Question

Question #3f7bf

Answer 1

c) 22920 years

Explanation:

I am not 100% on this solution, so someone more knowledgeable than I should definitely feel free to jump in and correct if and where necessary.

Assumption : The question doesn't make sense if `(C-14)/(C-12) = 1/15` as the natural ratio is much, much smaller. That would mean that the ratio would have increased in the sample - impossible. Therefore I have assumed that the question meant that the ratio `(C-14)/(C-12)` in the sample has fallen to `1/15` of the current ratio.

Given the above assumption we have this:
`((C-14)/(C-12))_s = ((C-14)/(C-12))_c × 1/15`

Where subscript s denotes the sample's ratio and c the current ratio.

The question then is: how many half lives are required to reduce the ratio by that factor. Mathematically that can be expressed like this:

`(1/2)^x = 1/15` where x is the number of half lives.

`⇒ log((1/2)^x) = log(1/15)`

`⇒ xlog(1/2) = log(1/15)`

`⇒ x = log(1//15)/log(1//2) = 3.907` half lives

The question provides the half life of carbon-14, so we multiply the number of half lives by the time for each half life to determine the total length of time:

`t = 3.907 × 5730 = 22,386` years

This is not exactly equal to either of the four options (not ideal!), but it is closest to option (c) 22,920 years.

Answer 2

(c) 22,920 years

Explanation:

If the ratio of `(C-14)/(C-12)` has decreased to 1/15 of initial then the abundance of C-14 has decreased to 1/15 of its initial value. Hence `N/N_0 =1/15` where N is the current abundance and `N_0` is the initial abundance.

Then use this equation: `N=N_0 e^(-λ t)`

`⇒ N/N_0 = e^(-λ t)`

`⇒ ln( N/N_0) = -λ t`

`⇒ t = ln( N/N_0) / (-λ)`

`⇒ t = ln( N_0/N) / (λ)`

Remember that tthe decay constant, λ is given by:
`λ = ln (2)/ t_½`

Then the equation becomes:
`⇒ t = (ln( N_0/N) × t_½) / (ln (2)`

Substitute in the values:
`⇒ t = (ln( 15/1) × 5730) / (ln (2)) = 22,386` years

Option (c) is closest to this value, so that is the solution.