Question

If \(N_{\mathrm{o}}\) is the initial number of nuclei, number of nuclei remaining undecayed at the end of \(n^{\text {th }}\) half-life is (a) \(2^{n} \cdot N_{\mathrm{o}}\) (b) \(2^{-n} \cdot N_{\mathrm{o}}\) (c) \(n^{2} \cdot N_{\mathrm{o}}\) (d) \(n^{-2} \cdot N_{\mathrm{o}}\)

Short answer

The number of nuclei remaining undecayed after n half-lives is (b) \( 2^{-n} \cdot N_{\mathrm{o}} \).

Step-by-step solution

Understanding Half-Life

The half-life of a substance is the time it takes for half of the initial number of radioactive nuclei to decay. After each half-life period, the number of remaining undecayed nuclei is halved.

Deriving the formula after n half-lives

If we denote the number of half-lives by n, then after the first half-life we have \( \frac{N_{\mathrm{o}}}{2} \) nuclei remaining, after the second half-life we have \( \frac{N_{\mathrm{o}}}{2^2} \) nuclei remaining, and so on. After n half-lives, the number of remaining nuclei will be \( \frac{N_{\mathrm{o}}}{2^n} \).

Simplifying the Expression

The expression for the number of remaining nuclei can also be written in exponential form as \( 2^{-n} \cdot N_{\mathrm{o}} \).

Choosing the Correct Option

Comparing the derived expression with the given options, we find that the correct answer is (b) \( 2^{-n} \cdot N_{\mathrm{o}} \).

Key concepts

Radioactive Decay

Radioactive decay is a fundamental process by which unstable atomic nuclei release energy by emitting radiation. This spontaneous transformation is an intrinsic property of certain isotopes, which are variants of elements with different numbers of neutrons. As we delve into the world of nuclear chemistry, it becomes evident that radioactive decay is guided by statistical laws.

Every radioactive isotope is characterized by its half-life—the time taken for half of its atoms to decay. This half-life can range from fractions of a second to millions of years, depending on the isotope. For students tackling half-life problems, the key concept is that no matter how many atoms you start with, after one half-life has passed, only half of them will remain undecayed.

Understanding this principle is crucial in fields such as dating archaeological finds (via carbon-14 dating), medical imaging, and even determining the age of the earth. When solving such problems, remember that the decay process is random for individual atoms, but predictable when looking at a large ensemble.

Exponential Decay

At the heart of understanding half-life problems lies the concept of exponential decay. This mathematical model is not only fundamental in nuclear chemistry but also applies to various phenomena in physics, biology, and finance. Exponential decay describes how quantities decrease over time at a rate proportional to their current value.

In the context of half-life, this decay process can be expressed using the formula \( 2^{-n} \cdot N_{\mathrm{o}} \), where \( N_{\mathrm{o}} \) is the original number of undecayed nuclei, and \( n \) is the number of half-lives elapsed. The negative exponent signifies that the quantity is decreasing, with each subsequent half-life reducing the remaining amount by half, hence 'exponential' decay. When dealing with half-life problems, always check for consistency in units (such as years, seconds, etc.) and be mindful that the decay follows an exponential trend, not a linear one.

Nuclear Chemistry

Nuclear chemistry is a fascinating field that explores atomic nuclei's changes and the energy created during these changes. It includes studying nuclear reactions, such as fission where a nucleus splits into smaller parts, and fusion where two nuclei combine to form a larger nucleus. Half-life problems are a vital part of this field because they help us understand the stability of different isotopes and the rates of radioactive decay.

For example, nuclear reactors rely on the fission of uranium-235, a process controlled by its half-life. In medicine, isotopes with shorter half-lives are often used in diagnostic imaging, as they decay quickly and minimize the patient's exposure to radiation. In your studies, focus on the practical applications of half-life in nuclear chemistry, like the management of nuclear waste, which requires knowledge of long-lived isotopes and their half-lives. Additionally, safety measures around radioactive materials are anchored in understanding decay rates and the protection required over various time scales.