Pre-Lab 13 Assignment: Radioactive Decay

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