Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

A radioactive substance has a half-life of 700 years. If there were 10 grams initially, how much would be left after 300 years?

Step-by-Step Solution

Answer

Approximately 7.42 grams remain after 300 years.

Step 1: Understanding Half-Life

The half-life of a substance is the time required for half of the substance to decay. In this problem, the half-life is 700 years. This means that every 700 years, the remaining amount of the substance will be halved.

Step 2: Determine Number of Half-Life Periods

To find out how many half-life periods have passed in 300 years, use the formula: \( n = \frac{t}{T_{1/2}} \), where \( t = 300 \) years and \( T_{1/2} = 700 \) years. Thus, \( n = \frac{300}{700} \approx 0.4286 \).

Step 3: Calculate Remaining Substance

The remaining amount of a substance after a certain number of half-lives can be calculated using the formula: \( A = A_0 \times (0.5)^n \). Here, \( A_0 = 10 \) grams and \( n \approx 0.4286 \). Substitute the values: \[ A = 10 \times (0.5)^{0.4286} \approx 10 \times 0.742 \approx 7.42 \text{ grams} \]

Step 4: Box the Final Answer

Conclude the calculation by clearly specifying the amount of substance remaining after 300 years is approximately 7.42 grams.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process by which unstable atomic nuclei lose energy by emitting radiation. This decay happens randomly, but at a predictable average rate, which is unique for each radioactive isotope. It involves the transformation of a radioactive parent isotope into a stable daughter isotope over time.
In this context, understanding the concept of half-life is crucial. Half-life is the time required for half of the radioactive substance to decay. For example, if a substance has an initial amount of 10 grams and a half-life of 700 years, in those 700 years, only 5 grams will remain.
Finding out how much of a substance remains after a certain period involves calculating how many half-life periods have elapsed. This understanding helps in various fields, such as carbon dating or nuclear medicine, where predicting the decay of radioactive materials is crucial.

Exponential Decay

Exponential decay describes a process where the quantity of a substance decreases at a rate proportional to its current value. This type of decay is fundamental in understanding half-life and radioactive decay. Exponential decay can be mathematically expressed with the formula:

• \( A = A_0 \times (0.5)^n \)
  

Where:

• \( A_0 \) is the initial quantity of the substance,
• \( n \) is the number of half-life periods, and
• \( A \) is the remaining quantity.

In our initial exercise, the exponential decay formula allows us to compute how much of a radioactive substance remains after a given period, taking into account how many half-life periods have occurred.
Using this technique efficiently determines the remaining amount of a substance without needing to count individual decays, which may not be feasible given the small scale and constant rate of decay in real-life scenarios.

Substance Decay Over Time

Substance decay over time is a concept that describes the gradual decrease in the amount of a substance due to various natural phenomena, like radioactive decay. When you track a substance's decline over time, you are essentially observing how many half-lives have passed.
To calculate this mathematically, the number of half-life periods (\( n \)) is determined by:

• \( n = \frac{t}{T_{1/2}} \), where \( t \) is the elapsed time and \( T_{1/2} \) is the half-life period.

Monitoring substance decay over time helps in predicting how long a certain amount will last or understanding environmental changes. This type of calculation becomes significant in fields such as archaeology, where knowing the half-life can help date historical artifacts, or in nuclear engineering, where knowing decay rates is necessary for safety and efficiency.