Question

An ancient scroll allegedly originates from Greek scholars in about 500 B.C. A measure of its carbon -14 decay rate gives a value that is 89% of that found in living organisms. How old is the scroll and could it be authentic?

Answer 1

We know that

half life of carbon-14, t_1/2 = 5730 yrs

Radio active decay is a first order reaction.

For first order recation,

half life t_1/2 = 0.693 /k where k is rate constant

k = 0.693/ t_{1/2 --- Eq (1)}

k = 1/t ln { [A]_o/[A]_t} -----Eq (2)

From Eqs (1) and (2),

0.693/ t_1/2 = (1/t) ln {[A]_o/ [A]_t} ------Eq (3)

t_1/2 = 5730 yrs

t = age of the object

Given that  [A]_t = 89 % of the carbon-14

  [A]_o = 100 % of the carbon-14

Substitute all the values in Eq (3),

0.693/ t_1/2 = (1/t) ln {[A]_o/ [A]_t}

0.693/ 5730 yrs = (1/t) In (100/89)

t = In (100/89) x 5730 yrs / 0.693

= 963.5 yrs

t = 963.5 yrs

Therefore, age of scroll = 963.5 yrs