Problem 34 - College Physics

Question

Nitrogen-13, with a half-life of \(10.0 \mathrm{~min}\), decays by beta emission. (a) Write down the decay equation to determine the daughter product and whether the beta particle is a positron or electron. (b) If a sample of pure \({ }^{13} \mathrm{~N}\) has a mass of \(1.50 \mathrm{~g}\) at a certain time, what is the activity 35.0 min later? (c) What percentage of the sample is \({ }^{13} \mathrm{~N}\) at this time? (d) What alternative process could have happened to the nitrogen-13? Write down its decay equation and determine the daughter product for this process.

Step-by-Step Solution

Answer

(a) \( ^{13}N \rightarrow ^{13}C + e^+ + \nu_e \). (b) Use \( A = A_0 e^{-\lambda t} \). (c) Calculate \( N/N_0 \) and convert to %. (d) Electron capture: \( ^{13}N + e^- \rightarrow ^{13}C + \nu_e \).

Step 1: Write the Beta Decay Equation for Nitrogen-13

Nitrogen-13 decays by beta-plus decay, which means it transforms by emitting a positron. The beta decay equation for \( ^{13}N \) can be written as \( ^{13}N \rightarrow ^{13}C + e^+ + u_e \). Here, \( e^+ \) is a positron and \( u_e \) is a neutrino. The daughter product is Carbon-13.

Step 2: Calculate Activity 35 Minutes Later

Activity (\(A\)) can be calculated using the formula \( A = A_0 e^{-\lambda t} \). The decay constant \( \lambda \) is calculated by \( \lambda = \frac{\ln(2)}{t_{1/2}} \), where \( t_{1/2} = 10 \) minutes. Calculate \( \lambda \), then substitute \( \lambda \) and \( t = 35 \) minutes into the activity formula to get the final activity.

Step 3: Determine Remaining Percentage of Nitrogen-13

The remaining percentage of \(^{13}N\) can be found using the formula \( N = N_0 e^{-\lambda t} \). First calculate \( N/N_0 \), which represents the fraction of \(^{13}N\) left. Then convert it into a percentage.

Step 4: Identify Alternative Decay Process for Nitrogen-13

An alternative decay process for \(^{13}N\) is electron capture. In this process, one proton in the nucleus captures an electron, transforming into a neutron. The reaction is \( ^{13}N + e^- \rightarrow ^{13}C + u_e \), with Carbon-13 as the daughter product.

Key Concepts

Nitrogen-13

Nitrogen-13 is a radioactive isotope of nitrogen. It is denoted as \(^ {13}N\), where the number 13 represents its atomic mass, which is the sum of its protons and neutrons. This isotope contains 7 protons and 6 neutrons.
Nitrogen-13 is commonly used in positron emission tomography (PET) scans in the medical field due to its short half-life and the emission of positrons. This makes it ideal for imaging and diagnosing medical conditions.
It typically originates from the bombardment of natural nitrogen with high-energy particles such as protons. This process is often performed in a cyclotron, a type of particle accelerator.

Half-life

The half-life of Nitrogen-13 is 10 minutes. This means that every 10 minutes, half of the atoms in a sample of \(^ {13}N\) will decay into more stable products.
The concept of a half-life is crucial in understanding radioactive decay as it describes the rate at which instability reduces.

• In 10 minutes, 50% of the original sample remains.
• In 20 minutes, 25% of the original sample is left.
• In 30 minutes, 12.5% remains, and so on.

This rapid decay makes Nitrogen-13 particularly useful in medical diagnostics, allowing healthy tissue to absorb radioactive isotopes quickly, without prolonged exposure.

Positron Emission

Positron emission is a type of beta decay where a proton inside the nucleus transforms into a neutron, emitting a positron and a neutrino. For Nitrogen-13, this process can be written as: \(^ {13}N \rightarrow ^ {13}C + e^+ + u_e \)
Here, \( e^+ \) represents the positron, which is essentially an anti-electron with the same mass as an electron but a positive charge.
During this process:

• The atomic number decreases by 1, turning the nitrogen into carbon.
• The mass number remains unchanged at 13.
• The emitted positron (along with the neutrino) travels a short distance before it encounters an electron, leading to annihilation and the release of gamma rays, which can be detected externally.

Such outcomes are what make positron emission so valuable in PET scans for mapping functional processes in the body.

Radioactive Decay

Radioactive decay is a natural process by which unstable atomic nuclei lose energy by emitting radiation. For isotopes like Nitrogen-13, this decay occurs via positron emission, transforming them into more stable elements like Carbon-13.
This transformation involves the emission of particles and energy. Understanding radioactive decay helps predict the behavior of radioactive isotopes over time.
Besides positron emission, Nitrogen-13 can also decay through a process known as electron capture, described by the reaction: \(^ {13}N + e^- \rightarrow ^{13}C + u_e \).

• In electron capture, a proton captures an electron and converts to a neutron.
• This also results in the formation of Carbon-13 as the daughter product.

The study of such decays provides insights into nuclear reactions and the stability of elements, assisting in various scientific and medical fields.