Question

In 2008 , crime scene investigators discover the bones of a person who appeared to have been the victim of a brutal attack that occurred a long time ago. They would like to know the year when the person was murdered. Using carbon dating, they determine that the rate of change of the \({ }_{6}^{14} \mathrm{C}\) is 0.268 Bq per gram of carbon. The rate of change of \({ }_{6}^{14} \mathrm{C}\) in the bones of a person who had just died is 0.270 Bq per gram of carbon. What year was the victim killed? The half-life of \({ }_{6}^{14} \mathrm{C}\) is \(5.73 \cdot 10^{3} \mathrm{yr}\).

Short answer

Answer: The murder likely occurred in the year 1712.

Step-by-step solution

Determine the decay constant

First, we need to determine the decay constant, k. We know that the half-life of \({}_{6}^{14}\mathrm{C}\) is \(5.73 \cdot 10^{3} \mathrm{yr}\). Since radioactive decay follows an exponential function, we can use the half-life formula: \(t_{1/2} = \frac{\ln{2}}{k}\).

Solve for the decay constant

Now, let's put the information given into the formula and solve for k: \(t_{1/2} = \frac{\ln{2}}{k} \Rightarrow 5.73 \cdot 10^{3} \mathrm{yr} = \frac{\ln{2}}{k}\) \(k = \frac{\ln{2}}{5.73 \cdot 10^{3}}\)

Write radioactive decay formula

We can now write the radioactive decay formula for the carbon-14: \(A_t = A_0 e^{-kt}\) Where: \(A_t\) is the rate of change of \({}_{6}^{14}\mathrm{C}\) at time t (in this case, 0.268 Bq/gram) \(A_0\) is the initial rate of change of \({}_{6}^{14}\mathrm{C}\) (in this case, 0.270 Bq/gram) \(t\) is the time passed since the murder (in years)

Solve for time (t) using decay formula

Insert the given values for \(A_t\), \(A_0\), and \(k\) into the decay formula and solve for t: \(0.268 = 0.270e^{-\frac{\ln{2}}{5.73 \cdot 10^{3}}t}\) \(\frac{0.268}{0.270} = e^{-\frac{\ln{2}}{5.73 \cdot 10^{3}}t}\) Now, take the natural logarithm of both sides: \(\ln{\frac{0.268}{0.270}} = -\frac{\ln{2}}{5.73 \cdot 10^{3}}t\) Now, solve for t: \(t = \frac{\ln{\frac{0.270}{0.268}}}{\frac{\ln{2}}{5.73 \cdot 10^{3}}}\) \(t \approx 295.65\) years

Determine the year of the murder

Since the crime scene was discovered in 2008 and we found that approximately 295.65 years have passed since the murder, we can find the year of the murder by subtracting this time from 2008: Year of the murder = \(2008 - 295.65 \approx 1712.35\) Since we cannot have a fraction of a year, round it down to the nearest whole number: Year of the murder ≈ 1712

Key concepts

Understanding Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. In the context of Carbon-14 ({}_{6}^{14}C{}), which is used in radiocarbon dating, the nucleus decays over time into a stable form. This decay happens at a predictable rate specific to each radioactive isotope, called a half-life. Understanding this concept allows scientists to date organic materials by measuring the amount of Carbon-14 remaining in a sample compared to a reference standard.

Considering the above exercise, knowing the rate of decay allows investigators to determine how much time has elapsed since the death of an organism. Thus, radioactive decay is foundational in Carbon-14 dating and helps unravel the history preserved in our environment.

Exponential Function in Decay Processes

An exponential function is a mathematical representation of the consistent proportional change over time. In the case of radioactive decay, this is expressed as A_{t} = A_{0} e^{-kt} where A_{t} is the amount of substance at time (t), A_{0} is the initial amount, (e) is the base of the natural logarithm, (k) is the decay constant, and (t) is time. This formula showcases how the amount of a radioactive isotope decreases exponentially over time.

The exponential nature of decay means that there is a rapid decrease initially, which slows down as time progresses. In the exercise, this function is essential for solving the mystery of the victim's year of death, giving us a clear path from the measured decay rates back to the time of the murder.

Half-life: The Timer of Radioactive Elements

Half-life is defined as the time required for half of the radioactive atoms in a sample to decay. It's a constant for each radioactive element and pivotal for dating techniques. A half-life is specific to the particular isotope and unaffected by external conditions like temperature and pressure. For Carbon-14 ({}_{6}^{14}C{}), its half-life is about 5,730 years.

By knowing the half-life, scientists can calculate the decay constant ((k)) using the formula ln(2)/k = t_{1/2}, which is vital in the exponential decay equation. In our crime scene investigation example, understanding the half-life concept and its relation to the decay constant enables forensic scientists to pinpoint the age of the bones found and, consequently, the time of the murder.

The Role of Natural Logarithm in Dating

In physics and mathematics, the natural logarithm (ln) is the logarithm to the base (e), where (e) is an irrational and transcendental constant approximately equal to 2.718281828459. The natural logarithm transforms multiplicative processes into additive ones, which is extremely useful when dealing with exponential growth or decay like in radioactive materials.

For the radioactive decay equation, taking the natural logarithm of both sides allows for solving the variable of time ((t)) more straightforwardly. This step is demonstrated in our example investigation, where the logarithm helped isolate (t) and thereby calculate the time elapsed since the victim's death. The natural logarithm's properties are therefore essential in extracting meaningful conclusions from the otherwise complex exponential decay data.