Problem 368 - Calculus Volume 1

Question

True or False? If true, prove it. If false, find the true answer. You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era \((146 \text { million }\) years to 65 million years ago), and you find by radiocarbon dating that there is 0.000001\(\%\) the amount of radiocarbon. Is this bone from the Cretaceous?

Step-by-Step Solution

Answer

False, the bone is too young to be from the Cretaceous period.

Step 1: Understand Radiocarbon Dating

Radiocarbon dating measures the amount of carbon-14 isotopes remaining in an object to estimate the time since the organism's death. As organisms die, carbon-14 starts to decay at a known rate, expressed by the half-life of approximately 5730 years.

Step 2: Calculate Time Elapsed Using Formula

The formula for radiocarbon dating is \[ N = N_0 \times (0.5)^{t/T} \]where \(N\) is the current amount of carbon-14, \(N_0\) is the initial amount, \(t\) is the time since death, and \(T\) is the half-life of carbon-14 (5730 years). Given \(N/N_0 = 0.000001\%\), we find the time \(t\).

Step 3: Convert Percentage to Decay Fraction

Given that the current amount \(N\) is 0.000001\% of \(N_0\), convert this percentage to a decimal: \[ \frac{0.000001}{100} = 0.00000001 \]This represents \(N/N_0\).

Step 4: Solve For Time Elapsed

Using the decay formula: \[ 0.00000001 = (0.5)^{t/5730} \]Take the natural logarithm of both sides to solve for \( t \). \[\ln(0.00000001) = \left(\frac{t}{5730}\right) \ln(0.5)\]Solving gives:\[ t = \frac{\ln(0.00000001)}{\ln(0.5)} \times 5730 \approx 191418 \text{ years} \]

Step 5: Conclusion Based on Era Comparison

191,418 years is far younger than the time range for the Cretaceous period, which ended 65 million years ago. Therefore, the bone cannot be from a Tyrannosaurus Rex.

Key Concepts

Decay Formula

The decay formula is a crucial part of radiocarbon dating. It helps to determine how long it has been since an organism died. When an organism dies, the carbon-14 in its body begins to decay at a set rate. The core formula used is \[N = N_0 \times (0.5)^{t/T}\]This equation calculates the remaining amount of carbon-14, denoted by \(N\), from the original amount \(N_0\). The \(t\) represents the time elapsed since the organism's death, and \(T\) is the half-life of carbon-14.

Understanding the Equation

- \(N\) is the detectable carbon-14 left in the sample.- \(N_0\) is the presumed initial carbon-14 present.- \(t\) is the time since the organism died.- \(T\), the half-life, is about 5730 years for carbon-14.So essentially, the formula expresses how much of a sample has decayed based on the half-life. To find \(t\), use the ratio \(N/N_0\) and solve the equation by taking the natural logarithm, making it possible to determine precisely how many years have passed since death.

Carbon-14

Carbon-14 is a radioactive isotope that is pivotal in dating organic materials. Unlike other carbon atoms, carbon-14 comprises 6 protons and 8 neutrons, making it unstable. When living organisms, including plants and animals, are alive, they continuously intake carbon-14 from the atmosphere and foods.

Role in Radiocarbon Dating

- Living beings maintain a constant ratio of carbon-12 and carbon-14. - At death, carbon-14 intake stops, and its decay begins. - Over time, the amount of carbon-14 diminishes due to radioactive decay. The concept behind its use in radiocarbon dating is that by measuring the remaining carbon-14 in a sample, scientists can calculate when the organism died. The changes in carbon-14 levels reveal the time lapse since death.

Half-life

Half-life is central to understanding radiocarbon dating. It measures the time required for half of the radioactive atoms in a sample to decay. For carbon-14, this is approximately 5730 years.

Understanding Half-life

- Half-life signifies a rate of decay. - The principle applies to any radioactive material, not just carbon-14. - After one half-life, 50% of the initial radioactive atoms remain. - After two half-lives, 25% remain, and so forth. For radiocarbon dating, knowing the half-life of carbon-14 enables calculations regarding the age of ancient artifacts. By understanding how much carbon-14 remains, and knowing its half-life, scientists track back to estimate when the organism died relative to the present time.