UCSB Science Line How are scientists able to figure out what the half life of an atom is when a half life can be billions of years? Do you ever completely get rid of radioactive atoms? How are we able to age the oldest rock that is 4 billion years old?
Question Date: 2011-04-18
Answer 1:

The equation describing radioactive decay is

decay-equation

Where t is time, N(t) is the amount of isotope remaining at time t, N0 is the initial amount of the isotope present, t1/2 is the half life, and = t1/2/ln(2). If you do not understand this equation, do not worry because I will explain everything you need to know.

The half life of a radioactive isotope is the amount of time it takes for half of the isotopes to decay. However, you do not need to wait the whole half life to measure decay. The table below shows how much of the radioactive isotope remains and how much has decayed after a given amount of time, expressed in units of the half life. I calculated the numbers in the table using the equation so you can forget about the equation for now.

Half life of a radioactive isotope
# of half-lives
elapsed
Fraction of
isotopes
remaining
Fraction of
isotopes
decayed
0 1.00 0.00
1/8 0.92 0.08
1/4 0.84 0.16
3/8 0.77 0.23
1/2 0.71 0.29
5/8 0.65 0.35
3/4 0.59 0.41
7/8 0.55 0.45
1 0.50 0.50
2 0.25 0.75
3 0.13 0.88
4 0.063 0.938
5 0.031 0.969
6 0.016 0.984
7 0.0078 0.9922
8 0.0039 0.9961
9 0.9980
10 0.0010 0.9990

Looking at the table, we see that after one half life, 1/2 of the radioactive atoms remain which means 1/2 of them have decayed. After only 1/2 of a half life, 0.29 (or 29%) of the radioactive atoms have decayed but 0.71 (or 71%) of them still remain. As the amount of time we wait gets smaller, the more atoms remain. The figure below shows the data from the chart in graphical form.

graph

Completely getting rid of radioactive atoms would take a very long time. From the table, you can see that after 10 half lives, 99.9% of the atoms have decayed. After many thousands of half lives, only a few radioactive atoms will be left. However, the concept of a half life only applies to large numbers of atoms because it is based on the probability of large systems. When we only have a few radioactive atoms remaining, we need to start thinking in terms of probability for one atom instead of purely in terms of half lives. The same equation governs the probability of decay for one atom so the table still applies, but we need to think about it a little differently. It works like this: If we have one radioactive atom and we wait for one half life, then there is a 50% chance it will decay. If we wait for two half lives, there is a 75% chance it will decay. If we wait for 10 half lives, there is a 99.9% chance it will decay. And so on. No matter how long we wait, we will never quite get to a 100% chance of decay. So to answer your question, if you wait enough half lives

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