Radiocarbon dating is a method used to determine the age of an ancient artifact or fossil by measuring the amount of carbon-14 it contains. Over time, carbon-14 decays at a predictable rate, helping scientists estimate the age of the sample.
To estimate the age of an old fossil using radiocarbon dating, scientists measure the remaining concentration of carbon-14 and compare it to the expected initial concentration. From this, they can calculate how many half-lives have passed and thus the age of the fossil.
### Numerical Problem
A certain radioactive substance has a half-life of 10 hours. If its initial number of atoms is [tex]\(6 \times 10^6\)[/tex], calculate the decay constant and the number of atoms remaining after 30 hours.
#### Step-by-Step Solution:
1. Determine the Half-Life and Initial Number of Atoms:
- Half-life [tex]\(T_{1/2}\)[/tex] = 10 hours
- Initial number of atoms [tex]\(N_0\)[/tex] = [tex]\(6 \times 10^6\)[/tex] atoms
2. Calculate the Decay Constant [tex]\( \lambda \)[/tex]:
- The decay constant [tex]\( \lambda \)[/tex] is related to the half-life by the formula:
[tex]\[
\lambda = \frac{0.693}{T_{1/2}}
\][/tex]
- Substituting the given half-life:
[tex]\[
\lambda = \frac{0.693}{10} = 0.0693 \, \text{hours}^{-1}
\][/tex]
3. Determine the Time Elapsed [tex]\(t\)[/tex]:
- Time elapsed [tex]\(t\)[/tex] = 30 hours
4. Calculate the Remaining Number of Atoms [tex]\(N(t)\)[/tex]:
- The formula to calculate the remaining number of atoms after time [tex]\(t\)[/tex] is:
[tex]\[
N(t) = N_0 \times e^{-\lambda t}
\][/tex]
- Plugging in the values:
[tex]\[
N(t) = 6 \times 10^6 \times e^{-0.0693 \times 30}
\][/tex]
- Evaluating the exponent:
[tex]\[
-0.0693 \times 30 = -2.079
\][/tex]
- Using this in the exponent:
[tex]\[
N(t) = 6 \times 10^6 \times e^{-2.079}
\][/tex]
- Calculating the exponential term:
[tex]\[
e^{-2.079} \approx 0.12506
\][/tex]
- Multiplying to find the remaining number of atoms:
[tex]\[
N(t) = 6 \times 10^6 \times 0.12506 \approx 750331.229 \, \text{atoms}
\][/tex]
##### Final Answer:
- The decay constant [tex]\( \lambda \)[/tex] is [tex]\(0.0693 \, \text{hours}^{-1}\)[/tex].
- The remaining number of atoms after 30 hours is approximately [tex]\(750331.229\)[/tex] atoms.
Therefore, the calculations show us that after 30 hours, only about 750,331 atoms out of the initial 6 million will remain.
