Find the age ttt of a sample, if the total mass of carbon in the sample is mcmcm_c, the activity of the sample is AAA, the current ratio of the mass of 14 6C 614C to the total mass of carbon in the atmosphere is rrr, and the decay constant of 14 6C 614C is λλlambda. Assume that, at any time, 14 6C 614C is a negligible fraction of the total mass of carbon and that the measured activity of the sample is purely due to 14 6C 614C. Also assume that the ratio of mass of 14 6C 614C to total carbon mass in the atmosphere (the source of the carbon in the sample) is the same at present and on the day when the number of 14 6C 614C atoms in the sample was set. Express your answer in terms of the mass mamam_a of a 14 6C 614C atom, mcmcm_c, AAA, rrr, and λλlambda. View Available Hint(s)

An artifact is found in a desert cave. The anthropologists who found this artifact would like to know its age. They find that the present activity of the artifact is 9.25 decays/s and that the mass of carbon in the artifact is 0.100 kg. To find the age of the artifact, they will need to use the following constants:

r=1.2

The activity of carbon 14 is

[tex]A=A_0e^{\lambda t}[/tex]

where,

[tex]A_0[/tex] is the initial activity of the compound

Solve for t

[tex]-\lambda t=In\frac{A}{A_0}[/tex]

[tex]t=-\frac{1}{\lambda} In(\frac{A}{A_0} )[/tex]

[tex]=-\frac{1}{\lambda} In(\frac{A}{\lambda r(\frac{m_c}{m_a} )} )[/tex]

since,

[tex]A_0=\lambda r(\frac{m_c}{m_a} )[/tex]

[tex]=-\frac{1}{\lambda} In(\frac{A\ m_a}{\lambda r m_c} )[/tex]

Now, the age of the artifact is

[tex]=-\frac{1}{\lambda} In(\frac{A\ m_a}{\lambda r m_c} )[/tex]

[tex]=-\frac{1}{1.21\times 10^{-4}} In(\frac{(9.25)(2.32\times 10^{-26}}{1.21\times 10^{-4}(\frac{1}{3.15569\times10^7} )(1.2\times 10^{-12})(0.100)}} )\\\\=6303.4 \ years[/tex]

to two significant figure = 6300 years