Problem 26 - Fundamentals of Physics

Question

The half-life of a radioactive isotope is 140 d. How many days would it take for the decay rate of a sample of this isotope to fall to one-fourth of its initial value?

Step-by-Step Solution

Answer

The decay rate falls to one-fourth of its initial value in 280 days.

Step 1: Understand the Concept of Half-life

The half-life of a substance is the time it takes for half of the material or its activity to decay. In this problem, the substance has a half-life of 140 days.

Step 2: Set Up the Decay Formula

The decay formula relating the decay constant \( k \), half-life \( t_{1/2} \), and decay is: \( N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \). We want to find \( t \) for which \( N(t) = \frac{1}{4}N_0 \).

Step 3: Set up the Equation for the Problem

Using the decay formula, \( \left(\frac{1}{2}\right)^{\frac{t}{140}} = \frac{1}{4} \). Since \( \frac{1}{4} = \left(\frac{1}{2}\right)^2 \), the equation simplifies to \( \left(\frac{1}{2}\right)^{\frac{t}{140}} = \left(\frac{1}{2}\right)^2 \).

Step 4: Solve the Equation

Because the bases are equal, equate the exponents: \( \frac{t}{140} = 2 \). Solve for \( t \) by multiplying both sides by 140, which gives \( t = 280 \) days.

Key Concepts

Half-life

The concept of half-life is fundamental in understanding radioactive decay. It refers to the amount of time it takes for half of a given quantity of a radioactive isotope to decay into a more stable form. In other words, if you start with a sample of 100 atoms, after one half-life, only 50 atoms of the original isotope remain, assuming no other processes affect this atom count. The remaining atoms would have decayed into another element or isotope.

Half-life is a constant value unique to each radioactive substance, and it helps scientists predict the rate at which a sample will decay.
For example:

• A shorter half-life means the substance decays rapidly.
• A longer half-life indicates a slower decay process.

The half-life of an element can be used in various fields such as archaeology for dating artifacts, medicine for cancer treatment, and nuclear energy production. By knowing the half-life, you can determine how long it will take for a substance to reach a certain level of radioactivity or to stabilize.

Decay Rate

Decay rate refers to the speed at which a radioactive isotope undergoes decay. This term is often used interchangeably with activity, which is a measure of how many decay events occur per unit of time.
Decay rate decreases over time as the radioactive material diminishes, but initially, it can be quite high. This rate is directly related to the amount of a substance and its half-life.

When you are considering a sample's

• Initial decay rate: This is when the sample has just been measured.
• Future decay rate: This is calculated by understanding how much of the substance remains after a certain period of time, usually measured in half-lives.

In our problem, the decay rate falls to one-fourth of its initial value over 280 days. Understanding decay rate is crucial to predicting how radioactive materials behave over time, which has significant implications for safety measures and environmental assessments.

Decay Constant

A decay constant is a number that gives more precise information about how rapidly a radioactive isotope decays. It represents the probability per unit time of an atom decaying. This constant is symbolized by the letter \(k\) and can be calculated using the half-life with the formula:
\[ k = \frac{\ln(2)}{t_{1/2}} \]where \(t_{1/2}\) is the half-life.

The value of the decay constant \(k\) is crucial for calculating decay processes because:

• It quantifies the likelihood of decay per unit time for a single remaining atom or unit of substance.
• It helps describe the exponential nature of radioactive decay mathematically.

The decay constant is vital for understanding how quickly a given radioactive isotope will diminish. In practical terms, knowing both the decay constant and the amount of substance can help scientists and engineers design systems that manage the decay products effectively.