Question

A living plant contains approximately the same fraction of carbon-14 4 as in atmospheric carbon dioxide. Assuming that the observed rate of decay of carbon-14 4 from a living plant is 13.6 counts per minute per gram of carbon, how many counts per minute per gram of carbon will be measured from a \(15,000\) -year-old sample? Will radiocarbon dating work well for small samples of 10 \(\mathrm{mg}\) or less? (For \(^{14} \mathrm{C}, t_{1 / 2}=5730\) years.)

Short answer

In a 15,000-year-old sample, the activity measured will be approximately 7.58 counts per minute per gram of carbon. Radiocarbon dating may not work well for small samples of 10 mg or less, as the number of carbon-14 atoms will be much fewer and difficult to accurately measure.

Step-by-step solution

Calculate the decay constant

Using the half-life formula, we can determine the decay constant (λ) for carbon-14: \[t_{1/2} = \frac{ln(2)}{\lambda}\] Rearrange the equation to solve for λ: \[\lambda = \frac{ln(2)}{t_{1/2}}\] Now, plug in the given half-life, \(t_{1/2}=5730\) years, to get the decay constant: \[\lambda = \frac{ln(2)}{5730} = 1.21 \times 10^{-4} \ \text{year}^{-1}\]

Determine the remaining carbon-14 atoms

Using the decay equation, we can find the number of remaining carbon-14 atoms in the 15,000-year-old sample: \[N(t) = N_0 e^{-\lambda t}\] Where \(N(t)\) is the number of remaining carbon-14 atoms at time t, \(N_0\) is the initial number of carbon-14 atoms, and t is the time in years. We want to find \(N(t)\) when \(t = 15,000\) years: \[N(15,000) = N_0 e^{- (1.21 \times 10^{-4})(15,000)}\]

Calculate the original number of carbon-14 atoms

We know the initial activity of the carbon-14 in the sample, so we can calculate the initial number of carbon-14 atoms using the given activity of 13.6 counts per minute per gram of carbon. The activity (A) is related to the decay constant and the number of atoms by: \[A = \lambda N\] Therefore, we can find the initial number of carbon-14 atoms: \[N_0 = \frac{A}{\lambda} = \frac{13.6}{1.21 \times 10^{-4}} = 1.13 \times 10^5\ \text{atoms}\]

Determine the counts per minute per gram of carbon

Now we can plug in the initial number of carbon-14 atoms into our decay equation and find the remaining number of carbon-14 atoms at 15,000 years: \[N(15,000) = (1.13 \times 10^5) e^{- (1.21 \times 10^{-4})(15,000)} = 6.26 \times 10^4\ \text{atoms}\] We can now find the activity in counts per minute per gram of carbon for the 15,000-year-old sample using the activity formula: \[A(t) = \lambda N(t) = (1.21 \times 10^{-4})(6.26 \times 10^4) = 7.58 \ \text{counts per minute per gram of carbon}\]

Discuss the viability of radiocarbon dating for small samples

Since the activity of carbon-14 in the 15,000-year-old sample is 7.58 counts per minute per gram of carbon, the activity is significantly lower than the initial activity of 13.6 counts per minute per gram of carbon. For small samples of 10 mg or less, the number of carbon-14 atoms will be much fewer, making it difficult to accurately measure the activity and radiocarbon date the sample. Therefore, radiocarbon dating may not work well for small samples of 10 mg or less.

Key concepts

Carbon-14 Decay

Carbon-14 decay is a natural process where the radioactive isotope carbon-14 ( ^{14} C) transforms into nitrogen-14 ( ^{14} N) over time. This occurs through a type of radioactive decay known as beta decay, where a neutron in a carbon-14 atom's nucleus is converted into a proton, releasing an electron (beta particle) and an anti-neutrino. As a result, the carbon-14 atom becomes a nitrogen-14 atom.

This process of decay is essential for radiocarbon dating because it is predictable and occurs at a steady rate over centuries. When a living organism dies, it stops absorbing carbon-14, and the existing carbon-14 in its body starts decaying. Scientists can estimate the time of death by measuring the remaining amount of carbon-14 in a sample. This predictable decay rate allows us to date samples that are thousands of years old with reasonable accuracy.

Half-Life Calculation

The half-life of a radioactive isotope is the time required for half of the isotope in a sample to decay. It's a constant value and is a fundamental concept in radiocarbon dating. For carbon-14, the half-life is approximately 5730 years.

Calculating the half-life involves using the formula:\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]Where t_{1/2} is the half-life, \ln(2) is the natural logarithm of 2 (approximately 0.693), and \lambda is the decay constant. By rearranging this formula, we can derive the decay constant needed for other calculations in radiocarbon dating. The half-life provides a timeline for how much time has passed since the specimen died, thus enabling scientists to approximate its age. This calculation is crucial to confidently applying radiocarbon dating.

Decay Constant

The decay constant, represented by the symbol \lambda, is a unique value that quantifies the probability of decay of a single atom per unit time. In the case of carbon-14, it's essential when calculating how many atoms have decayed from a sample over a specific period.

The decay constant can be found using the half-life formula rearranged to:\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]For carbon-14, with a half-life of 5730 years, the decay constant is calculated as approximately 1.21 \times 10^{-4} per year. This means that for each year, roughly 0.0121% of the remaining carbon-14 atoms decay. Understanding the decay constant allows scientists to make precise predictions about the rate of decay over time, which is key when determining the age of archaeological and geological samples.