Problem 33 - Chemistry The Molecular Science

Question

After 2 hours, tantalum- 172 has \(\frac{1}{16}\) of its initial radioactivity. Calculate its half-life (s).

Step-by-Step Solution

Answer

The half-life of tantalum-172 is 1800 seconds.

Step 1: Understanding the Half-Life Concept

The half-life of a radioactive element is the time it takes for half of the initial amount of the substance to decay. In this problem, we need to find the half-life of tantalum-172.

Step 2: Relating Decay to Time

We know that after 2 hours, only \(\frac{1}{16}\) of the original amount remains. This implies that 4 half-lives have elapsed (since \((\frac{1}{2})^4 = \frac{1}{16}\)).

Step 3: Setting Up the Equation

The number of half-lives \(n\) is given by the equation: \((\frac{1}{2})^n = \frac{1}{16}\). Solving, we have:\(n = 4\).

Step 4: Calculating the Half-Life

Since 4 half-lives correspond to 2 hours, we can calculate the half-life as follows: Half-life \(= \frac{2 \text{ hours}}{4} = \frac{1}{2} \text{ hour} = 1800 \text{ seconds}\).

Key Concepts

half-life calculation

Half-life calculation is a fundamental concept in understanding radioactive decay. It involves determining the time required for half of the radioactive atoms present in a sample to decay. This decay follows an exponential pattern. For any radioactive substance, the half-life stays constant regardless of the initial amount present. When calculating half-life, you typically compare how much of the substance remains after a certain period.

• Start by identifying the fraction of the substance remaining. In this exercise, \(\frac{1}{16}\) of the original tantalum-172 remained after 2 hours.
• Recognize how this fraction relates to half-lives. Here, \(\left(\frac{1}{2}\right)^4 = \frac{1}{16}\), indicating four half-lives have occurred.
• Finally, divide the total time by the number of half-lives to find the time of a single half-life: \(\frac{2\ \text{hours}}{4} = \frac{1}{2}\ \text{hour}\). To convert to seconds, multiply by 3600, resulting in 1800 seconds.

Remember, the consistency of half-life is critical for using this calculation in real-world applications like carbon dating, medical treatments, and nuclear energy management.

tantalum-172

Tantalum-172 is an isotope of the element tantalum, known for its radioactive properties. Isotopes, in general, share the same number of protons but differ in the number of neutrons. This specific isotope has an unstable nucleus, leading it to undergo radioactive decay. In practice, studying isotopes like tantalum-172 helps scientists understand a broad range of phenomena, from the structure of atoms to the mechanisms of nuclear reactions.
Tantalum itself is a rare, hard, and corrosion-resistant metal. When isolated, tantalum is often used in electronics, as it can efficiently transport electric current and heat. However, tantalum-172, in particular, due to its radioactive nature, is generally confined to academic and controlled environments for research purposes.
Scientists measure the radioactive decay of tantalum-172 to conduct experiments in nuclear chemistry and to test theories of nuclear physics. This work contributes to a larger understanding of radioactive processes and their implications in various scientific and practical fields.

nuclear chemistry

Nuclear chemistry is the branch of chemistry dealing with the chemical processes in radioactive substances and phenomena. This area focuses on isotopes and their decay, including the radiation they emit and the underlying mechanisms. Nuclear reactions are not only fascinating but also impactful, influencing fields as diverse as medicine, energy production, and archaeology.
A key aspect of nuclear chemistry involves understanding and harnessing radioactive decay. This includes studying isotopes like tantalum-172, exploring half-lives, and examining the resulting changes in atomic nuclei.

• Nuclear chemists might work on developing methods to date ancient artifacts through radioactive dating systems, using measured isotopes.
• In the medical field, radioactive isotopes are used for diagnosis and treatment, such as using radiation to target and destroy cancer cells.
• In terms of energy, nuclear chemistry is critical in managing power plants and exploring new nuclear fusion technologies.

The principles of nuclear chemistry help in safely managing radioactive materials and mitigating potential risks posed by their use. This knowledge continues to grow as we explore the possibilities, benefits, and responsibilities of working with nuclear substances.