Question

What is the age BP of a bone fragment that shows an average of 2.9 disintegrations/min per gram of carbon in 2005 ? 'The carbon in living organisms undergoes an average of \(15.3\) disintegrations/min per gram, and the half-life of \({ }^{14} \mathrm{C}\) is 5715 years.

Short answer

The bone fragment is approximately 13716 years old BP.

Step-by-step solution

Understanding the Problem

We need to determine the age Before Present (BP) of a bone fragment based on its disintegration rate of carbon isotopes. Given data include the disintegration rate of modern carbon (15.3 dpm/g) and the rate of the bone fragment (2.9 dpm/g). The half-life of Carbon-14 ( ^{14}C) is 5715 years.

Applying the Exponential Decay Formula

The age of the bone fragment can be calculated using the radioactive decay formula: \[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]where \(N(t)\) is the disintegration rate at time \(t\), \(N_0\) is the initial disintegration rate (15.3 dpm/g), and \(T_{1/2}\) is the half-life (5715 years).

Rearranging the Formula to Solve for Time

First, rearrange the decay equation to solve for \(t\): \[ t = T_{1/2} \times \frac{\log(\frac{N(t)}{N_0})}{\log(0.5)} \]

Substituting the Known Values

Plug in the known values to find the age:\[ t = 5715 \times \frac{\log(\frac{2.9}{15.3})}{\log(0.5)} \]

Performing the Calculations

Calculate the fraction:\( \frac{2.9}{15.3} \approx 0.1895 \)Then, calculate the logarithms:\( \log(0.1895) \approx -0.7223 \) and \( \log(0.5) \approx -0.3010 \).

Calculating the Age

Now, compute the time \( t \) using the values obtained:\[ t = 5715 \times \frac{-0.7223}{-0.3010} \approx 5715 \times 2.40 \approx 13716 \text{ years} \]

Conclusion

The bone fragment is approximately 13716 years old BP (Before Present, referring to 1950 as the base year).

Key concepts

Exponential Decay

In the context of radiocarbon dating, exponential decay describes the process by which the number of radioactive atoms decreases over time. Specifically, it follows a predictable pattern where a substance loses half of its quantity in a consistent time period, known as its half-life.
The formula used in these calculations is:

• \( N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \)

This means the current amount \(N(t)\) can be determined if we know the initial amount \(N_0\), the half-life \(T_{1/2}\), and the time elapsed \(t\).
This pattern allows scientists to make precise predictions about how much radioactive material would remain after a certain period, which is essential for dating ancient objects accurately.

Half-Life

The half-life is the time required for a quantity to reduce to half its initial value. It's a critical factor in calculating the age of archaeological samples using radiocarbon dating.
For Carbon-14, the isotope used in radiocarbon dating, the half-life is 5715 years.
This means that every 5715 years, half of the Carbon-14 present in a material will have decayed into Nitrogen-14.

• This property of Carbon-14 makes it particularly useful for dating materials that are thousands of years old.
• It helps researchers estimate the age of organic materials by measuring how much Carbon-14 remains.

Knowing the half-life allows us to use decay equations to find the age of objects efficiently.

Disintegration Rate

Disintegration rate refers to the number of radioactive decay events occurring per minute. In radiocarbon dating, it is often expressed in disintegrations per minute per gram of carbon (dpm/g).
This rate decreases over time as carbon isotopes decay.
Living organisms, which constantly replenish Carbon-14, have a relatively high disintegration rate of about 15.3 dpm/g.
However, once an organism dies, the intake of Carbon-14 stops, and its amount decreases, as reflected by the drop in disintegration rate.

• By comparing the current disintegration rate of a sample to that of living organisms, scientists can estimate the time since the death of the organism.
• This process allows for calculating the age of ancient biological materials.

Understanding the disintegration rate is vital for accurate radiocarbon dating.

Carbon-14 Isotope

Carbon-14 is a radioactive isotope of carbon used extensively in radiometric dating due to its radioactive properties. It has 6 protons and 8 neutrons, giving it an atomic mass of 14.
Carbon-14 is naturally occurring in the environment, formed high in the atmosphere when cosmic rays interact with nitrogen.
Living organisms continuously take in Carbon-14 through the food chain.

• Once an organism dies, it stops absorbing Carbon-14, and the isotope begins to decay at a known rate.
• This decay forms the basis of radiocarbon dating, as the amount of Carbon-14 left in a sample can tell us how long ago the organism died.

Because of its half-life, Carbon-14 is best suited for dating objects up to about 50,000 years old, beyond which its concentration becomes too low to measure accurately.