Problem 57 - Physics for Scientists and Engineers with Modern Physics

Question

(II) The activity of a radioactive source decreases by \(2.5 \%\) in 31.0 hours. What is the half-life of this source?

Step-by-Step Solution

Answer

The half-life of the radioactive source is approximately 827 hours.

Step 1: Understand the Concept of Half-Life

The half-life of a substance is the time it takes for half of the radioactive particles in a sample to decay. We can relate this to the decay constant, which indicates how quickly a substance decays.

Step 2: Use the Exponential Decay Formula

The activity of a radioactive substance over time can be modeled using the exponential decay formula: \[ N(t) = N_0 e^{-\lambda t} \]where \( N(t) \) is the activity at time \( t \), \( N_0 \) is the initial activity, and \( \lambda \) is the decay constant.

Step 3: Set Up the Problem with Given Decrease

It is given that the activity decreases by 2.5\% in 31.0 hours. Therefore, \( N(31) = 0.975 N_0 \). Substituting in the decay formula:\[ 0.975 N_0 = N_0 e^{-\lambda \times 31} \]We can divide both sides by \( N_0 \) to simplify:\[ 0.975 = e^{-\lambda \times 31} \]

Step 4: Solve for the Decay Constant \( \lambda \)

Take the natural logarithm of both sides to isolate \( \lambda \):\[ \ln(0.975) = -\lambda \times 31 \]Solving for \( \lambda \):\[ \lambda = -\frac{\ln(0.975)}{31} \]

Step 5: Calculate the Half-Life Using \( \lambda \)

The half-life \( T_{1/2} \) is related to the decay constant by the equation:\[ T_{1/2} = \frac{\ln(2)}{\lambda} \]Substitute the value of \( \lambda \) from the previous step to find the half-life.

Step 6: Perform the Calculation

First calculate \( \lambda = -\frac{\ln(0.975)}{31} \) which gives \( \lambda \approx 0.0008376\). Now calculate the half-life:\[ T_{1/2} = \frac{\ln(2)}{0.0008376} \approx 827.08 \text{ hours} \]

Key Concepts

Exponential Decay Formula

Radioactive substances decay exponentially over time. This means that at any given time, a fixed percentage of the substance decays.
This process is expressed with the exponential decay formula:

• \( N(t) = N_0 e^{-\lambda t} \)
• \( N(t) \) is the activity (or amount) at time \( t \).
• \( N_0 \) is the initial activity (or amount).
• \( \lambda \) is the decay constant, indicating the rate of decay.

The formula allows us to calculate how much of a radioactive substance remains after a certain amount of time has passed.
By inputting the known values of time and decay constant, we can predict the activity at any future point.
It is widely used in fields such as physics, archaeology, and medicine to understand how quickly materials deplete.

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This spontaneous process changes the parent atom into a different element or isotope.
In the context of this exercise, the rate at which the activity decreases (2.5% in 31 hours) reflects this fundamental process.

• Radioactive substances are everywhere, from the smoke detectors in homes to medical isotopes used in hospitals.
• Each radioactive element has a characteristic decay rate, which is quantified by its half-life.
• Over time, these elements transform, sometimes turning into stable elements through various types of decay such as alpha, beta, or gamma decay.

Understanding radioactive decay is crucial for handling radioactive materials safely, as well as utilizing them effectively in technology and research.
By calculating the half-life, scientists can determine how long a radioactive substance will remain active, which is vital for both safety measures and scientific analysis.

Decay Constant

The decay constant \( \lambda \) is a vital parameter in the mathematics of radioactive decay. It represents the probability per unit time that a given atom will decay.
A higher decay constant means a substance decays more quickly.To find \( \lambda \), you can rearrange the exponential decay formula given a specific time period and a measured decrease.

• If 2.5% of a substance decays in 31 hours, it provides a specific case to calculate \( \lambda \).
• Solving \( \ln(0.975) = -\lambda \times 31 \) gives the decay constant \( \lambda \).
• In this exercise, \( \lambda \approx 0.0008376 \) per hour was found.

The decay constant plays a crucial role in determining the half-life of a radioactive element through the relation \( T_{1/2} = \frac{\ln(2)}{\lambda} \).
This connection helps predict how long it takes for half of a substance to decay, which is a fundamental concept in radioactive studies.