CK-12-Physics - Intermediate

http://www.ck12.org Chapter 25. Nuclear Physics

FIGURE 25.4

Carbon dating

Half-life

Each radioactive nucleus decays independently and randomly. Each isotope also has a distincthalf-lifethat defines
how quickly it decays. The half-life of a nucleus is defined as follows: Over the period of time of one half-life, each
nucleus has a 50% chance of decaying.

Radioactive isotopes have different half-lives. For example, polonium-210, mentioned above, has a half-life of
about 138 days. Uranium-238 has a half-life of over four billion years. Some radioactive isotopes have half-lives of
thousandths of seconds, and even smaller. The meaning of half-life is the same, however. No matter what the time
duration for the half-life is, after that time interval passes, one-half of the original sample remains.

Let us use the symbolTofor the half-life of any radioactive isotope andNofor initial massNfor the remaining mass
of the radioactive isotope. We can then construct the following table.

TABLE25.1:

Time(t)in units of half-livesTo Sample amount(N)

0 No

1 To^12 No

2 To^14 No

3 To^18 No

4 To 161 No

Note that based on the defintion of a half-life, the table describes an exponential function.

A graph corresponding to the table is provided below,Figure25.5. It is assumed that whent= 0 ,No=10 grams.
Note that the time is expressed in units of half-life,n.

FIGURE 25.5

The decay of a radioactive material in

units of half-lives.

An equation describing the data in the table (and the graph) is

N=

( 1

2

)n

No(Equation A)

wherenis the number of half-lives. For those who have studied precalculus, we state an equivalent equation using
the numbere

N=Noe(

− (^0) T. 693
o t)Equation B, where -0.693 is an approximation of ln^12.
For a demonstration of exponential decay, follow the link below.
http://demonstrations.wolfram.com/ExponentialDecay/

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