Problem 26 - College Physics

Question

Prove that the number \(N\) of radioactive nuclei remaining in a sample after an integer number \((n)\) of half-lives has elapsed is \(N=\frac{N_{o}}{2^{n}}=\left(\frac{1}{2}\right)^{n} N_{0} .\) Here \(N_{0}\) stands for the initial number of nuclei.

Step-by-Step Solution

Answer

The formula is proved by observing the pattern of decay per half-life.

Step 1: Understanding the Problem

We need to show that the remaining number of radioactive nuclei, after certain half-lives, can be represented by the formula \( N = \frac{N_0}{2^n} = \left(\frac{1}{2}\right)^n N_0 \). Here, \( N_0 \) is the initial number of nuclei, and \( n \) is the number of half-lives.

Step 2: Understanding Half-Life

A half-life is the amount of time required for half of the radioactive nuclei in a sample to decay. After one half-life, the quantity of the radioactive substance will be half of the initial amount, that is \( \frac{N_0}{2} \).

Step 3: Expression After One Half-Life

After one half-life, the remaining number of nuclei \( N \) is given by \( N = \frac{N_0}{2} \).

Step 4: Expression After Two Half-Lives

After a second half-life passes, the remaining nuclei become half of \( \frac{N_0}{2} \), which is \( \frac{N_0}{4} \). This can be rewritten as \( \frac{N_0}{2^2} = \left(\frac{1}{2}\right)^2 N_0 \).

Step 5: Expression for n Half-Lives

By following this pattern, the remaining number of radioactive nuclei after \( n \) half-lives would be \( \frac{N_0}{2^n} \). Therefore, \( N = \frac{N_0}{2^n} = \left(\frac{1}{2}\right)^n N_0 \).

Step 6: Final Verification

Each half-life results in the remaining nuclei being half of those from the previous half-life, which aligns perfectly with the formula given. Therefore, the proof that \( N = \frac{N_0}{2^n} = \left(\frac{1}{2}\right)^n N_0 \) is complete.

Key Concepts

Half-Life

In the realm of radioactive decay, the term "half-life" is used to describe the time necessary for half of the radioactive nuclei in a sample to undergo decay. This concept is crucial because it provides a measure of the rate at which a radioactive substance diminishes over time.
After one half-life, a substance's quantity is halved. For example, if you start with an initial number of nuclei, represented as \( N_0 \), then after one half-life, the amount of nuclei remaining will be \( \frac{N_0}{2} \).
In essence, with each half-life, you continuously halve the number of remaining radioactive nuclei. This behavior helps in predicting how much of a radioactive sample will remain after a given period. Understanding half-life is fundamental to grasping broader concepts of radioactive decay and is universally applicable across various radioactive materials.

Exponential Decay

Exponential decay is a mathematical concept that describes a process where the quantity decreases at a rate that is proportional to its current value. In the context of radioactive decay, this means that the number of radioactive nuclei decreases by a constant percentage over equal time intervals, specifically related to the half-life.
The number of remaining nuclei after an integer number \( n \) of half-lives is not just halved once but reduces exponentially. This can be captured using the formula \( N = \frac{N_0}{2^n} \), which demonstrates how the quantity shrinks exponentially over time.
Exponential decay is a hallmark of radioactive decay, reflecting the natural pattern in which radioactive nuclei disintegrate. This formula elegantly illustrates how each passing half-life results in an exponential reduction in the number of radioactive atoms.

Radioactive Nuclei

Radioactive nuclei, also known as radioactive isotopes or radionuclides, are unstable atoms that release energy by emitting radiation. These nuclei undergo decay, transforming into different, more stable atoms over time.
The primary characteristic of radioactive nuclei is their instability, which drives the process of radioactive decay. This instability results from a delicate balance of forces within the atomic nucleus, often due to an excess of protons or neutrons.
During each half-life, radioactive nuclei lose part of their population through decay. This process continues until they transform into stable nuclei, at which point they cease to be radioactive. Understanding the behavior of radioactive nuclei is essential for grasping the principles of radioactive decay and its applications in fields such as nuclear physics, medicine, and environmental science.

Mathematical Proof

Proving the formula for radioactive decay involves understanding how the number of remaining nuclei decreases with each half-life. Starting with a known quantity \( N_0 \) and the concept of half-life, we can establish a mathematical relationship for any number of half-lives \( n \).
After the first half-life, the nuclei remaining is \( \frac{N_0}{2} \). After the second half-life, it becomes \( \frac{N_0}{4} = \frac{N_0}{2^2} \). This pattern continues as \( \frac{N_0}{2^n} \) after \( n \) half-lives, showing an exponential relationship.
Thus, the formula \( N = \frac{N_0}{2^n} = \left(\frac{1}{2}\right)^n N_0 \) emerges as a consistent rule for determining the remaining radioactive nuclei. This proof not only mathematically demonstrates the exponential decay nature of radioactive substances but also provides a reliable method for predictions related to radioactive decay over time.