Measuring Radioactivity

The radioactivity of a substance is measured by how many decays per unit time occur. Two popular units of the activity are used: the curie (Ci), defined as

1 Ci = 3.7 x 10¹⁰ decays/s (8)

and the becquerel (Bq), defined as

1 Bq = 1 decay/s (9)

Note that, by themselves, these units do not measure accurately how dangerous a given amount of radiation might be for humans - for medical purposes other units of radioactivity reflecting this aspect are more appropriate.

It is found that, if a given radioactive substance at a certain time contains N nuclei, then at a short time $\Delta$t later a certain number $\Delta$N have decayed which is given by

 

$$\Delta \lambda \Delta$$ (10)

where $\lambda$ is called the decay constant. Note that $\lambda$ has units of inverse time, or s ^{- 1}. The decay rate, R , is defined as the number of decays per unit time:

$$\left\vert \frac{\Delta N}{\Delta t} \right\vert \lambda$$ (11)

For finite times it is found that the number of nuclei present after a time t is given by

 

$$\lambda$$ (12)

where N₀ is the number of nuclei present at t = 0 and e = 2.71828... is the base of the ``natural'' logarithms (compared to the base 10 of the ``common'' logarithms). Thus, from Eq. (29.12), in a time t = $\lambda$ a fraction e ^{- 1} = 0.36787944... of a substance remains.

It is sometimes convenient to introduce the half-life, T_1/2 , of a substance, defined as the time over which exactly one half of a substance remains. From Eq. (29.12) one finds

 

$$\lambda {\frac{N}{N_0}} \equiv {\textstyle\frac{1}{2}} \Rightarrow {\frac{\ln 2}{\lambda}}$$ (13)

where ln 2 = 0.693... is the natural logarithm of 2 (note ln e = 1 , which is analogous to log 10 = 1 for the common logarithm). In terms of the half-life Eq. (29.12) can be written as

$${\frac{N}{N_0}} \left(\frac{1}{2}\right)$$ (14)

Thus, after a period of one half-life ${\frac{1}{2}}$ of a substance remains, after another half life ${\frac{1}{2}}$ x ${\frac{1}{2}}$ = ${\frac{1}{4}}$ of a substance remains, and so on. Half-lives of substances range from tiny fractions of a second to millions of years.