Problem 43 - Fundamentals of Physics

Question

A radioactive sample intended for irradiation of a hospital patient is prepared at a nearby laboratory. The sample has a half-life of \(83.61 \mathrm{~h}\). What should its initial activity be if its activity is to be \(7.4 \times 10^{8} \mathrm{~Bq}\) when it is used to irradiate the patient \(24 \mathrm{~h}\) later?

Step-by-Step Solution

Answer

The initial activity should be approximately \(8.8 \times 10^8\) Bq.

Step 1: Understand the half-life concept

The half-life of a radioactive material is the time it takes for the activity to reduce to half its initial value. For this problem, the half-life is given as 83.61 hours. We need to calculate the initial activity given the future activity after a certain duration.

Step 2: Use the decay formula

The decay of radioactive material can be described by the formula: \[ A = A_0 \left( \frac{1}{2} \right)^\frac{t}{T_{1/2}} \]where \(A\) is the activity at time \(t\), \(A_0\) is the initial activity, \(T_{1/2}\) is the half-life, and \(t\) is the time elapsed.

Step 3: Substitute known values

We know that the time \(t\) is 24 hours, the future activity \(A\) is \(7.4 \times 10^8\) Bq, and the half-life \(T_{1/2}\) is 83.61 hours. Substitute these into the decay formula:\[ 7.4 \times 10^8 = A_0 \left( \frac{1}{2} \right)^\frac{24}{83.61} \]

Step 4: Solve for initial activity \(A_0\)

To find \(A_0\), rearrange the equation:\[ A_0 = \frac{7.4 \times 10^8}{\left( \frac{1}{2} \right)^\frac{24}{83.61}} \]Calculate \(\left( \frac{1}{2} \right)^\frac{24}{83.61}\), then divide \( 7.4 \times 10^8 \) by the result.

Step 5: Perform the calculation

First, calculate the decay factor:\[ \left( \frac{1}{2} \right)^\frac{24}{83.61} \approx 0.8405 \]Then, calculate the initial activity:\[ A_0 = \frac{7.4 \times 10^8}{0.8405} \approx 8.8 \times 10^8 \text{ Bq} \]

Key Concepts

half-life

The concept of half-life is a core principle in the study of radioactive decay. It refers to the time it takes for half of a given amount of radioactive material to decay and lose its activity. In simpler terms, if you have a certain quantity of a radioactive substance, after one half-life, only half of it will remain radioactive.
This period is constant for any given isotope and does not depend on how much material you have. For example, uranium-238 has a half-life of 4.5 billion years, no matter if you start with a pound or a ton. Understanding the half-life is crucial in various fields, from medical radiation therapy to nuclear power generation.
This concept helps scientists and engineers predict the behavior of radioactive materials over time, aiding in both effective use and safe disposal.

radioactivity

Radioactivity involves the unexpected and spontaneous emission of particles and/or electromagnetic radiation from unstable nuclei. Atoms with an excess of nuclear energy or whose nuclei are not stable often set out to achieve stability by releasing this energy.
When a nucleus decays, it can emit different types of radiation, such as alpha particles, beta particles, or gamma rays. Each type of radiation has its own penetrating power and effect on matter.
The rate at which these emissions occur is known as the activity of the material. Scientists measure radioactivity in Becquerels (Bq), indicating one decay per second, to determine just how radioactive a sample is. Determining this is essential for applications like medical imaging or radiation therapy in hospitals.

activity calculation

When dealing with radioactive materials, calculating their activity is essential to ensuring both safety and effectiveness, especially when they're used in medical treatment. Activity refers to the number of decays or disintegrations happening per second in a radioactive sample.
The formula to calculate activity after some time has passed is:\[ A = A_0 \left( \frac{1}{2} \right)^\frac{t}{T_{1/2}} \]In this, \( A \) is the activity after time \( t \), \( A_0 \) is the initial activity, and \( T_{1/2} \) is the half-life of the material.
This calculation helps scientists adjust the initial amount of a radioactive substance to ensure it has the desired activity level at a specific future time, like when making sure a sample's activity is correct before it's used for a patient.

decay formula

The decay formula is a mathematical representation that describes how the activity of a radioactive material decreases over time. This formula is pivotal for predicting how radioactive materials behave and is especially useful for practical applications, such as timing irradiations in medical settings.
The decay formula expresses this process as:\[ A = A_0 \left( \frac{1}{2} \right)^\frac{t}{T_{1/2}} \]- \( A \) represents the activity at time \( t \),- \( A_0 \) is the initial activity,- \( t \) is the time that has passed,- \( T_{1/2} \) is the half-life.Using this formula, you can solve for the initial activity if other values are known, as shown in the problem from the original exercise. Such calculations ensure radiation sources perform as expected when needed, making this formula a vital tool in nuclear chemistry and medical physics.