Question

¹⁴C decays by emitting a β^- particle. This decay process has a half-life of 5.72×10³ yr. Answer the following questions about this process and report all answers to three significant figures.  

A. Determine the initial activity (in dis/yr) of a 5.81 mg sample of this isotope.

B. Determine the time (in yr) that it takes for the activity of ¹⁴C to decrease to 8.39×10⁷ dis/yr.

Answer 1

half life time is $5.72 \times 10^{3}$ yr

decay constant will be:-

$\lambda = 0.693/5.72 \times 10^{3}$

$\lambda = 1.21 \times 10^{-4}$

A) initial activity will be given as:-

$A=\lambda \times N$

A= activity

N = number of undecayed nuclei

here N = $5.81\times 6.023\times 10^{23}/14$

$=24.9\times 10^{19}$

puttinG values of both we get:-

$\lambda =24.9\times 10^{19}\times 1.21\times 10^{-4}$

$\lambda =30.1\times 10^{15}$ dis/yr

B) the time can be calculated as:-

$8.39 \times 10^{7}= 30.1\times 10^{15}\times e^{(-1.21 \times 10^{-4})\times t}$

$2.78 \times 10^{-9}= e^{(-1.21 \times 10^{-4})\times t}$

taking natural log both sides we have :-

$ln(2.78) -9\times 2.303= {(-1.21 \times 10^{-4})\times t}$

$1.02 -20.7= {(-1.21 \times 10^{-4})\times t}$

$t= 1.63 \times 10^{5}$ Yr