Half-Life in Radioactive Decay

During radioactive decay, unstable atomic nuclei lose energy by emitting radiation in the form of particles or electromagnetic waves. Over time, the number of undecayed atoms decreases, and the sample becomes less radioactive. The type of atom that has an unstable nucleus and loses energy by emitting radiation is called a radioisotope, and the substance emitting the radiation is the radioactive substance. In radioactivity, half-life describes the time it takes for half of the atoms in a sample of a radioactive substance to decay.

The half-life is a fixed characteristic of that substance and is independent of the amount of material present. It means that whether you have a large chunk of radioactive material or just a tiny sample, the time it takes for half of the atoms to decay will always be the same.

Half-Life Formula

The half-life of a radioactive substance, denoted by the symbol T_1/2, can be represented mathematically. If you start with a certain number of radioactive atoms, after one half-life, half of those atoms will have decayed. After two half-lives, only a quarter of the original atoms will remain, and so on.

The relationship between the number of undecayed atoms (N) at a given time and the original number of atoms (N₀) can be expressed using the following formula:

\[N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \]

Where:

– N(t) is the number of undecayed atoms at time t.

– N₀ is the original number of atoms.

– t is the elapsed time.

– t_1/2 is the half-life of the substance.

This formula shows how the number of undecayed atoms decreases exponentially over time, with each half-life reducing the remaining number of radioactive atoms by half. In practical terms, knowing the half-life of a substance allows scientists to predict how long it will take for a certain amount of radioactive material to decay.

For instance, in carbon dating, scientists determine the age of a sample by measuring the amount of Carbon-14 remaining. By knowing the half-life of Carbon-14 (about 5,730 years), they can calculate how many years have passed since the sample started decaying.

Radioactive Decay Graph

A radioactive decay graph is a visual representation of how the amount of a radioactive substance decreases over time. The graph typically plots the quantity of the substance on the vertical axis (y-axis) against time on the horizontal axis (x-axis). Let us take the example of cobalt-60, a commonly used radioactive element in medical and industrial applications.

The decay graph typically shows a smooth, exponential decline over time. Initially, the decay happens rapidly as a large number of atoms are present, but as time progresses and fewer undecayed atoms remain, the rate of decay slows down. Cobalt-60 has a half-life of approximately 5.27 years, meaning that every 5.27 years, the amount of radioactive substance present is reduced by half.

Half-Life Values

The following table shows the half-lives of a few well-known radioisotopes that are widely studied.

Radioisotope	Symbol	Half-Life
Hydrogen – 3	³H	12.3 y
Carbon – 14	¹⁴C	5730 y
Potassium – 40	⁴⁰K	1.26 X 10⁹ y
Chromium – 51	⁵¹Cr	27.7 d
Strontium – 90	⁹⁰Sr	29.1 y
Iodine – 131	¹³¹I	8.04 d
Radon – 222	²²²Rn	3.823 d
Uranium – 235	²³⁵U	7.04 X 10⁸ y
Uranium – 238	²³⁸U	4.47 X 10⁹ y
Plutonium – 243	²³⁸Pt	5 h
Americium – 245	²⁴⁵Am	432.7 y

Example Problems and Solutions

Problem 1: A wooden artifact was found in an archaeological dig, and scientists determined that it contains 25% of its original carbon-14 content. Given that the half-life of carbon-14 is 5,730 years, how old is the artifact?

Solution:

If 25% of the original carbon-14 remains, this means that half of the carbon-14 decayed once to 50%, and then half of that decayed again to 25%. Therefore, the artifact has undergone 2 half-lives.

\[ \text{Age} = \text{Number of half-lives} \times \text{Half-life of carbon-14} \\ \text{Age} = 2 \times 5,730 \text{ years} \\ \text{Age} = 11,460 \text{ years} \]

The artifact is approximately 11,460 years old.

Problem 2: A patient receives a dose of iodine-131 for thyroid treatment, and the initial dose contains 100 mg. If the half-life of iodine-131 is 8 days, how much of it will remain in the patient’s body after 24 days?

Solution:

Given

Initial dose = 100 mg.

Half-life of iodine-131 = 8 days.

Time elapsed = 24 days.

The number of half-lives is given by:

\[ \text{Number of half-lives} = \frac{\text{Elapsed time}}{\text{Half-life}} \\ => \text{Number of half-lives} = \frac{24 \text{ days}}{8 \text{ days}} = 3 \]

After each half-life, the amount of iodine-131 is halved:

\[ \text{Remaining amount} = 100 \text{ mg} \times \left(\frac{1}{2}\right)^3 = 12.5 \text{ mg} \]

After 24 days, 12.5 mg of iodine-131 will remain in the patient’s body.