Question

Bandages can be sterilized by exposure to gamma radiation from cobalt-60, which has a half-life of 5.27 y. How much of a 10.0-mg sample of cobalt-60 is left after one half-life? Two half-lives? Three half-lives?

Short answer

After one half-life (5.27 years), 5.0 mg of the initial 10.0 mg cobalt-60 sample remains. After two half-lives (10.54 years), 2.5 mg of the initial sample remains. Finally, after three half-lives (15.81 years), 1.25 mg of the initial sample remains.

Step-by-step solution

Identify initial amount and half-life

The initial amount of the cobalt-60 sample is 10.0 mg and its half-life is 5.27 years.

Calculate remaining amount after one half-life

After one half-life, half of the initial sample will remain. By dividing the initial amount by 2, we get the remaining amount: Remaining amount after one half-life = Initial amount / 2 \(= \frac{10.0\,\text{mg}}{2}\) \(= 5.0\,\text{mg}\)

Calculate remaining amount after two half-lives

After two half-lives, the sample will go through another half-life, meaning the remaining amount will be divided by 2 again: Remaining amount after two half-lives = Remaining amount after one half-life / 2 \(= \frac{5.0\,\text{mg}}{2}\) \(= 2.5\,\text{mg}\)

Calculate remaining amount after three half-lives

After three half-lives, the sample will go through one more half-life, so once again, we divide the remaining amount by 2: Remaining amount after three half-lives = Remaining amount after two half-lives / 2 \(= \frac{2.5\,\text{mg}}{2}\) \(= 1.25\,\text{mg}\)

Present the results

After one half-life, 5.0 mg of the initial 10.0 mg cobalt-60 sample remains. After two half-lives, 2.5 mg of the initial sample remains. Finally, after three half-lives, 1.25 mg of the initial sample remains.

Key concepts

Radioactive Decay

Radioactive decay is a fundamental process through which unstable atomic nuclei release energy by emitting radiation in the form of particles or electromagnetic waves. This spontaneous transformation occurs as atoms strive to reach a more stable nuclear state.

In the context of our exercise, Cobalt-60 demonstrates this process by emitting gamma radiation, which can penetrate and sterilize materials such as medical bandages. The concept of a half-life, which is the time required for half of a radioactive substance to decay, is crucial in understanding radioactive decay. For Cobalt-60, with a half-life of 5.27 years, this implies that after 5.27 years, only half of the original sample remains due to radioactive decay.

The step-by-step solution provided in the exercise helps visualize this concept by showing the decrease in mass from the original 10 mg of Cobalt-60 over subsequent half-lives.

Nuclear Chemistry

Nuclear chemistry is the study of changes in atomic nuclei. It encompasses both the natural transformations of elements through radioactive decay, as mentioned earlier, and artificial reactions driven by human made processes in nuclear reactors or accelerators.

Processes such as fission, fusion, and radioactive decay fall under the umbrella of nuclear chemistry. In medical applications, nuclear chemistry plays a vital role not only in sterilization but also in diagnostic and treatment procedures through the use of radioisotopes—like using Cobalt-60 for sterilization in our given problem. The predictable pattern of decay and energy emission from radioisotopes makes nuclear chemistry a highly valuable field in science and medicine. Understanding the principles of nuclear reactions and half-life calculations is integral to safely handling and applying radioactive materials.

Cobalt-60 Properties

Cobalt-60 is a synthetic radioactive isotope of cobalt, known for its application in radiation therapy and industrial radiography. It's not found naturally and is produced by the neutron irradiation of Cobalt-59 in nuclear reactors.

Cobalt-60 decays by beta decay to a stable nickel isotope, during which it emits gamma rays. These gamma rays form the basis of Cobalt-60's use in sterilization and cancer treatment because of their ability to destroy bacteria and tumor cells. In our exercise, the initial 10.0 mg sample of Cobalt-60 loses half its mass after each 5.27-year interval due to its radioactive properties. These properties of Cobalt-60, including its half-life, make it a powerful tool in the fields mentioned above, but also necessitate stringent safety protocols to protect against its intense radiation.