Apiece of fossilized wood has a carbon-14 radioactivity that is 1/4 that of new wood. the half-life of carbon-14 is 5730 years. how old is the cloth?

1.146 x 10⁴ year.

Explanation:

The decay of carbon-14 is a first order reaction.The rate constant of the reaction (k) in a first order reaction = ln (2)/half-life = 0.693/(5730 year) = 1.21 x 10⁻⁴ year⁻¹.The integration law of a first order reaction is:

kt = ln [A₀]/[A]

k is the rate constant = 1.21 x 10⁻⁴ year⁻¹.

t is the time = ??? years.

[A₀] is the initial percentage of carbon-14 = 100.0 %.

[A] is the remaining percentage of carbon-14 = 1/4[A₀] = 25.0 %.

∵ kt = ln [Ao]/[A]

∴ (1.21 x 10⁻⁴ year⁻¹)(t) = ln (100.0%)/[25.0 %]

(1.21 x 10⁻⁴ year⁻¹)(t) = 1.386.

∴ t = 1.386/ (1.21 x 10⁻⁴ year⁻¹) =  1.146 x 10⁴ year.

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