Problem 11 - Essential Calculus Early Transcendentals

Question

Scientists can determine the age of ancient objects by the method of radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, \(^{14} \mathrm{C},\) with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates \(^{14} \mathrm{C}\) through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14 \(\mathrm{C}\) begins to decrease through radioactive decay. Therefore the level of radioactivity must also decay exponentially. A parchment fragment was discovered that had about 74\(\%\) as much \(^{14}\) C radioactivity as does plant material on the earth today. Estimate the age of the parchment.

Step-by-Step Solution

Answer

The parchment is approximately 2606 years old.

Step 1: Understand the Exponential Decay Formula

The amount of a radioactive substance decreases over time following the formula \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the quantity remaining at time \( t \), \( N_0 \) is the initial quantity, and \( k \) is the decay constant.

Step 2: Determine the Decay Constant

The decay constant \( k \) can be calculated using the half-life of the isotope. The formula for \( k \) is \( k = \frac{\ln(2)}{T_{1/2}} \). For carbon-14, the half-life \( T_{1/2} \) is 5730 years, so \( k = \frac{\ln(2)}{5730} \approx 1.2097 \times 10^{-4} \text{ years}^{-1} \).

Step 3: Set Up the Equation for the Parchment's Remaining 14C

Since the parchment has 74\(\%\) of the original radioactivity, we can express it as a fraction: \( N(t) = 0.74 N_0 \). Substitute into the decay formula: \( 0.74 N_0 = N_0 e^{-kt} \).

Step 4: Solve for Time \( t \)

Cancel \( N_0 \) from both sides: \( 0.74 = e^{-kt} \). Take the natural logarithm of both sides: \( \ln(0.74) = -kt \). Substitute the decay constant \( k \) calculated earlier: \( \ln(0.74) = -1.2097 \times 10^{-4} t \).

Step 5: Calculate \( t \)

Solve the equation: \( t = \frac{\ln(0.74)}{-1.2097 \times 10^{-4}} \approx 2606 \) years.

Key Concepts

Exponential Decay

Exponential decay is a process where the quantity of a substance decreases at a rate proportional to its current amount. This concept is commonly seen in radioactive substances and is expressed through the exponential decay formula:

• \( N(t) = N_0 e^{-kt} \)

Here, \( N(t) \) is the quantity at time \( t \), \( N_0 \) is the initial quantity, and \( k \) is the decay constant. This formula indicates that as time progresses, the amount of the radioactive substance diminishes exponentially rather than linearly.
This kind of decay is characterized by a rapid depletion at the beginning, which slows over time. This feature is vital for understanding how substances like carbon-14 break down, providing key insights into dating ancient artifacts.

Half-Life of Isotopes

The half-life of an isotope is the time required for half the quantity of a radioactive substance to break down and decay. It is a significant concept for determining how long it will take for radioactive material to diminish to half its original amount.

• For carbon-14, the half-life is approximately 5730 years.

Knowing the half-life is crucial for calculating the decay constant \( k \), using the formula:

• \( k = \frac{\ln(2)}{T_{1/2}} \)

Understanding the half-life allows scientists to predict the age of artifacts by observing the remaining amount of the isotope after a certain period has elapsed.

Carbon-14 Decay

Carbon-14, a radioactive isotope of carbon, plays a central role in radiocarbon dating. It is formed in the atmosphere and absorbed by living organisms during their lifecycles. Once an organism dies, it stops absorbing carbon-14, and the existing carbon-14 begins to decay. The decay law for carbon-14 is observed by exponential decay, where the proportion of carbon-14 reduces by half over its half-life. This predictable decay pattern allows scientists to measure the remaining carbon-14 in an object and compare it to the current level of carbon-14 in the environment to ascertain when the organism perished. This process helps in studying historical artifacts and understanding the timeline of historical events through the lens of chemistry and physics.

Age Estimation

Age estimation using radiocarbon dating involves measuring the percentage of carbon-14 remaining in an object. This measurement is then used to trace back to when the organism was alive. Let's break down the steps:

• Determine the percentage of carbon-14 remaining in the sample. For instance, if a sample has 74% of its original carbon-14, it indicates some time has elapsed since death.
• Apply the decay formula: \( 0.74 = e^{-kt} \), where \( k \) is the decay constant.
• Calculate \( t \) to find out how long it has been since the organism's death using the natural logarithm: \( t = \frac{\ln(0.74)}{-k} \).

By performing these calculations, scientists estimate the age of the object, unlocking a tangible link to the past. This allows us to paint a vivid picture of times long gone, based on the decay of a simple element.