Problem 59 - Calculus for Biology and Medicine

Question

The half-life of \(\mathrm{C}^{14}\) is 5730 years. If a sample of \(\mathrm{C}^{14}\) has a mass of 20 micrograms at time \(t=0\), how much is left after 2000 years?

Step-by-Step Solution

Answer

Approximately 15.68 micrograms of \(\mathrm{C}^{14}\) remains after 2000 years.

Step 1: Understand the Half-Life Concept

The half-life of a substance is the time it takes for half of the substance to decay. For \(\mathrm{C}^{14}\), the half-life is 5730 years, meaning every 5730 years, the amount of \(\mathrm{C}^{14}\) will be halved.

Step 2: Use the Half-Life Equation

The general decay formula is \[ A(t) = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]where - \( A(t) \) is the amount remaining at time \( t \), - \( A_0 \) is the initial amount,- \( t \) is the time elapsed,- \( T_{1/2} \) is the half-life.In this case, \( A_0 = 20 \) micrograms, \( t = 2000 \) years, and \( T_{1/2} = 5730 \) years.

Step 3: Plug Values into the Equation

Substitute the values into the decay formula:\[ A(2000) = 20 \left( \frac{1}{2} \right)^{\frac{2000}{5730}} \].This equation will help us determine how much of the \(\mathrm{C}^{14}\) remains after 2000 years.

Step 4: Calculate the Exponent

Calculate the exponent first: \[ \frac{2000}{5730} \approx 0.349 \].This represents the fraction of the half-life periods that have elapsed after 2000 years.

Step 5: Evaluate the Expression

Now evaluate the expression \[ \left( \frac{1}{2} \right)^{0.349} \approx 0.784 \].Thus, only approximately 78.4% of the original amount remains after 2000 years.

Step 6: Determine Final Amount

Finally, calculate the remaining mass:\[ A(2000) = 20 \times 0.784 \approx 15.68 \] micrograms.Therefore, approximately 15.68 micrograms of \(\mathrm{C}^{14}\) is left after 2000 years.

Key Concepts

Decay Formula

In science, the decay formula is a crucial mathematical equation used to predict how much of a substance remains after a given period, considering its decay rate. Understanding this formula can be very helpful in fields like archaeology, geology, and physics. It is especially useful when determining the remaining amount of radioactive substances.
The basic decay formula is:\[ A(t) = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] Here are what the symbols stand for:

• \( A(t) \): the amount of substance remaining after time \( t \).
• \( A_0 \): the initial amount of the substance.
• \( t \): the time that has passed.
• \( T_{1/2} \): the half-life of the substance.

The formula effectively tells us how many half-lives, or periods of decay, have elapsed and adjusts the initial amount accordingly.
Think of it as a way to monitor how much of a substance loses its effect over time. Each half-life reduces the original amount by half, and the formula allows us to calculate this reduction for any period.

Exponential Decay

Exponential decay describes a process where the quantity of a substance reduces systematically over time at a consistent rate. It's called "exponential" because the amount decreases exponentially—the more there is, the more that decays in a given period.
In the context of our decay formula, the exponential aspect comes from the exponent \( t/T_{1/2} \). This indicates the number of half-lives that have already passed. Each whole half-life that elapses causes the remaining amount to be halved.
Here’s why exponential decay is interesting and widely applicable in nature:

• Natural processes often mimic exponential decay, including radioactive decay, cooling of objects, and population decline in environments.
• In this pattern, large amounts decay faster than smaller ones; the bigger the "hill," the steeper the slope downwards.

Understanding exponential decay can improve one's ability to make predictions about natural phenomena and resource management. The predictability makes it a choice model for analyzing decay in substances scientifically.

Carbon-14

Carbon-14, or \( \mathrm{C}^{14} \), is a radioactive isotope of carbon. It's particularly vital for radiocarbon dating, a technique commonly used to determine the age of ancient organic materials. Archaeologists and geologists rely on it to date artifacts and geological events.
This isotope is constantly produced through the interaction of nitrogen in the atmosphere with cosmic rays. Living organisms absorb \( \mathrm{C}^{14} \) during their lifetime. However, once they die, they stop absorbing it, and the \( \mathrm{C}^{14} \) begins to decay with a known half-life of 5730 years.
Here’s why \( \mathrm{C}^{14} \) is so significant:

• Its predictable decay allows scientists to calculate how long it has been since the organism's death, by measuring the remaining \( \mathrm{C}^{14} \) in samples.
• The technique requires only small sample sizes due to \( \mathrm{C}^{14} \)'s sensitive measurement capabilities.

This isotope’s decay not only helps the scientific community uncover historical timelines but also enriches our understanding of natural history, climate change, and archaeological sites. Despite being just one of many isotopes of carbon, \( \mathrm{C}^{14} \) plays a crucial role in scientific discovery and research.