Question

The age of any remains from a once-living organism can be determined by radiocarbon dating, a procedure that works by determining the concentration of radioactive \({ }^{14} \mathrm{C}\) in the remains. All living organisms contain an equilibrium concentration of radioactive \({ }^{14} \mathrm{C}\) that gives rise to an average of \(15.3\) nuclear decay events per minute per gram of carbon. At death, however, no additional \({ }^{14} \mathrm{C}\) is taken in, so the concentration slowly drops as radioactive decay occurs. What is the age of a bone fragment from an archaeological dig if the bone shows an average of \(2.3\) radioactive events per minute per gram of carbon? For \({ }^{14} \mathrm{C}, t_{1 / 2}=5715\) years.

Short answer

The bone fragment is approximately 15650 years old.

Step-by-step solution

Understanding the Problem

We need to determine the age of a bone fragment based on the count of radioactive decay events it displays, using the known half-life of carbon-14 ( $t_{1/2}=5715 ext{ years} $). We have the initial decay rate ( $R_0=15.3 ext{ events per minute per gram} $) and the current decay rate ( $R=2.3 ext{ events per minute per gram} $).

Understanding Radioactive Decay Equation

The radioactive decay can be described by the equation \(R=R_0 \times e^{-\lambda t}\), where \(egin{align*}\)\lambda\( is the decay constant,\)t\( is the time (age) we need to determine.\)egin{align*}lambda &= \frac{\text{ln}2}{t_{1/2}}\(Using the half-life $$t_{1/2}=5715\) the decay constant \(\lambda\) is found first.

Calculate the Decay Constant

Using the half-life of carbon-14, calculate \(\lambda$$\lambda = \frac{\ln(2)}{5715}\)Calculating gives:\lambda \approx 1.213 imes 10^{-4} ext{ year}^{-1}.

Rearrange the Decay Equation

Rearrange the equation \(R=R_0 \times e^{-\lambda t}\) to solve for \(t\):\(t = \frac{\ln(R_0/R)}{\lambda}\).

Substitute Given Values

Substitute the known values into the equation:$$t = \frac{\ln(15.3/2.3)}{\lambda}\(This continues to give:\)t = \frac{\ln(15.3/2.3)}{1.213 imes 10^{-4}}$.

Calculate the Age

Calculate\(egin{align*}\ln(15.3/2.3) & \approx \ln(6.652)t ≈ \frac{1.8978}{1.213 imes 10^{-4} }& \approx 15650.1\text{ years}.}\)

Conclusion

Thus, the bone fragment is approximately $15650$ years old.

Key concepts

carbon-14

Carbon-14, often abbreviated as C-14, is a radioactive isotope of carbon. It is crucial in the process of radiocarbon dating, which is widely used to determine the age of archaeological and geological samples. C-14 is formed in the atmosphere when cosmic rays interact with nitrogen atoms. Once formed, carbon-14 combines with oxygen to create carbon dioxide, which is absorbed by living organisms during respiration. This results in a constant level of C-14 in all living things throughout their lifetime.
However, once an organism dies, it stops absorbing carbon dioxide and hence C-14. The C-14 present in the organism begins to decay at a known rate, allowing scientists to use its current state to gauge the time elapsed since the organism's death. Unlike its more stable counterparts, such as carbon-12 or carbon-13, carbon-14 has a unique and predictable rate of decay, making it invaluable in dating organic remains.

• Formed in the atmosphere
• Constant presence in living organisms
• Decays after organism's death

radioactive decay

Radioactive decay is a process by which a radioactive isotope loses energy by emitting radiation. In the context of carbon-14, this decay transforms C-14 into nitrogen-14 over time. The process follows a predictable mathematical pattern described by an exponential decay function.
This decay process is what allows radiocarbon dating to be possible. By measuring the current rate of C-14 decay, and knowing its initial concentration, scientists can determine how long it has been since the organism stopped replenishing its carbon-14, which is after its death. This is because the rate at which these radioactive isotopes decay is constant and can be mathematically expressed.
The equation used in radiocarbon dating is as follows: \[ R = R_0 \times e^{-\lambda t} \]where:

• \( R \) is the remaining decay rate of C-14
• \( R_0 \) is the initial decay rate
• \( \lambda \) is the decay constant, which relates to its half-life
• \( t \) is the time, or age one wishes to calculate

half-life

The half-life of a radioactive isotope is the time taken for half of its atoms to decay. For carbon-14, this period is approximately 5715 years. This means that every 5715 years, half of the carbon-14 in a sample will have decayed into nitrogen-14.
This characteristic is pivotal in radiocarbon dating because it provides a measurable timeframe over which decay occurs. Knowing the half-life allows scientists to establish the decay constant, a key value used in the calculation of the age of a sample. The half-life is determined experimentally and is unique to each radioactive isotope.
The decay constant \( \lambda \) can be calculated using the half-life with the formula: \[ \lambda = \frac{\ln(2)}{t_{1/2}} \]This constant helps predict the amount of radioactive material that will remain after a certain time, enabling the dating of ancient organic remains.

archaeological dating

Archaeological dating is a field that aims to establish the age of artifacts, structures, and remains found at various dig sites. Radiocarbon dating, which utilizes carbon-14, is one of the most frequently used methods for dating organic material such as bones, wood, and charcoal.
This method requires measuring the remaining C-14 in a sample and comparing it to what would have been its original concentration when the organism was living. By applying the principles of radioactive decay and understanding the half-life of carbon-14, scientists can estimate an artifact's or remain's age with considerable accuracy.
Radiocarbon dating has provided insights into various ancient civilizations by dating their organic remains, helping archaeologists to reconstruct historical timelines and understand ancient ways of life.

• Establishes the age of organic remains
• Uses carbon-14 to determine the time of death
• Helps reconstruct timelines of ancient societies