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

2578.99 years

Step-by-step 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]=(11.20\ per\ minute)/(15.3\ per\ minute\ per\ gram)* [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* t)`

`\frac {0.7320[A_0]}{[A_0]}=e^(-0.000120968* t)`

`0.7320=e^(-0.000120968* t)`

`ln\ 0.7320=-0.000120968* t`

t = 2578.99 years