Question

A patient is given \(0.080 \mathrm{mg}\) of technetium- \(99 \mathrm{~m}\), a gamma emitter with a half life of \(6.0\) hours. How long would it take for the technetium to reach \(1 / 8\) of its initial amount? (Assume no excretion of the nuclide from the body.)

Short answer

It would take 18 hours for the technetium to reach 1/8 of its initial amount.

Step-by-step solution

Understand the Half-Life Concept

The half-life of a substance is the time it takes for half of the substance to decay. In each half-life period, the amount of the substance is reduced by half.

Determine the Number of Half-Lives for Decay to 1/8

Since we need to know when the technetium will reach 1/8 of its initial amount, we recognize that this will be after 3 half-lives. This is because (1/2)^3 equals 1/8.

Calculate the Total Time Elapsed

If one half-life is 6 hours, then 3 half-lives would take 3 times as long, which is 3 times 6 hours.

Final Calculation

Multiply the number of half-lives by the length of one half-life: 3 half-lives * 6 hours/half-life = 18 hours.

Key concepts

Nuclear Chemistry

At its core, nuclear chemistry involves the study of the chemical and physical properties of elements as influenced by changes in the structure of the atomic nucleus. It deals primarily with radioactivity, nuclear processes, and properties.

Radioisotopes, such as technetium-99m used in medical imaging, are a focal point of nuclear chemistry. These unstable atoms emit radiation as they transform into more stable forms. Understanding and harnessing the decay of radioisotopes is key to numerous applications, from dating archaeological finds with carbon-14 to treating cancer with targeted radioactive compounds.

In the given exercise, technetium-99m's significant attribute is its half-life, which determines the time span that it is useful for diagnostic imaging. The precise calculation of how much of the substance remains after a certain period is fundamental to safely and effectively using radioisotopes in medicine.

Radioactive Decay

Radioactive decay is a random process where unstable atomic nuclei lose energy by emitting radiation in the form of alpha particles, beta particles, and gamma rays. This process is spontaneous and changes the original atom into a different element or a different isotope.

Gamma emitters like technetium-99m decay by emitting gamma radiation, which is a high-energy form of electromagnetic radiation. This feature allows medical professionals to track the movement of these atoms inside the human body using special cameras.

In the context of the exercise, understanding that the decay is an ongoing process provides insight into why, over time, the amount of the original substance decreases. The exercise guides through a classic calculation to determine how long the technetium-99m remains at a level viable for medical imaging before it diminishes to the point where it's no longer detectable.

Exponential Decay

Exponential decay describes the process by which a quantity diminishes at a rate proportional to its current value. Mathematically, it is described by an exponential function, which in the case of radioactive decay is often written as \( N(t) = N_0 \times (\frac{1}{2})^{(t/T_{1/2})} \), where \( N(t) \) is the amount of substance at time \( t \), \( N_0 \) is the initial amount, and \( T_{1/2} \) is the half-life.

For the provided exercise, exponential decay is central to understanding the reduction in the amount of technetium-99m over time. After each half-life period, rather than losing a fixed amount of substance, the material loses half of whatever quantity is present. As a result, the quantity of radioactive isotope decreases rapidly at first, and then more slowly. It's the reason why three half-lives, for instance, reduce the initial amount to \( (\frac{1}{2})^3 \) or one-eighth, rather than simply halving it three times independently. In terms of patient safety and effective treatment, this allows for the calculation of time intervals over which the radioactive material is useful and safe.