Question

Carbon-14 dating can be used to estimate the age of formerly living materials because the uptake of carbon-14 from carbon dioxide in the atmosphere stops once the organism dies. If tissue samples from a mummy contain about \(81.0 \%\) of the carbon-14 expected in living tissue, how old is the mummy? The half- life for decay of carbon-14 is 5730 years.

Short answer

The mummy is approximately 1676 years old.

Step-by-step solution

Understand the concept of half-life

Half-life is the amount of time required for half of a quantity of a radioactive isotope to decay. Given that Carbon-14 has a half-life of 5730 years, after 5730 years, only half of the original Carbon-14 would remain in the sample.

Calculate the number of half-lives elapsed

Since the tissue contains about 81.0% of the original Carbon-14, we can use the formula for exponential decay, which is \( N(t) = N_0 (1/2)^{t/T} \) where \( N(t) \) is the remaining quantity of the substance, \( N_0 \) is the initial quantity, \( T \) is the half-life, and \( t \) is the time elapsed. Set \( N(t)/N_0 = 0.81 \) and solve for \( t \) using \( T = 5730 \) years.

Solving for the age of the mummy

The equation \( 0.81 = (1/2)^{t/5730} \) can be solved for \( t \) by taking the logarithm of both sides. Use the property that \( \log(a^b) = b \cdot \log(a) \) to isolate \( t \) on one side of the equation. Then, calculate \( t \) using the properties of logarithms.

Calculate the natural logarithm of both sides

Take the natural logarithm of both sides to get \( \ln(0.81) = (t/5730) \cdot \ln(1/2) \) and then solve for \( t \).

Find the value of \( t \) representing the age of the mummy

Rearrange the equation from Step 4 to solve for \( t \) as \( t = \frac{\ln(0.81)}{\ln(1/2)} \cdot 5730 \) years. Calculate this value to find the age of the mummy.

Key concepts

Half-life

The term 'half-life' is fundamental to understanding the puzzle of how old our mummified friend is. It describes the unique stopwatch that ticks for radioactive elements like Carbon-14 (C-14). Picture a room full of couples dancing, and every 5730 years, half of these couples decide to leave the party—it's a slow event, but eventually, the room empties out. In science terms, a half-life is the time it takes for half of a given amount of a radioactive isotope to transform into something else. Here's where our numerical dance starts: after one half-life (5730 years), only 50% of the original C-14 would remain in our mummy. But our mummy still has 81% of its C-14, meaning less than a half-life has passed since our mummy joined the afterlife gala.

Radioactive Isotope Decay

Let's delve into what happens during radioactive isotope decay. When we say an isotope, like Carbon-14, is radioactive, we're talking about its instability and eagerness to break down. Each C-14 atom in our mummy yearns to shed its radioactive personality and become Nitrogen-14, a stable, non-radioactive isotope. This transformation is a random event for each atom, but when we look at many atoms together, they show a predictable pattern. This pattern, or rate, of decay allows us to use C-14 like a molecular stopwatch. As the decades roll on, C-14 lessens in its post-mortem disco, decreasing exponentially, which leads us to the next big concept—how to mathematically describe this decay.

Exponential Decay Formula

The formula that narrates the tale of our mummy's age is the exponential decay formula. It is the pattern we use to understand how materials like C-14 vanish over time. It's mathematically represented as \( N(t) = N_0 (1/2)^{t/T} \), where \( N(t) \) is the party-going C-14 that's still dancing at time \( t \), \( N_0 \) is the original number of dancers (C-14 atoms) our mummy started with, \( T \) is the time it takes for half the dancers to leave (the half-life), and \( t \) is the total time elapsed—basically, how long the mummy has been enjoying its eternal slumber. Since we know the remaining percentage of the original C-14, we just have to crank the numbers through this formula to get our answer.

Logarithmic Calculations

Finding the mummy's age is like unlocking a cryptic code; this is where logarithmic calculations make a grand entrance. They help us unravel the exponential formula we just mentioned. By taking the logarithm of each side of our decay formula, we can solve for \( t \), the elapsed time since our mummy ceased swapping carbon dance partners. The equation \( \text{ln}(0.81) = (t/5730) \times \text{ln}(1/2) \) is a stroke of mathematical magic that allows us to isolate \( t \). This steps away from the realm of percentages and deep into the world of decay times. Once we put these numbers through the logarithmic mill, we uncover the mummy's age, unwrapping the mystery that we began with, and giving our mummy its chronological place in history.