What is the age in years of a bone in which the 14C/12C ratio is measured to be 4.45×10-132 Express your answer as a number of years.
The Correct Answer and Explanation is:
To determine the age of the bone, we use the concept of radiocarbon dating, which relies on the decay of carbon-14 ((^{14}C)) to carbon-12 ((^{12}C)) over time. The ratio of (^{14}C/^{12}C) in a sample decreases over time as carbon-14 decays, and by measuring this ratio, we can calculate the age of the sample.
The ratio of (^{14}C/^{12}C) in a living organism is roughly constant, but as the organism dies, the (^{14}C) begins to decay, while the (^{12}C) remains constant. The half-life of carbon-14 is about 5730 years, which means that after 5730 years, half of the original amount of (^{14}C) in the sample will have decayed.
The general equation for the decay of carbon-14 is given by:
[
N(t) = N_0 e^{-\lambda t}
]
where:
- (N(t)) is the amount of (^{14}C) remaining after time (t),
- (N_0) is the initial amount of (^{14}C),
- (\lambda) is the decay constant, and
- (t) is the time that has passed.
We can also express the decay of carbon-14 in terms of its half-life using the following relationship between (\lambda) and the half-life (t_{1/2}):
[
\lambda = \frac{\ln(2)}{t_{1/2}}
]
Substituting the half-life of carbon-14 ((5730 \, \text{years})) into this equation:
[
\lambda = \frac{\ln(2)}{5730} \approx 1.21 \times 10^{-4} \, \text{per year}
]
The ratio of the (^{14}C/^{12}C) in the sample is given as (4.45 \times 10^{-13}), which is a fraction of the original ratio. To find the age of the sample, we rearrange the decay formula:
[
\frac{N(t)}{N_0} = e^{-\lambda t}
]
Taking the natural logarithm of both sides:
[
\ln \left( \frac{N(t)}{N_0} \right) = -\lambda t
]
Substituting the values:
[
\ln \left( 4.45 \times 10^{-13} \right) = -1.21 \times 10^{-4} t
]
[
-28.92 = -1.21 \times 10^{-4} t
]
Solving for (t):
[
t = \frac{-28.92}{-1.21 \times 10^{-4}} \approx 2.39 \times 10^5 \, \text{years}
]
Thus, the age of the bone is approximately 239,000 years.
This calculation demonstrates the application of radiocarbon dating to determine the age of ancient organic materials based on the measurement of remaining carbon-14.