Question

The half-life of tritium (hydrogen-3) is \(12.3 \mathrm{yr}\). If \(56.2 \mathrm{mg}\) of tritium is released from a nuclear power plant during the course of an accident, what mass of this nuclide will remain after \(12.3\) yr? After 100 yr?

Short answer

After 12.3 years, there will be \(28.1 \mathrm{mg}\) of tritium remaining, and after 100 years, there will be approximately \(0.205 \mathrm{mg}\) of tritium remaining.

Step-by-step solution

Identify the variables

We are given: - The initial mass of tritium (m₀) = 56.2 mg - The half-life of tritium (t₁/₂) = 12.3 years We need to find: - The mass of tritium remaining after 12.3 years, (m₁₂₃) - The mass of tritium remaining after 100 years, (m₁₀₀)

Calculate the number of half-lives

First, we need to find out how many half-lives have passed in those time periods. To do so, we can divide the time by the half-life of tritium. Number of half-lives (n) = time elapsed (t) / half-life (t₁/₂) For 12.3 years: Number of half-lives (n₁₂₃) = 12.3 years / 12.3 years = 1 For 100 years: Number of half-lives (n₁₀₀) = 100 years / 12.3 years ≈ 8.13

Calculate the remaining mass using the formula

To find the remaining mass of tritium, we can use the formula: Remaining mass (m) = Initial mass (m₀) × (1/2) ^ Number of half-lives (n) For 12.3 years: m₁₂₃ = 56.2 mg × (1/2) ^ 1 m₁₂₃ = 56.2 mg × 0.5 m₁₂₃ = 28.1 mg For 100 years: m₁₀₀ = 56.2 mg × (1/2) ^ 8.13 m₁₀₀ = 56.2 mg × 0.00366 m₁₀₀ ≈ 0.205 mg So, after 12.3 years, there will be 28.1 mg of tritium remaining, and after 100 years, there will be approximately 0.205 mg of tritium remaining.

Key concepts

Nuclear Chemistry

Nuclear chemistry is a subfield of chemistry that deals with radioactivity, nuclear processes, and the properties and reactions of unstable atomic nuclei. One of the most studied aspects of nuclear chemistry is the behavior of radioactive isotopes, which are variants of chemical elements that have an unstable nucleus. These isotopes emit radiation as they transform into more stable forms—a process known as radioactive decay.

Radioactive isotopes have many applications, from medical imaging and cancer treatments to archaeological dating and power generation. However, they also pose significant risks due to their emitting radiation that can cause damage to living tissues and the environment. Understanding the behavior of these isotopes, including their decay over time, is essential for their safe and effective use.

An isotope's half-life is a key concept in nuclear chemistry. It is the amount of time required for half of the isotope to decay. The half-life of an isotope like tritium, which is a radioactive form of hydrogen, provides critical information about how long it will remain active and how it should be handled safely in both nuclear power plants and medical scenarios.

Radioactive Decay

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable one. This transformation can occur through various modes, such as alpha decay, beta decay, or gamma emission. The decay process results in the emission of particles and/or energy, and it is fundamental to the nature of radioactive substances.

For instance, tritium undergoes beta decay, during which it emits an electron and a neutrino to transform into helium-3. The rate of decay for a radioisotope is characterized by its half-life. The mathematical understanding of radioactive decay is crucial for measuring the age of archaeological findings, managing nuclear waste, and ensuring safety around radioactive sources.

Several units are used to measure radioactivity, including becquerels and curies, which express the number of decays per second. Health and environmental regulations concerning the exposure to radioactive materials are based on these units, and they are crucial when addressing the aftermath of a nuclear accident, such as the release of tritium from a nuclear facility.

Half-life Calculation

Calculating the half-life of a radioactive substance is a quantitative way of understanding how long it takes for half of a sample of the substance to decay. It's a concept widely used across fields like nuclear chemistry, environmental science, and medicine to predict the behavior of radioactive materials over time.

For example, tritium has a half-life of 12.3 years. This means that every 12.3 years, any given amount of tritium will be reduced to half of its original mass through radioactive decay. The formula to calculate the remaining mass of a radioactive substance after a certain period is given by: \( m = m_0 \times (\frac{1}{2})^n \), where \( m_0 \) is the initial mass, \( n \) is the number of half-lives that have passed, and \( m \) is the remaining mass.

By using this formula, we can determine the remaining mass of tritium or any other radioactive isotope after any given time period, provided we know its initial mass and half-life. These calculations are indispensable in a variety of practical scenarios, such as planning the long-term storage of nuclear waste, assessing environmental safety risks, and designing treatments in nuclear medicine.