Problem 35 - Physics

Question

The number of radioactive nuclei present at the start of an experiment is \(4.60 \times 10^{15}\). The number present twenty days later is \(8.14 \times 10^{14}\). What is the half-life (in days) of the nuclei?

Step-by-Step Solution

Answer

The half-life of the radioactive nuclei is approximately 8 days.

Step 1: Understand the problem

We need to find the half-life of the radioactive nuclei. We know the initial amount and the amount remaining after 20 days. The half-life is the time it takes for half of the radioactive nuclei to decay.

Step 2: Write the decay formula

The number of radioactive nuclei at any time can be calculated using the formula: \[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}} \] where \( N(t) \) is the number of nuclei at time \( t \), \( N_0 \) is the initial number of nuclei, and \( T \) is the half-life.

Step 3: Insert known values

Insert the known values into the decay formula. Here, \( N_0 = 4.60 \times 10^{15} \), \( N(t) = 8.14 \times 10^{14} \), and \( t = 20 \) days. \[ 8.14 \times 10^{14} = 4.60 \times 10^{15} \times \left(\frac{1}{2}\right)^{\frac{20}{T}} \]

Step 4: Simplify and solve for half-life

Divide both sides of the equation by \( 4.60 \times 10^{15} \) to isolate the exponential term: \[ \frac{8.14 \times 10^{14}}{4.60 \times 10^{15}} = \left(\frac{1}{2}\right)^{\frac{20}{T}} \] Calculate the value on the left to get approximately 0.1770. Therefore, \[ 0.1770 = \left(\frac{1}{2}\right)^{\frac{20}{T}} \] Take the natural log of both sides: \[ \ln(0.1770) = \frac{20}{T} \cdot \ln\left(\frac{1}{2}\right) \]

Step 5: Solve for T

Calculate the natural logs: \( \ln(0.1770) \approx -1.733 \) and \( \ln\left(\frac{1}{2}\right) \approx -0.693 \). Rearrange to find \( T \): \[ T = \frac{20 \cdot (-0.693)}{-1.733} \] Calculate \( T \): \[ T \approx \frac{-13.86}{-1.733} \approx 8.0 \] Thus, the half-life is approximately 8 days.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation in the form of particles or electromagnetic waves. This process causes these nuclei to transform into different elements or isotopes over time. One of the key characteristics of radioactive decay is its random nature; we cannot predict when a specific nucleus will decay. However, we can determine the decay rate for a large collection of nuclei, referred to as the substance's half-life.

During radioactive decay, different types of emissions may occur, such as:

• Alpha decay: the nucleus emits an alpha particle (two protons and two neutrons).
• Beta decay: a neutron in the nucleus converts into a proton, emitting a beta particle (electron) and an antineutrino, or a proton converts into a neutron, emitting a positron and a neutrino.
• Gamma decay: the nucleus releases energy in the form of gamma rays, without changing its structure in terms of protons and neutrons.

These decay processes are essential in fields like radiometric dating, medical imaging, and nuclear power generation.

Exponential Decay Formula

The exponential decay formula is a mathematical representation of the rate at which a quantity decreases over time. In the context of radioactive decay, this formula helps to calculate the remaining number of radioactive nuclei after a given time, taking into account the half-life of the material.

The formula for exponential decay is given by:\[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}} \]where:

• \( N(t) \) is the number of nuclei at time \( t \).
• \( N_0 \) is the initial number of nuclei when \( t = 0 \).
• \( T \) is the half-life of the substance.

This equation is a powerful tool in nuclear physics as it allows scientists and engineers to predict how long radioactive materials will remain active, which is crucial for safety in nuclear plants and for understanding the age of archaeological finds based on carbon dating.

Natural Logarithm

The natural logarithm is an essential mathematical function used to solve exponential equations, particularly in the realm of radioactive decay. It helps transform the exponential decay equation into a simple linear form that can be easily solved.

The natural logarithm function, denoted as \( \ln(x) \), is based on the constant \( e \approx 2.718 \). This base makes it specially suited for problems involving continuous growth or decay. For instance, in solving for half-life, taking the natural log of both sides of the decay equation linearizes it:\[ \ln(N(t) / N_0) = \frac{t}{T} \cdot \ln\left(\frac{1}{2}\right) \]The natural logarithm plays a critical role in breaking down complex decay calculations into manageable steps, assisting students and scientists in obtaining accurate measurements and predictions.

Nuclear Physics

Nuclear physics is the branch of physics focused on understanding the atomic nucleus, including its components like protons and neutrons, their interactions, and the forces that bind them together. It encompasses topics such as nuclear reactions, radioactive decay, and nuclear energy. In nuclear physics, understanding how the nucleus behaves is crucial for applications such as nuclear power generation, where nuclear fission is controlled to release energy. Additionally, knowledge of radioactive decay helps in medical diagnostics through techniques like PET scans, and in archaeological dating. Key insights about the atomic nucleus in nuclear physics include:

• The concept of the strong nuclear force, which binds protons and neutrons together despite the repulsive electromagnetic force between protons.
• The understanding of nuclear stability and how additional neutrons or protons can affect it, leading to different isotopes and decay pathways.
• Techniques for measuring and interpreting half-life, which allow precise calculations for various nuclear processes.

Fundamentally, nuclear physics not only explores the tiny components within atoms but also vastly impacts technological and scientific advancements globally.