Pre-Lab 13 Assignment: Radioactive Decay

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Radioactive Decay v 0.1 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 = 0.693 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 ∆N ∆t = ∆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
Radioactive Decay v 0.1 4 3 Activity (×10 10 Decays/sec) 2 1 0 0 5 10 15 20 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
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Pre-Lab 13 Assignment: Radioactive Decay

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