Question

Radioactive nitrogen-13 has a half-life of 10. minutes. Calculate the mass of this isotope that remains after an hour in a sample that originally contained \(96 \mathrm{mg}\).

Short answer

1.5 mg remains after 60 minutes.

Step-by-step solution

Understand the Problem

We are given a radioactive isotope, nitrogen-13, with a half-life of 10 minutes. The initial mass is 96 mg. Our task is to determine how much of this isotope remains after 60 minutes.

Calculate the Number of Half-lives

Find the total time and divide by the half-life to determine how many half-lives occur in 60 minutes. \[ \text{Number of half-lives} = \frac{60 \text{ minutes}}{10 \text{ minutes/half-life}} = 6 \text{ half-lives} \]

Use the Half-life Formula

Use the formula for radioactive decay: \[ A = A_0 \left( \frac{1}{2} \right)^n \] Where \(A\) is the amount remaining, \(A_0\) is the initial amount (96 mg), and \(n\) is the number of half-lives (6).

Calculate the Remaining Mass

Substitute the values into the formula: \[ A = 96 \times \left( \frac{1}{2} \right)^6 \] Compute the power and multiplication to determine the remaining mass.

Perform the Calculation

Calculate \( \left( \frac{1}{2} \right)^6 \) which is \( \frac{1}{64} \). Now, multiply by the initial mass: \[ A = 96 \times \frac{1}{64} = 1.5 \text{ mg} \] Thus, 1.5 mg of nitrogen-13 remains after an hour.

Key concepts

Half-life calculation

A half-life is the time taken for half of a radioactive substance to decay. It’s an essential concept in understanding how radioactive materials transform over time. To calculate the remaining mass of a substance after a given time, we use the half-life calculation.

The formula often used is:

• \[ A = A_0 \left( \frac{1}{2} \right)^n \]

Where:

• \( A \) is the remaining mass after a specific period.
• \( A_0 \) is the initial mass.
• \( n \) is the number of half-lives passed.

Consider that each half-life represents a reduction of the original mass by half. In the presented exercise, finding how many half-lives have passed is crucial. This number helps in plugging into the formula to find out how much of the radioactive substance will remain. The calculation involves understanding the logarithmic scale of decay, which makes radioactive decay predictable yet uniquely fascinating.

Nitrogen-13 isotope

Nitrogen-13 is a radioactive isotope of nitrogen. In its natural state, nitrogen is stable, but isotopes like nitrogen-13 are unstable and radioactive. Nitrogen-13 is used predominantly in medical imaging, particularly in positron emission tomography (PET) scans which help in detecting diseases.

Here's why nitrogen-13 is significant:

• It decays rapidly, with a half-life of about 10 minutes, making calculations practical and results prompt.
• Its decay process ultimately leads to a stable form, boosting its utility in short-term medical assessments.

This exercise on nitrogen-13 helps learners understand the dynamics of short-lived isotopes in practical scenarios. Working through decay problems with nitrogen-13 offers insight into both theoretical flow and practical application in the healthcare field.

Radioactive isotopes

Radioactive isotopes, or radionuclides, are atoms with excess nuclear energy making them unstable. This instability leads to radioactive decay where atoms release energy to transform into stable forms. Radioactive isotopes are found naturally and are also synthetically produced for various applications.

Some key points about radioactive isotopes:

• They have unique half-lives which help determine their rate of decay.
• They are used in medicine, industry, and scientific research.
• Understanding how they decay enables better control and application in their respective fields.

The problem involving nitrogen-13 is a classic example of how radioactive isotopes can be quantitively analyzed to predict their remaining mass over time. This understanding is crucial, not just for academic purposes, but also for practical applications in fields like health and nuclear science. Tracking and predicting the decay of radioactive isotopes ensures safe and effective use in varied domains.