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Carbon - 14

If the equilibrium concentration of carbon 14 in living plants gives 16 disintegrations per minute per gramme of carbon, estimate the age of a piece of timber if 2.0g of carbon prepared from it gives 15 disintegrations per minute. (The half-life of carbon 14 is 5.57 x 103 years.)

 

14g carbon 14 contains NA atoms

therefore a 2g sample contains (6.02 x 1023 ) / 7 atoms

N = (6.02 x 1023 ) / 7

half-life = 5.57 x 103 years = 5.57 x 103 x 365.25 x 24 = 4.88 x 107 hours = 2.93 x 109 minutes

l = ln2/half-life = 0.693 / 2.93 x 109 = 2.37 x 10-10 minutes -1

dN/dt = -lN

Age is zero when dN/dt = 16 min-1 g-1

Age is t when dN/dt = 15 min-1 for a 2g sample = 7.5 min-1 g-1

the ratio of dN/dt to N is constant therefore 16/N0 = 7.5/Nt

and Nt /N0 = 7.5/16

So,

e-lt = 7.5/16

Taking logs

-lt = ln(7.5/16)

t = -ln(7.5/16)/l

t = 0.758/(2.37 x 10-10)

=3.2 x 109 minutes

= 6080 years

= 6100 years (2 sig figs)