Problem 42 - Chemistry Structure and Dynamics

Question

Measurements on the linen wrappings from the Book of Isaiah in the Dead Sea Scrolls suggest that the scrolls contain about \(79.5 \%\) of the \({ }^{14} \mathrm{C}\) expected in living tissue. How old are these scrolls? \(\left({ }^{14} \mathrm{C}: t_{1 / 2}=5730 \mathrm{yr}\right)\)

Step-by-Step Solution

Answer

The scrolls are approximately 1920 years old.

Step 1: Understand the Problem

We need to determine the age of the scrolls given that they have 79.5% of the original \(^{14}C\) remaining and knowing the half-life of \(^{14}C\) is 5730 years.

Step 2: Use the Exponential Decay Formula

The formula for exponential decay is \( N(t) = N_0 e^{-\lambda t} \), where \( N(t) \) is the amount remaining at time \( t \), \( N_0 \) is the initial amount, and \( \lambda \) is the decay constant. To relate the decay constant to half-life, we use \( \lambda = \frac{\ln(2)}{t_{1/2}} \).

Step 3: Calculate the Decay Constant

Using the half-life of \(5730\) years, compute the decay constant: \( \lambda = \frac{\ln(2)}{5730} \approx 0.000121\; \text{per year}.\)

Step 4: Set Up the Decay Equation

Since the scrolls have 79.5% of \(^{14}C\) left, we set \( \frac{N(t)}{N_0} = 0.795 \). This gives us \( 0.795 = e^{-0.000121 t} \).

Step 5: Solve for Time, \( t \)

Take the natural logarithm of both sides to solve for \( t \): \[ \ln(0.795) = -0.000121 t \] \[ t = \frac{\ln(0.795)}{-0.000121} \approx 1920 \text{ years}. \]

Step 6: Conclusion

The calculation estimates the scrolls to be approximately 1920 years old.

Key Concepts

Exponential Decay

Radiocarbon dating is based on the principle of exponential decay, a process where the quantity of a radioactive isotope decreases over time in a predictable way. Imagine a bucket with a small, constant leak. Just like water slowly leaving the bucket, radioactive isotopes like carbon-14 ({}^{14}C) decrease at a consistent rate.
In mathematical terms, this process is described by the exponential decay formula: \( N(t) = N_0 e^{-\lambda t} \), where:

• \( N(t) \) is the amount of isotope remaining after time \( t \).
• \( N_0 \) is the initial amount of the isotope.
• \( \lambda \) is the radioactive decay constant.

Exponential decay is characterized by its smooth curve that never quite touches zero, meaning complete decay takes an infinite amount of time.
This ensures that even very small amounts of {}^{14}C in ancient objects can be measured, allowing us to estimate their age reliably.

Half-Life of Carbon-14

The half-life of a radioactive substance is the time it takes for half of the initial quantity to decay. For {}^{14} C, this period is 5730 years.
What's fascinating about half-life is that it remains constant, irrespective of the amount of substance you start with. So, whether you have a gram or a milligram, after 5730 years, half of it will still remain.
Understanding half-life helps in radiocarbon dating because it provides a time-scale for the exponential decay function. If you measure an object to only have a certain percentage of its original {}^{14} C, you can use the half-life to calculate how many years have passed.
This is especially useful in archaeology for dating artifacts, like the linen wrappings of the Dead Sea Scrolls, giving us a glimpse into history by determining their age.

Radioactive Decay Constant

The radioactive decay constant, \( \lambda \), is a key part of understanding how quickly a radioactive substance decays. It's directly linked to the half-life of the isotope, using the formula: \( \lambda = \frac{\ln(2)}{t_{1/2}} \).
This formula highlights that the decay constant depends on the half-life. For {}^{14}C, with a half-life of 5730 years, \( \lambda \) is approximately 0.000121 per year.
So, every year, about 0.0121% of {}^{14}C in a sample will decay. The smaller the value of \( \lambda \), the slower the decay process.
Knowing \( \lambda \) allows scientists to calculate the age of carbon-containing materials by plugging this value into the exponential decay formula, and comparing the current {}^{14}C level to its original amount in living organisms. This is the underlying science that enabled us to estimate that the Book of Isaiah is approximately 1920 years old.