Carbon-14 has a half-life of 5,730 y. How much of a 144g sample of carbon-14 will remain after 1.719x10 ^4 y.
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To determine how much of a 144g sample of carbon-14 will remain after 1.719 x 10^4 years, you can use the formula for exponential decay:
\[N(t) = N_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{T}}\]
Where:
- \(N(t)\) is the remaining amount after time \(t\).
- \(N_0\) is the initial amount.
- \(t\) is the time that has passed.
- \(T\) is the half-life.
In this case, \(N_0\) is 144g, \(t\) is 1.719 x 10^4 years, and \(T\) is the half-life of carbon-14, which is 5,730 years.
Plug these values into the formula:
\[N(t) = 144g \cdot \left(\frac{1}{2}\right)^{\frac{1.719 \times 10^4\text{ years}}{5,730\text{ years}}}\]
Now, calculate:
\[N(t) = 144g \cdot \left(\frac{1}{2}\right)^{\frac{3}{2}}\]
\[N(t) = 144g \cdot \left(\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\right)\]
\[N(t) = 144g \cdot \frac{1}{8}\]
Now, multiply 144g by 1/8 to find the remaining amount:
\[N(t) = \frac{144g}{8} = 18g\]
So, after 1.719 x 10^4 years, only 18g of the 144g sample of carbon-14 will remain.
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