Question

A sample of curium has an activity of \(1,600 \mathrm{~Bq}\). If the half-life of curium is \(24.0 \mathrm{~s}\), how long before its activity is \(25.0 \mathrm{~Bq}\) ?

Short answer

The activity will reach 25 Bq after 144 seconds.

Step-by-step solution

Understand the Problem

We are given a radioisotope, curium, with an initial activity of 1600 Bq. We need to determine the time it takes for the activity to decrease to 25 Bq given that the half-life is 24 seconds.

Use the Exponential Decay Formula

The exponential decay formula is given by \( A = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \), where \( A \) is the final activity, \( A_0 \) is the initial activity, \( t \) is the time, and \( T_{1/2} \) is the half-life of the substance.

Plug in the Known Values

Substitute \( A = 25 \ \text{Bq} \), \( A_0 = 1600 \ \text{Bq} \), and \( T_{1/2} = 24 \ \text{s} \) into the formula: \[ 25 = 1600 \times \left(\frac{1}{2}\right)^{\frac{t}{24}} \]

Solve for Time (t)

First, divide both sides by 1600 to isolate the exponential term: \[ \frac{25}{1600} = \left(\frac{1}{2}\right)^{\frac{t}{24}} \]Simplify the left side: \[ 0.015625 = \left(\frac{1}{2}\right)^{\frac{t}{24}} \]Take the logarithm of both sides to solve for \( t \): \[ \log(0.015625) = \left(\frac{t}{24}\right) \times \log\left(\frac{1}{2}\right) \]Re-arrange to find \( t \): \[ t = 24 \times \frac{\log(0.015625)}{\log\left(\frac{1}{2}\right)} \]

Calculate the Exact Time

Use a calculator to compute the values:- \( \log(0.015625) \approx -1.806 \)- \( \log(0.5) \approx -0.301 \)Substitute these values back:\[ t = 24 \times \frac{-1.806}{-0.301} \approx 144.0 \ \text{s} \]

Key concepts

Half-life

The concept of half-life is a core idea in understanding radioactive decay. Half-life is defined as the time required for half of a radioactive substance to decay. It is a constant value for each radioisotope, meaning that no matter how much or how little of the material there is, it will always take the same amount of time for half of it to decay.
For example, if you start with 1,000 atoms of a radioisotope with a half-life of 10 years, after 10 years, you will have 500 atoms left. After another 10 years (20 years total), you'll have 250 atoms. This pattern continues as the material decays.

Key points about half-life:

• It is unique to each radioisotope.
• It measures the rate at which a radioisotope decays.
• It helps in predicting how long a radioactive material will remain active.

Exponential Decay Formula

The exponential decay formula is a mathematical representation used to describe the process of decay in radioactivity. It shows how quickly the activity of a radioactive substance decreases over time. The formula is: \[A = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\]Where:

• \(A\) is the final activity,
• \(A_0\) is the initial activity,
• \(t\) is the time elapsed,
• \(T_{1/2}\) is the half-life of the substance.

This formula is derived from the principle that radioactive substances follow a consistent decay pattern, halving their activity over each half-life period. As you calculate different values using this formula, you can determine how long it will take for a sample's radioactivity to reach a specific level.
Look at the given exercise: you have the values needed to plug into the formula and solve for the time when the activity crosses a certain threshold. Such calculations are crucial in fields like nuclear medicine and environmental science.

Radioisotopes

Radioisotopes are isotopes of elements that are unstable and emit radiation as they decay. They have the same number of protons, but a different number of neutrons compared to the stable isotopes of the same element. This instability leads them to transform into a more stable form by releasing energy in the form of particles or electromagnetic waves.
Radioisotopes have a wide range of applications:

• Nuclear Medicine: Used in diagnosing and treating diseases.
• Archaeological Dating: Helpful in determining the age of archaeological finds.
• Energy Generation: Utilized as a source of power in nuclear reactors.

In the context of the exercise, understanding radioisotopes like curium is essential for managing nuclear materials and ensuring safe and efficient use in various technological applications. Different radioisotopes have different half-lives and decay paths, impacting how they can be used or stored safely.