Question

If \(8 \mathrm{~g}\) of a radioactive isotope has a halflife of \(10 \mathrm{~h}\). The half-life of \(2 \mathrm{~g}\) of the same substance is (a) \(2.5 \mathrm{~h}\) (b) \(5 \mathrm{~h}\) (c) \(10 \mathrm{~h}\) (d) \(40 \mathrm{~h}\)

Short answer

The half-life remains constant regardless of the amount of substance, so the answer is (c) 10 hours.

Step-by-step solution

Understand the concept of half-life

The half-life of a radioactive substance is the time it takes for half of the substance to decay. It is a constant value for a given substance, which means it is independent of the amount of the substance present.

Apply the concept to the given problem

Since the half-life is a constant value for a radioactive isotope, the amount of the substance present does not affect its half-life. Therefore, whether we have 8 grams or 2 grams of the substance, the half-life remains the same.

Choose the correct answer

Given that the half-life of the isotope (8 g) is 10 hours, the half-life for 2g of the same isotope is also 10 hours. So the correct answer is (c) 10 hours.

Key concepts

Radioactivity

Imagine a world where some elements have an unstable nucleus that's always looking to find balance. That's the reality of radioactivity. At its core, radioactivity is the process by which an unstable atomic nucleus loses energy by emitting radiation. This can happen in several ways, such as through the release of alpha particles, beta particles, or gamma rays.

Elements that exhibit this behavior are known as radioactive isotopes, and they are found both in nature and can be created artificially in laboratories. A radioactive isotope, also called a radioisotope, undergoes a spontaneous transformation into a more stable form, a process we refer to as radioactive decay.

Understanding radioactivity is important not just in nuclear chemistry but also in a multitude of fields like medicine for diagnostic imaging, archaeology for carbon dating, and even in space exploration for powering spacecraft. It's fascinating, yet it must be approached with caution due to the potential risks associated with radiation exposure.

Half-Life Calculation

When we talk about the half-life of a radioactive substance, we're referring to the time it takes for one half of any sample of the substance to decay. It's like a stopwatch for radioactivity that, once you start it, tells you how long you have to wait until only half of your original sample remains. Calculating the half-life of a substance is crucial in various applications, from nuclear medicine to environmental science.

Decay Formula

To put it into a mathematical form, the amount of substance left after a certain number of half-lives can be determined by the formula: \( N = N_0 \times (\frac{1}{2})^{\frac{t}{T}} \), where \(N\) is the remaining quantity of the substance, \(N_0\) is the original quantity, \(t\) is the time elapsed, and \(T\) is the half-life of the substance.

It's critical to unravel that half-life is independent of the initial amount of substance; it remains constant regardless of how much substance you start with. This means that whether you have a mountain or a molehill of a radioactive isotope, the time it will take for half of it to decay is the same.

Nuclear Chemistry

Nuclear chemistry is like the rulebook for the nucleus of an atom. It's a field where chemists look at the heart of the atom and see how it interacts, transforms, and sometimes even splits apart or combines with another to release staggering amounts of energy. This branch of chemistry focuses on reactions that involve changes in nuclear composition as well as the study of the nature and properties of the atomic nucleus.

Nuclear chemistry has given us power production through nuclear reactors, medical advancements through radiation therapy, and even the ability to understand the history of ancient objects through radiocarbon dating. However, it is also the science behind nuclear weapons, making it a field that carries significant weight and responsibility.

One major aspect of nuclear chemistry is understanding and controlling the rate of nuclear reactions, which is where the knowledge of radioactive half-life becomes incredibly useful. By studying half-lives, nuclear chemists can predict the behavior of radioactive materials over time, which is essential for safety and practicality in all applications involving radioactivity.