Question

The half-life of carbon-14 is 5370 years. The carbon-14 levels in a fossil indicate that 6 half-lives have passed. How old is the fossil?
32,220 years
75,200 years
50,000 years
35,000 years

Answer 1

On half life is 5370 years; 6 half lives have passed. You just multiply,

5370*6 = 32,220 years

Answer 2

Answer: 35,000 years

Explanation: Radioactive process follow first order kinetics.

To calculate the rate constant, we use the formula:

$k=\frac{0.693}{t_{1/2}}$

$k=\frac{0.693}{5370years}}$

$k=1.2\times 10^{-4} years^{-1}$

Also : $a=\frac{a_o}{2^n}$

where,  

a = amount of reactant left after n-half lives

$a_o$ = Initial amount of the reactant

n = number of half lives = 6

Putting values in above equation, we get:

$2^6=\frac{a_0}{a}$

$64=\frac{a_0}{a}$

The rate expression for first order kinetics is given by:

$t=\frac{2.303}{k}\log\frac{a_0}{a}$

where,

k = rate constant

t = time taken for decay process

$a_0$  = initial amount of the reactant

a = amount left after decay process

Putting values in above equation, we get:

$t=\frac{2.303}{1.2\times 10^{-4}}\log{64}$

$t=35000years$