Finding the age of the rock given half life

Question

 Once the C-14 activity in a fossil has dropped below 99.8% of it’s original value, it can no longer be accurately measured. What is the age of the oldest fossil we could accurately date using carbon dating? The half-life of carbon-14 is 5730 years17. Once the C-14 activity in a fossil has dropped below 99.8% of it’s original value, it can no longer be accurately measured. What is the age of the oldest fossil we could accurately date using carbon dating? The half-life of carbon-14 is 5730 years

Answer 1

I agree with the previous answer, but if you need it stated with rate laws:

 

All radioactive decay is first order.

The half life is related to the rate constant for a first order reaction by the equation t_1/2 = ln2

                                                                                                                               k

Substituting in the half life for C¹⁴,   5730 yrs = ln 2              k = 1.2097 x 10^-4  yr^-1

                                                                      k

 

The first order rate equation is  ln [A_t] = -k t    If 99.8% has been lost, the remainder [A_t] is 0.002 and [A_t]/[A₀] =       

                                                [A₀]             0.002/1 = 0.002

 

   ln 0.002  = -1.2097 x 10^-4 t

 

     t  = ln 0.002           = 51373 yrs     This should probably be reported as ~50,000 years or ~51,000 yrs based on 

           -1.2097E-4                              the significant figure rules for logarithms.

Answer 2

At maximum age, the activity will have dropped to (100 - 99.8)% of its original value, which is 0.002 times its original value.  Given a half-life of 5730 years, the maximum time t in years can be found from

 

0.002  = 1 (1/2)^t/5730

 

log(0.002) = (t/5730) log(0.5)

 

t = 5730 log(0.002) / log(0.5) years ≅ 51,374 years