Problem 53 - Precalculus

Question

A radioactive isotope of beryllium-11 decays to borom-11 with a half-life of 13.81 seconds. Beryllium is given to patients that suffer Chronic Beryllium Disease (CBD). If 800 mg are given to a CBD patient, how much beryllium is present after 2 minutes? Round your answer to the nearest milligram.

Step-by-Step Solution

Answer

Approximately 2 mg of beryllium remains after 2 minutes.

Step 1: Understand the Concept of Half-Life

Half-life is the time required for a quantity to reduce to half its initial amount. Here, the half-life of beryllium-11 is given as 13.81 seconds.

Step 2: Convert Time to Seconds

We need to determine how much beryllium is left after 2 minutes. Since half-life is in seconds, convert 2 minutes to seconds: \( 2 \text{ minutes} = 2 \times 60 = 120 \text{ seconds} \).

Step 3: Calculate the Number of Half-Lives

The number of half-lives that occur in 120 seconds is calculated by dividing the total time by the half-life:\[ n = \frac{120 \text{ seconds}}{13.81 \text{ seconds/half-life}} \approx 8.69 \].

Step 4: Apply the Half-Life Formula

Use the formula to compute the remaining quantity after several half-lives:\[ N = N_0 \times \left(\frac{1}{2}\right)^n \]where \( N_0 = 800 \text{ mg} \) and \( n \approx 8.69 \).

Step 5: Calculate Remaining Beryllium

Substitute the known values in the half-life formula:\[ N \approx 800 \times \left(\frac{1}{2}\right)^{8.69} \].

Step 6: Evaluate the Expression

Using a calculator, evaluate the expression:\[ N \approx 800 \times 0.003 \approx 2.4 \text{ mg} \]. Round 2.4 mg to the nearest milligram.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. During this process, the original nucleus, known as the 'parent', transforms into a different atom, referred to as the 'daughter'. This transformation happens because the nucleus releases particles, which can be in the form of alpha, beta, or gamma rays. These emitted particles carry away energy, leading to the gradual breakdown of the original atom.

There are some important points to keep in mind about radioactive decay:

• It is a random process, meaning that we cannot predict exactly when a particular atom will decay, though we can measure the rate at which a group of atoms will decay.
• The rate of decay is characterized by the half-life, which is the time it takes for half of a sample of radioactive atoms to decay.
• This process is key in many scientific and medical applications, such as radiometric dating and cancer treatment.

Exponential Decay

Exponential decay is a specific manner in which the quantity of something decreases over time. In the case of radioactive materials, this means that the amount of the substance diminishes at a rate that is proportional to its current value. This leads to a rapid decrease at first, which slows as the amount lessens.

The formula for exponential decay is often written as: \[ N = N_0 \times \left(\frac{1}{2}\right)^n \]where:

• \( N \) is the remaining quantity after time \( n \).
• \( N_0 \) represents the initial amount.
• \( n \) is the number of elapsed half-lives.

This equation illustrates how exponential decay functions by halving the amount during each half-life period. In practical examples, it helps in calculating how much material stays after a given time, especially in cases like medical dosages or nuclear waste management.

Beryllium-11

Beryllium-11 is a radioactive isotope of the element beryllium. Its instability leads it to decay into another element, boron-11, through radioactive decay processes. This isotope is particularly interesting in scientific and medical contexts, such as its use in certain medical treatments.

Here is some key information about beryllium-11:

• It has a relatively short half-life of about 13.81 seconds, meaning it decays quickly compared to other isotopes.
• It is used in research and some medical treatments, like providing relief for Chronic Beryllium Disease (CBD).
• Understanding how it decays helps in effectively managing its use, especially in determining the safe dosage and accumulation in medical patients.

By knowing how beryllium-11 behaves and decays, scientists and doctors can strategize its use in treatments, ensuring both effectiveness and safety for patients.