Question

Art historians can measure the date of a painting by measuring the ratio of 14C to 12C atoms in the fibers of the canvas, comparing with the accepted modern value. If the ratio is measured to be 1.2×10-12 , how old is the canvas?

Please explan the equations you use, thank you

Answer 1

The time elapsed before half the active nuclei decay is called half life(t_1/2) .let us consider there is mass of M₀ at t=0.so after t_1/2 it will be M₀/2. so from the equation of decay we may write that

M₀/2=M₀ e^(-kt_1/2) (k is called decay constant)

or, e^(kt_1/2) =2

or, kt_1/2 =ln 2

or, t_1/2=ln2/k   

as the activity () is proportional to M the activity also reduces to half its value in one half life.

so we have M=M₀ e^{-(ln2) t/t1/2} = [puttng the value of k and e^ln2 =2 from law of log]

so, M=M₀(1/2)t/t1/2    it is called half life equation

in our problem the ratio(M/M₀) is given by 1.2*10-12 so

M/M₀ = (1/2)t/t1/2

or, ln(M/M₀) =ln (1/2)t/t1/2 [taking ln in both side]

or,ln (1.2*10-12) = t/t_1/2 ln (1/2) [law of log]

or,t/t_1/2  = (-27.44869) / (-0.69314) =39.6004

or,t =t_1/2 *39.6004 we know t_1/2 of carbon is5730 yr

so t=5730*39.6004 =226910 yr