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 0.712mmol 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 0.712mmol 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, and 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 0.712/1.000. This means that the amount of carbon-14 that has decayed is 1.000 - 0.712 = 0.288 mmol.
We can set up the equation as follows:
0.712 = 1.000 * (1/2)^(t/5731)
To solve for t, we can take the natural logarithm of both sides:
ln(0.712) = ln(1.000 * (1/2)^(t/5731))
This simplifies to:
ln(0.712) = (t/5731) * ln(1/2)
Solving for t gives:
t = 5731 * ln(0.712) / ln(1/2)
Calculating this gives:
t ≈ 1900 years
So, the rock is approximately 1900 years old.
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