Problem 43
Question
Phosphorus-32 is commonly used in nuclear medicine for the identification of malignant tumors. It decays to sulphur-32 with a half-life of 14.29 days. If a patient is given \(3.5 \mathrm{mg}\) of phosphorus- \(32,\) how much phosphorus-32 will remain after 1 month (i.e. 30 days)?
Step-by-Step Solution
Verified Answer
0.8127 mg of Phosphorus-32 will remain after 30 days.
1Step 1: Determine the Number of Half-Lives
First, calculate the number of half-lives that occur in 30 days. The half-life of Phosphorus-32 is 14.29 days. Therefore, the number of half-lives in 30 days is calculated as follows:\[\text{Number of half-lives} = \frac{30 \text{ days}}{14.29 \text{ days/half-life}} \approx 2.10\]This means that about 2.10 half-lives will have passed in 30 days.
2Step 2: Use the Half-Life Formula
The formula for remaining quantity after a certain number of half-lives is given by:\[N = N_0 \left(\frac{1}{2}\right)^n\]where \(N_0\) is the initial quantity (3.5 mg), \(n\) is the number of half-lives (2.10), and \(N\) is the remaining quantity.
3Step 3: Calculate the Remaining Quantity
Substitute the known values into the formula:\[N = 3.5 \times \left(\frac{1}{2}\right)^{2.10} \approx 3.5 \times 0.2322 \approx 0.8127 \text{ mg}\]Therefore, approximately 0.8127 mg of Phosphorus-32 will remain after 30 days.
Key Concepts
Half-Life CalculationNuclear MedicineRadioactive Decay
Half-Life Calculation
Understanding half-life is essential when studying radioactive substances. The half-life refers to the time it takes for half of a given amount of a radioactive element to decay into another form. This concept is key in calculating how much of a radioactive substance remains after a certain period.
For phosphorus-32, used in nuclear medicine, the half-life is 14.29 days. This means that every 14.29 days, only half of the phosphorus-32 remains. For example, if you start with 3.5 mg of phosphorus-32, after one half-life (14.29 days), you'll have 1.75 mg remaining.
In practice, you often calculate how much is left after multiple half-lives. For our exercise, to find out how much phosphorus-32 remains after 30 days, we first calculate how many half-lives fit into 30 days.
For phosphorus-32, used in nuclear medicine, the half-life is 14.29 days. This means that every 14.29 days, only half of the phosphorus-32 remains. For example, if you start with 3.5 mg of phosphorus-32, after one half-life (14.29 days), you'll have 1.75 mg remaining.
In practice, you often calculate how much is left after multiple half-lives. For our exercise, to find out how much phosphorus-32 remains after 30 days, we first calculate how many half-lives fit into 30 days.
- Divide 30 by 14.29 which gives roughly 2.10 half-lives.
- Apply the formula: \(N = N_0 \left(\frac{1}{2}\right)^n\), where \(N_0\) is the initial quantity (3.5 mg), and \ n\ is 2.10.
Nuclear Medicine
Nuclear medicine is a specialty that uses small amounts of radioactive materials, like phosphorus-32, to diagnose and treat diseases. It offers unique insights because radioactive substances behave differently than non-radioactive substances.
In nuclear medicine, radioactive materials are carefully chosen based on their properties, including their half-life. A longer half-life means the material stays active for a longer period, which is useful for prolonged treatment. Shorter half-lives are chosen for brief diagnostics to minimize radiation exposure.
Doctors use phosphorus-32 to highlight certain medical conditions like malignant tumors. When introduced into the body, phosphorus-32 targets fast-growing cells, providing important information about cancer spread, while gradually decaying to a stable state.
This decay is planned precisely to ensure it supports patient health without causing harm. The balance between treatment effectiveness and patient safety is a principal guideline in nuclear medicine.
In nuclear medicine, radioactive materials are carefully chosen based on their properties, including their half-life. A longer half-life means the material stays active for a longer period, which is useful for prolonged treatment. Shorter half-lives are chosen for brief diagnostics to minimize radiation exposure.
Doctors use phosphorus-32 to highlight certain medical conditions like malignant tumors. When introduced into the body, phosphorus-32 targets fast-growing cells, providing important information about cancer spread, while gradually decaying to a stable state.
This decay is planned precisely to ensure it supports patient health without causing harm. The balance between treatment effectiveness and patient safety is a principal guideline in nuclear medicine.
Radioactive Decay
Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. During this process, an element transforms into another element or a different isotope of the same element.
Phosphorus-32, for instance, decays into sulfur-32 by beta decay, releasing a beta particle in the process. This change is part of what allows it to be used in medical applications, as its decay is predictable and measurable.
Radioactive decay is spontaneous and random, but on a macroscopic level, it can be predicted using statistical methods. The half-life is crucial here as it quantifies the rate of decay. Every isotope has its unique half-life, defining how fast it changes.
Understanding this concept helps us not only in medical fields but also in diverse applications like carbon dating in archaeology and managing nuclear power. Recognizing the patterns and principles of decay allows for safe and effective uses of radioactivity.
Phosphorus-32, for instance, decays into sulfur-32 by beta decay, releasing a beta particle in the process. This change is part of what allows it to be used in medical applications, as its decay is predictable and measurable.
Radioactive decay is spontaneous and random, but on a macroscopic level, it can be predicted using statistical methods. The half-life is crucial here as it quantifies the rate of decay. Every isotope has its unique half-life, defining how fast it changes.
Understanding this concept helps us not only in medical fields but also in diverse applications like carbon dating in archaeology and managing nuclear power. Recognizing the patterns and principles of decay allows for safe and effective uses of radioactivity.
Other exercises in this chapter
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