When an Individual Radioactive Atom Will Decay

Introduction 

The purpose of this study is to predict when an individual radioactive atom will decay. It is how many atoms on average will decay during a specific time interval. The following is the hypothesis for the study: 

Null hypothesis (H0):  All of the counts are not be identical because of the random nature of radioactivity. 

The alternative hypothesis (H1):  All of the counts are identical because of radioactivity's random nature. 

Since we have thirty (30) different one-minute measurements of a radioactive source, if they were all identical, you would suspect problems with your detector. Even if the number of counts recorded were not identical but still exhibited minimal variation, the detector could have problems. 

It is very difficult to predict which radioactive atoms will decay during a specific time interval in radioactivity. Hence, it will be reasonably accurate to predict the exact number of radioactive atoms will decay during a certain time interval. In measuring radiation, a measurement will be more precise when the number of measured counts is large. However, there are two ways the number of counts can accumulate. That is with high counting rates and long counting times. Measuring 10,000 counts per minute (CPM) for one minute will yield a total of 10,000 counts, while measuring 1,000 CPM for ten minutes will result in the same total. In both cases, a large number of counts will be accumulated. It is easy to understand both measurements of background radiation and measurements of radioactive sources under these terms. 

Procedure 

Apparatus  

High Voltage, Signal Cable, high voltage cable, amplifier, and counter  

Steps   

The following is the procedure that was used in the experiment. First, we set the counting time on the counter to 60 seconds. Secondly, we measured the radiation emitted from the Cs-137 source. Thirdly, record the counts collected. Then clear the counter. The sixth step was to repeat steps 2 - 4 for a total of 30 measurements. Seventh, measure the radiation emitted by the source for 6 seconds. Lastly, repeat step 6 for a 60 seconds measurement. Also, repeat step 6 for a 300 seconds measurement. Repeat step 6 for a 600 seconds measurement. 

Data 

Table 1. Chi-square statistic data for ten measurements of a polonium source
Measurement	Counts (C)			) 2
1	10200	10073.67	126.33	15960.11
2	9959	10073.67	-114.67	13148.44
3	9977	10073.67	-96.67	9344.44
4	10163	10073.67	89.33	7980.44
5	10176	10073.67	102.33	10472.11
6	10061	10073.67	-12.67	160.44
7	10124	10073.67	50.33	2533.44
8	9972	10073.67	-101.67	10336.11
9	10060	10073.67	-13.67	186.78
10	10142	10073.67	68.33	4669.44
11	9997	10073.67	-76.67	5877.78
12	10193	10073.67	119.33	14240.44
13	10087	10073.67	13.33	177.78
14	10099	10073.67	25.33	641.78
15	10174	10073.67	100.33	10066.78
16	9992	10073.67	-81.67	6669.44
17	10149	10073.67	75.33	5675.11
18	10064	10073.67	-9.67	93.44
19	10171	10073.67	97.33	9473.78
20	10181	10073.67	107.33	11520.44
21	10182	10073.67	108.33	11736.11
22	10027	10073.67	-46.67	2177.78
23	10012	10073.67	-61.67	3802.78
24	9996	10073.67	-77.67	6032.11
25	10033	10073.67	-40.67	1653.78
26	9987	10073.67	-86.67	7511.11
27	10083	10073.67	9.33	87.11
28	9934	10073.67	-139.67	19506.78
29	10070	10073.67	-3.67	13.44
30	9945	10073.67	-128.67	16555.11
Total	302210.00		0.00	208304.67

Mean = 302210.00 / 30 = 10073.67 

= 10073.67 

 = 208304.67 / 10073.67 = 20.68 

= 20.68 

Critical value =  = 42.557 

Critical value =  = 17.708 

The fractional standard deviation is calculated as: 

  where C = 302210 

C 1/2 =   

  = 0.00182 

= 0.182 % 

Figure 1 : Graph for the number of counts and measurements 

The graph above shows the trend for the number of counts with measurements of the polonium source. The trendline indicates that an increase in the number of measurements results in a rise in the number of counts. The highest number of counts was experienced when there was one measurement of 10200, while the lowest number of counts was experienced with 30 measurements. 

Discussion 

The computed Chi-square value,  = 20.68, and the two critical values are 17.708 and 42.552. We will reject the null hypothesis since the computed chi-square value lies between the two critical values (17.708 < 20.68 < 42.552). This implies that all of the counts are identical because of the random nature of radioactivity. Therefore, the detector passed the Chi-square test. 

The fractional standard deviation measures the precision of the measurement. The fractional standard deviation is 0.182 %, implying that we are 68 % confident that the true mean will fall between 301660 and 302760 counts . 

Conclusion 

The study's purpose was to examine how many atoms on average will decay during a certain time interval. We tested the hypothesis of whether all of the counts are identical. We established that the detector passed the Chi-square test. Therefore, the experiment supported the hypothesis. Using the fractional standard deviation, we found out that at 68 % confident that the true mean will fall between 301660 and 302760 counts.