Question

Iron- 59 in the form of iron(II) citrate is used in iron metabolism studies. Its half-life is 44.5 days. If you start with \(0.56 \mathrm{mg}\) iron- 59 , calculate the mass (mg) that remains after 1 year.

Short answer

Approximately 0.00145 mg of Iron-59 remains after 1 year.

Step-by-step solution

Understanding Half-life Concept

The half-life of a radioactive substance is the time required for half of the material to decay. For Iron-59, it takes 44.5 days for half of any given amount to decay.

Converting Year to Days

To calculate the remaining mass of Iron-59 after a year, we first need to convert the duration into days. Since 1 year equals 365 days (ignoring leap years for simplicity), we have 365 days.

Determine Number of Half-life Periods

Calculate the number of half-life periods in 365 days by dividing the total days by the half-life of Iron-59. \[\text{Number of half-lives} = \frac{365 \text{ days}}{44.5 \text{ days/half-life}} \approx 8.20\]

Calculating Remaining Mass

Use the half-life formula to calculate the remaining mass of a substance. If the initial mass is \(M_0 = 0.56 \, \text{mg}\), after \(n\) half-lives, the remaining mass \(M\) is given by: \[M = M_0 \times \left( \frac{1}{2} \right)^n = 0.56 \times \left( \frac{1}{2} \right)^{8.20}\]Computing the expression yields the remaining mass.

Final Calculation

Calculate \( \left( \frac{1}{2} \right)^{8.20} \) and then multiply with the initial mass. Using a calculator or computational tool:\[M \approx 0.56 \times 0.002594 \approx 0.00145 \, \text{mg}\]

Key concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei release energy and transform into more stable forms. This occurs by emitting radiation such as alpha, beta, or gamma rays until a stable state is reached. This process follows a predictable pattern described by the concept of half-life.
The half-life is the time required for half of a given quantity of a radioactive substance to decay. It doesn't matter how much of the substance you start with—after one half-life, only half remains.
This concept is crucial in fields like archeology for dating ancient objects and in medicine for nuclear treatments. Understanding the half-life helps scientists and professionals gauge how long a radioactive substance will be active, which is vital for safety and application purposes.

Iron-59

Iron-59 is an isotope of iron that is frequently used in medical and biological studies, particularly in tracing iron metabolism in humans. Unlike normal iron, Iron-59 is radioactive, meaning it undergoes decay over time.
Its half-life is 44.5 days, which means that after this period, any given amount of Iron-59 will have diminished by half due to radioactive decay. In medical applications, for instance, Iron-59 can help in researching how the body processes iron, offering insights into conditions like anemia.

• Iron-59 emits beta radiation, essential for detecting its presence in biological systems.
• Ineffective for long-term studies due to its relatively short half-life.

The careful handling and use of Iron-59 illustrate the balance required in harnessing radioactive materials for beneficial purposes.

Nuclear Chemistry

Nuclear chemistry involves the study of nuclear reactions and properties, focusing on changes in atomic nuclei. At its core, this branch of chemistry examines how nuclear substances transform and the energy involved in these processes.
A key concept within nuclear chemistry is understanding radioactive decay and isotope half-lives. Nuclear reactions, like those seen with Iron-59, showcase how nuclei become stable over time through decay.
Nuclear chemistry applies in various real-world situations like energy production, medical imaging, and cancer treatment.

• Power generation in nuclear reactors.
• Development of nuclear medicines and radiopharmaceuticals.
• Environmental radiochemistry for contamination tracking.
• Nuclear weapons technology and safety measures.

Through nuclear chemistry, we gain insights into the powerful energies locked within atomic nuclei, allowing advancement in technology, medicine, and scientific understanding.