The half life for the radioactive decay of carbon-14 to nitrogen-14 is ×5.73103 years.Suppose nuclear chemical analysis shows that there is 0.590mmol of nitrogen-14 for every 1.000mmol of carbon-14 in a certain sample of rock. Calculate the age of the rock.Round your answer to 2 significant digits.
Question
The half life for the radioactive decay of carbon-14 to nitrogen-14 is ×5.73103 years.Suppose nuclear chemical analysis shows that there is 0.590mmol of nitrogen-14 for every 1.000mmol of carbon-14 in a certain sample of rock. Calculate the age of the rock.Round your answer to 2 significant digits.
Solution
To solve this problem, we need to use the formula for radioactive decay, which is:
N = N0 * (1/2)^(t/T)
where: N is the final amount of the substance N0 is the initial amount of the substance t is the time that has passed T is the half-life of the substance
In this case, we know that the ratio of nitrogen-14 to carbon-14 in the rock is 0.590/1.000. This means that the amount of carbon-14 has decreased to 1.000 - 0.590 = 0.410 mmol, while the amount of nitrogen-14 has increased to 0.590 mmol.
So, we can set up the equation as follows:
0.410 = 1.000 * (1/2)^(t/5730)
To solve for t, we can take the natural logarithm of both sides:
ln(0.410) = ln(1.000 * (1/2)^(t/5730))
This simplifies to:
ln(0.410) = (t/5730) * ln(1/2)
Solving for t gives:
t = 5730 * (ln(0.410) / ln(1/2))
Calculating this gives:
t ≈ 8267 years
So, the rock is approximately 8300 years old, rounded to two significant digits.
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