# Calculations Involving Half-Life and Decay-Rates

## The half-life of a radionuclide is the time taken for half the radionuclide's atoms to decay.

#### Key Points

• The half-life is related to the decay constant as follows: <equation contenteditable="false">$t_{1/2} = \tau ln2$.

• The relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent while those that radiate weakly endure longer.

• Half-lives of known radionuclides vary widely, from more than 1019 years, such as for the very nearly stable nuclide 209 Bi, to 10-23 seconds for highly unstable ones.

#### Terms

• the time required for half of the nuclei in a sample of a specific isotope to undergo radioactive decay

• A radionuclide is an atom with an unstable nucleus, characterized by excess energy available to be imparted either to a newly created radiation particle within the nucleus or via internal conversion.

#### Examples

• A sample of 14C, whose half-life is 5730 years, has a decay rate of 14 disintegrations per minute (dpm) per gram of natural carbon. An artefact is found to have radioactivity of 4 dpm per gram of its present C. How old is the artefact? We have:$N = N_{0}e^{-t/\tau }$where$N/N_{0}=4/14 \approx 0.286$$\tau = T_{1/2}/ln2 \approx 8267 \text{ years}$.$t = -\tau ln N/N_{0} \approx 10360 \text{ years}$

#### Figures

A simulation of many identical atoms undergoing radioactive decay, starting with four atoms (left) and 400 atoms (right). The number at the top indicates how many half-lives have elapsed

2. ##### Half-life

Part of a series of videos on physics problem-solving. The problems are taken from "The Joy of Physics." This one deals with radioactive half-life. The viewer is urged to pause the video at the problem statement and work the problem before watching the rest of the video.

The half-life of a radionuclide is the time taken for half of the radionuclide's atoms to decay. For the case of one-decay nuclear reactions:

$N = N_{0}e^{-\lambda t} = N_{0}e^{-\tau /t}$

The half-life is related to the decay constant as follows:

$N = N_{0}/2$

$t = T_{1/2}$

$t_{1/2} = ln2/\lambda = \tau ln2$

A half-life must not be thought of as the time required for exactly half of the entities to decay. Figure 1 is a simulation of many identical atoms undergoing radioactive decay. Note that after one half-life there are not exactly one-half of the atoms remaining; there are only approximately one-half left because of the random variation in the process. However, with more atoms (the boxes on the right), the overall decay is smoother and less random-looking than with fewer atoms (the boxes on the left), in accordance with the law of large numbers.

The relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent while those that radiate weakly endure longer. Half-lives of known radionuclides vary widely, from more than 1019 years, such as for the very nearly stable nuclide 209 Bi, to 10−23 seconds for highly unstable ones.

The factor of ln(2) in the above equations results from the fact that the concept of "half-life" is merely a way of selecting a different base other than the natural base e for the lifetime expression. The time constant τ is the e-1-life, the time until only 1/e remains -- about 36.8 percent, rather than the 50 percent in the half-life of a radionuclide. Therefore, τ is longer than t1/2. The following equation can be shown to be valid:

$N(t) = N_{0}e^{-t/\tau } = N_{0}2^{-t/t_{1/2}}$

Since radioactive decay is exponential with a constant probability, each process could just as easily be described with a different constant time period that (for example) gave its 1/3-life (how long until only 1/3 is left), or its 1/10-life (how long until only 1/10 is left), and so on. Therefore, the choice of τ and t1/2 for marker-times is only for convenience and for the sake of uploading convention. These marker-times reflect a fundamental principle only in that they show that the same proportion of a given radioactive substance will decay over any time period you choose.

Mathematically, the nth life for the above situation would be found by the same process shown above -- by setting $N = N_{0}/n$ and substituting into the decay solution, to obtain:

$t_{1/n} = \frac{lnn}{\lambda } = \tau lnn$

Figure 2

#### Key Term Glossary

atom
The smallest possible amount of matter which still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.
##### Appears in these related concepts:
base
A nucleotide's nucleobase in the context of a DNA or RNA biopolymer.
##### Appears in these related concepts:
decay
to change by undergoing fission, by emitting radiation, or by capturing or losing one or more electrons
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
##### Appears in these related concepts:
nuclide
A nuclide (from "nucleus") is an atomic species characterized by the specific constitution of its nucleus -- i.e., by its number of protons (Z), its number of neutrons (N), and its nuclear energy state.
##### Appears in these related concepts:
period
The duration of one cycle in a repeating event.
##### Appears in these related concepts:
Period
The period is the duration of one cycle in a repeating event.