Answer:
1844 years
Explanation:
¹⁴C follows a first-order beta-decay according to the following equation.
¹⁴C ⇒ ¹⁴N + β⁻
We can calculate the concentration of ¹⁴C after some time using the following expression.
[tex]ln(\frac{[C]_{t}}{[C]_{0}} )=-k.t[/tex]
where,
[C]t is the concentration of ¹⁴C after some time
[C]₀ is the original concentration of ¹⁴C
k is the rate constant
t is the time elapsed
We can calculate the rate constant if we know the half-life (t1/2) using the following expression.
[tex]k=\frac{ln2}{t_{1/2}}[/tex]
Half-life of ¹⁴C is 5730 years. Then,
[tex]k=\frac{ln2}{5730y}=1.210\times 10^{-4} y^{-1}[/tex]
The elapsed time when the concentration of ¹⁴C is 80% of original is:
[tex]ln(\frac{0.8[C]_{0}}{[C]_{0}} )=-1.210\times 10^{-4} y^{-1} \times t\\t = 1844y[/tex]
