreement with our hypothesis that approximately 50% of the coins should decay in the first throw. A trial chosen randomly for instance number 34 on table 2 indicate that 5 coins had heads up meaning 5 coins decayed during the first trial. This value is equivalent to 62.5%. In this random trial, the expected outcome of 50% percent was not attained. The first half-life of the pennies was expected to be 50% and this did not happen. The expected time for three half-lives is not realized based on the calculations obtained . One source of error is the sampling error. This type of error is unavoidable because tossing the coin is a game of probability. It is not easy to get the same result in a number trials. The best way to minimize this type of error is to perform many trials. In radioactivity, it is not possible to know which atom is going to decay, but we are sure of the percentage to decay. This concept applies in this experiment because we expect 50% to decay in the first throw, but we cannot identify the coins that will have heads up (decayed). The curve of the number of coins left versus number of trials is to the left because it is the direct opposite of the one for the one for accumulated coins. This is can be described that as the accumulated coins (decayed atoms) increase in value, the amount of the coins left (remaining atoms) decrease in quantity. Again the value obtained almost equals 52% obtained from experiments on lab one. This implies that both experiments are similar from the point that the set hypothesis is supported.
Conclusion
The common hypothesis for the two experiments that 50% of the coins decay (heads up) in the first throw supported. In the first experiment 52% decayed while in the second experiment, 50.25% decayed. This implies that the hypothesis is supported by the findings of the two experiments.

References
D.K. Jha (2004),Radioactivity And Radioactive Decay, Discovery Publishing House.
L'Annunziata, Michael F. Handbook of Radioactivity Analysis. Oxford: Academic, 2012. Print.
Serway, Raymond A, Clement J. Moses, and Curt A. Moyer. Modern Physics. Belmont, CA: Thomson Brooks/Cole, 2005. Internet resource.

Appendix
Table 1
|Trial number |Number Decayed frist time |Accumulated total of coins decayed |Coins left |
|0 |0 |0 |100 |
|1 |52 |52 |48 |
|2 |22 |74 |26 |
|3 |17 |91 |9 |
|4 |3 |94 |6 |
|5 |2 |96 |4 |
|6 |2 |98 |2 |
|7 |1 |99 |1 |
|8 |0 |99 |1 |
|9 |1 |100 |0 |

Table 2
|Trial Number |Number of Decayed frist |Number throws to get 1 or |Trial Number |Number of Decayed frist throw |Number throws to get 1 or 0 |
| |throw |0 coins left | | |coins left |
|1 |3 |3 |26 |5 |4 |
|2 |5 |2 |27 |7 |1 |
|3 |7 |1 |28 |1 |2 |
|4 |4 |3 |29 |5 |3 |
|5 |6 |2 |30 |2 |4 |
|6 |3 |4 |31 |4 |3 |
|7 |2 |3 |32 |1 |5 |
|8 |4 |3 |33 |6 |2 |
|9 |3 |2 |34 |5 |3 |
|10 |4 |4 |35 |3 |4 |
|11 |5 |3 |36 |5 |2 |
|12 |2 |5 |37 |2 |4 |
|13 |5 |4 |38 |7 |1 |
|14 |2 |3 |39 |3 |2 |
|15 |6 |2 |40 |5 |3 |
|16 |3 |3 |41 |6 |2 |
|17 |5 |3 |42 |1 |4 |
|18 |6 |2 |43 |5 |2 |
|19 |1 |5 |44 |2 |4 |
|20 |2 |4 |45 |6 |2 |
|21 |4 |3 |46 |3 |3 |
|22 |3 |2 |47 |5 |2 |
|23 |2 |4 |48 |6 |2 |
|24 |6 |2 |49 |4 |3 |
|25 |4 |3 |50 |5 |3 |
| | | | | | |

Table 3
|Number of Decayed frist throw |Freq |
|1 |4 |
|2 |8 |
|3 |8 |
|4 |7 |
|5 |12 |
|6 |8 |
|7 |3 |

Table 4

|Number throws to get 1 or 0 coins left |Freq |
|1 |3 |
|2 |16 |
|3 |17 |
|4 |15 |
|5 |3 |