Problem 43 - Chemistry: The Central Science in SI Units, Global Edition

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 mg of phosphorus-32, how much phosphorus-32 will remain after 1 month (i.e. 30 days)?

Step-by-Step Solution

Answer

After 1 month (30 days), approximately \(1.4 \, mg\) of phosphorus-32 will remain.

Step 1: List the given information

We are given the following information: - Initial amount of phosphorus-32 (A₀) = 3.5 mg - Half-life (half_life) = 14.29 days - Time passed (time) = 1 month = 30 days

Step 2: Set up the formula

The radioactive decay formula is given by: Amount remaining (A) = A₀ × (1/2)^(time / half_life) Let's use the given information to plug into the formula.

Step 3: Plug in the given values

Substituting the given values into the radioactive decay formula, we get: A = 3.5 × (1/2)^(30/14.29)

Step 4: Calculate (1/2)^(time / half_life)

(1/2)^(30/14.29) can be calculated as: (1/2)^(2.1) ≈ 0.40

Step 5: Calculate the remaining amount of phosphorus-32

Now that we have the value of (1/2)^(30/14.29), we can find the remaining amount of phosphorus-32 as follows: A ≈ 3.5 × 0.40 = 1.4 mg

Step 6: Final Answer

After 1 month (30 days), approximately 1.4 mg of phosphorus-32 will remain.

Key Concepts

Phosphorus-32

Phosphorus-32 is a radioactive isotope of phosphorus that plays a significant role in various scientific fields, especially nuclear medicine and biological research. It emits beta particles during its decay process to become sulfur-32.
This transition is helpful because the emitted radiation can be detected, which allows scientists and medical professionals to track and measure biological processes.
Some key facts about Phosphorus-32 include:

• Atomic Number: 15
• Nuclear Decay: Beta decay
• Uses: Phosphorus-32 is extensively used in tracer studies for metabolic research, and most notably in treating leukemia and controlling eye disease related to blood vessel growth.

It is crucial when handling Phosphorus-32 to follow appropriate safety protocols due to the radiation it emits, ensuring limited exposure to protect both the patient and medical personnel.

Half-life

The half-life of a radioactive substance is the time it takes for half of the substance to decay. It is a crucial concept in understanding how quickly a radioactive isotope transforms into another element.
For Phosphorus-32, the half-life is approximately 14.29 days. This means that after 14.29 days, half of any given amount of Phosphorus-32 will have decayed into sulfur-32.
Over another 14.29 days, half of what remains will decay again, and so forth.The concept of half-life can be defined mathematically with the following formula:\[ A = A_0 \times \left( \frac{1}{2} \right)^{\frac{t}{\text{half-life}}} \]Where:

• \( A \) is the amount remaining after time \( t \)
• \( A_0 \) is the initial amount
• \( t \) is the elapsed time

Understanding half-life makes it easier to calculate how much of the radioactive material remains after a certain period of time. This is pivotal in contexts like dose calculation in nuclear medicine.

Nuclear Medicine

Nuclear medicine is a specialized area of medical practice that uses radioactive isotopes for diagnosis, treatment, and research. Unique in its approach, it provides information about both the function and structure of virtually every major organ system within the body.
Phosphorus-32 plays a critical role in nuclear medicine by acting as a tracer. These tracers emit radiation that can be captured using imaging devices like gamma cameras. Highlighted roles of nuclear medicine include:

• Detection of malignant tumors: Phosphorus-32 can help identify aggressive cancerous cells by observing their metabolic activity.
• Treatment: It is used to deliver radiation in a focused manner to target specific areas within the body, minimizing damage to surrounding healthy tissue.
• Research: Radioisotopes help researchers study biochemical and physiological processes in detail.

This field, due to its reliance on radioactivity, requires stringent safety measures to protect both patients and healthcare workers. Nonetheless, nuclear medicine offers critical insights and therapeutic avenues not available through other means, enhancing the understanding and management of various medical conditions.