Question

A wooden tool is found at an archaeological site. Estimate the age of the tool using the following information: A 100 gram sample of the wood emits 1120 β-particles per minute from the decay of carbon-14. The decay rate of carbon-14 in living trees is 15.3 per minute per gram. Carbon-14 has a half-life of 5730 years.

Answer 1

Answer:

2578.99 years

Explanation:

Given that:

100 g of the wood is emitting 1120 β-particles per minute

Also,

1 g of the wood is emitting 11.20 β-particles per minute

Given, Decay rate = 15.3 % per minute per gram

So,

Concentration left can be calculated as:-

C left = $[A_t]=\frac{11.20\ per\ minute}{15.3\ per\ minute\ per\ gram}\times [A_0]= 0.7320[A_0]$

Where,  

$[A_t]$ is the concentration at time t

$[A_0]$ is the initial concentration

Also, Half life of carbon-14 = 5730 years

$t_{1/2}=\frac {ln\ 2}{k}$

Where, k is rate constant

So,  

$k=\frac {ln\ 2}{t_{1/2}}$

$k=\frac {ln\ 2}{5730}\ years^{-1}$

The rate constant, k = 0.000120968 year⁻¹

Time =?

Using integrated rate law for first order kinetics as:

$[A_t]=[A_0]e^{-kt}$

So,  

$\frac {[A_t]}{[A_0]}=e^{-0.000120968\times t}$

$\frac {0.7320[A_0]}{[A_0]}=e^{-0.000120968\times t}$

$0.7320=e^{-0.000120968\times t}$

$ln\ 0.7320=-0.000120968\times t$

t = 2578.99 years