Problem 12 - Calculus with Concepts in Calculus

Question

Richard Leakey's "1470" skull, found in Kenya, is reputed to be \(1,800,000\) years old. Show that the percentage of \(\mathrm{C}^{14}\) remaining now would be negligible, and hence that in this instance dating by means of \(\mathrm{C}^{14}\) would be meaningless.

Step-by-Step Solution

Answer

Carbon-14 dating is ineffective for a 1.8 million-year-old sample as the remaining C-14 amount is negligible.

Step 1: Understanding Carbon-14 Decay

Carbon-14 has a known half-life of approximately 5730 years. This means that every 5730 years, half of the Carbon-14 in a sample decays.

Step 2: Calculate Number of Half-Lives

For a sample that is 1,800,000 years old, we calculate the number of half-lives by dividing 1,800,000 by 5730. This gives us approximately 314 half-lives.

Step 3: Calculate Remaining Carbon-14

The amount of Carbon-14 remaining after a given number of half-lives can be calculated as \( \left(\frac{1}{2}\right)^{n} \), where \( n \) is the number of half-lives. Substituting \( n = 314 \), the remaining Carbon-14 would be \( \left(\frac{1}{2}\right)^{314} \).

Step 4: Evaluate Percentage of Remaining Carbon-14

\( \left(\frac{1}{2}\right)^{314} \) results in a value extremely close to zero because Carbon-14 decays exponentially. Thus, the percentage of Carbon-14 remaining is negligible.

Step 5: Concluding on Carbon-14 Dating Validity

With a negligible amount of Carbon-14 remaining, dating a 1,800,000-year-old sample using this method is meaningless. Carbon-14 dating is effective for much younger fossils, typically up to 50,000 years old.

Key Concepts

Radioactive Decay

Radioactive decay is a fundamental concept in understanding how elements transform over time. It involves the process where unstable atoms lose energy by emitting radiation. For Carbon-14, a radioactive isotope of carbon, this decay process means it slowly turns into Nitrogen-14, its more stable form. This happens at a fixed rate, known as its decay rate. During this transformation, the amount of the radioactive material decreases exponentially over time. Distinct from chemical reactions, radioactive decay occurs at a predictable and constant rate, with each decay event reducing the number of unstable nuclei.

• Every element has a unique decay constant that determines its rate of decay.
• In radioactive decay, the rate is measured by observing the time it takes for half of the radioactive substance to change into another element.
• This measurable rate is what makes radiometric dating like Carbon-14 dating possible.

Understanding radioactive decay helps us track how long a substance has been around, a concept crucial for dating ancient objects and determining their age.

Half-Life

Half-life is a crucial measure in the study of radioactive decay. It's the time required for half of the radioactive atoms in a sample to decay. For Carbon-14, this period is approximately 5730 years. If you begin with 100% of Carbon-14, after one half-life (5730 years), you will have about 50% of the original Carbon-14 left. The concept of half-life allows us to mathematically determine how much of a substance remains after any period and is fundamental to carbon dating.

• Half-life remains constant regardless of the amount of substance.
• This constant allows scientists to reliably measure and predict the decay of radioactive elements.
• The knowledge of half-lives helps in calculating the ages of fossils and archeological specimens.

In the context of very old samples like "1470" skull found, we compute multiple half-lives to understand the exponential decay in Carbon-14 over ancient timelines.

Exponential Decay

Exponential decay is a mathematical concept describing the rapid decrease of a quantity over time. It applies when the decrease is directly proportional to the current amount—in the case of Carbon-14, it describes how rapidly its amount diminishes over successive half-lives. The formula for exponential decay is expressed as:\[N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\]where:

• \(N(t)\) is the quantity of the substance at time \(t\).
• \(N_0\) is the initial quantity.
• \(T_{1/2}\) is the half-life.

For the "1470" skull, it's mentioned that over 314 half-lives have passed, implying an almost complete decay of Carbon-14. This is because each half-life exponentially decreases the quantity of Carbon-14, making it nearly zero after many cycles. Therefore, the method of Carbon-14 dating becomes ineffective for such ancient specimens, as the remaining quantity is too minute to accurately measure or result in meaningful dating outcomes. Understanding exponential decay is central to explaining why Carbon-14 dating is best for items less than 50,000 years old.