Problem 31 - Physics

Question

How many half-lives are required for the number of radioactive nuclei to decrease to one-millionth of the initial number?

Step-by-Step Solution

Answer

20 half-lives are required.

Step 1: Understand the half-life concept

A half-life is the time required for half of the radioactive nuclei in a sample to decay. After one half-life, the amount of radioactive nuclei is reduced to 0.5 of the original amount.

Step 2: Set up the equation for reduction due to half-lives

If you have an initial amount of radioactive nuclei, denoted as \(N_0\), after \(n\) half-lives, the remaining quantity \(N\) is given by the equation: \(N = N_0 \times \left( \frac{1}{2} \right)^n\).

Step 3: Substitute the condition into the equation

According to the problem, we want the remaining number of nuclei to be \(\frac{1}{10^6}\) of the initial amount. Hence, we can write the equation as: \(N_0 \times \left( \frac{1}{2} \right)^n = \frac{N_0}{10^6}\).

Step 4: Simplify the equation

Simplify the equation by dividing both sides by \(N_0\) to get: \(\left( \frac{1}{2} \right)^n = \frac{1}{10^6}\).

Step 5: Solve for \(n\) using logarithms

Take logarithms on both sides to solve for \(n\):\[ n \cdot \log\left( \frac{1}{2} \right) = \log\left( \frac{1}{10^6} \right) \]Rearrange to find \(n\):\[ n = \frac{\log\left( \frac{1}{10^6} \right)}{\log\left( \frac{1}{2} \right)} \]Remember that \(\log\left( \frac{1}{10^6} \right) = -6\) and \(\log\left( \frac{1}{2} \right) \approx -0.3010\).

Step 6: Calculate the value

Plug in the values:\[ n = \frac{-6}{-0.3010} \approx 19.9 \]Since \(n\) must be a whole number, we round up to 20.

Key Concepts

Radioactive Decay

Radioactive decay is a process in which unstable atomic nuclei lose energy by emitting radiation. This decay occurs in a predictable and measurable pattern, often characterized by the half-life, which is the time it takes for half of the radioactive nuclei in a sample to decay. As radioactive materials decay, they transform into different elements or isotopes.
Some important aspects of radioactive decay are:

• The decay process is random for each atom, but predictable overall for a large number of atoms.
• Different isotopes have different decay rates, defined by their unique half-lives.
• Decay can involve various types of emissions, such as alpha particles, beta particles, or gamma rays.

Understanding radioactive decay and half-lives is essential in nuclear physics, as it allows scientists to predict how long a radioactive substance will remain active by comparing the number of half-lives it undergoes. This concept is widely used in fields such as carbon dating, nuclear power generation, and medical imaging.

Logarithms

Logarithms are a mathematical tool used to solve equations involving exponential relationships. They convert multiplication operations into addition, which simplifies the handling of exponential equations. In the context of radioactive decay, logarithms help solve the equation related to half-lives by reducing complex exponential expressions to a more manageable form.
Here's how logarithms are applied in this context:

• Using the property that \(^x = a \), taking the logarithm gives \( = \text{log}(a)/\text{log}(x) \).
• This property is crucial to finding the number of half-lives needed for decay.
• By taking the logarithm of both sides of the equation that describes radioactive decay, you simplify it to find the number of cycles needed.

Logarithms make it easier to interpret and work with the exponential decay patterns observed in radioactive substances, thus providing key insights into the rate of decay and timelines for decay processes.

Nuclear Physics

Nuclear physics is the field of science that deals with the fundamental properties and behaviors of atomic nuclei. It explores how atomic nuclei are structured and how they interact, including the processes that result in radioactive decay.
Key topics in nuclear physics include:

• Understanding nuclear reactions and the forces that hold nuclei together.
• Exploring the behavior of subatomic particles like protons and neutrons.
• Applications of nuclear physics, such as energy production in nuclear reactors, medical treatments, and radiation detection.

Nuclear physics helps us comprehend how and why radioactive decay happens, providing a framework for technologies that tap into nuclear reactions for various practical applications. This includes designing measures to safely handle and dispose of radioactive materials, leveraging nuclear energy for electricity production, and innovating in the realm of nuclear medicine.