Question

How many half lives would it take for a sample of carbon-14 to be reduced to ${}^{1}/{}_{32}$of its original mass?

Answer 1

Hint:Half life is calculated for the compounds that undergo decay after completing their full life. Generally, calculated for isotopes of any element.

Complete step-by-step answer:We have been given a sample of carbon-14, reduced to ${}^{1}/{}_{32}$of its original mass. We have to calculate how many half lives it had taken to reach ${}^{1}/{}_{32}$of its original mass.
We have to apply a simple logic in this question. Half life is not to be calculated, only the number of half lives is to be determined. So, assuming half of the carbon-14 is decayed at half life of $\dfrac{1}{2}$ we have to calculate, how many times of this half life is reduces the mass of carbon-14 by$\dfrac{1}{32}$of the original mass. So, we have to keep reducing this $\dfrac{1}{2}$ by half till we find $\dfrac{1}{32}$.
Half life of carbon-14 = $\dfrac{1}{2}\,of\,its\,life\,$
Half of $\dfrac{1}{2}$is = $\dfrac{\dfrac{1}{2}}{2}$ =$\dfrac{1}{4}\,\,\,\,\,$
Now, half of $\dfrac{1}{4}$= $\dfrac{\dfrac{1}{4}}{2}$= $\dfrac{1}{8}\,\,$
Half, of $\dfrac{1}{8}\,\,$= $\dfrac{\dfrac{1}{8}}{2}$=$\dfrac{1}{16}\,\,$
Half, of $\dfrac{1}{16}\,\,$= $\dfrac{\dfrac{1}{16}}{2}$=$\dfrac{1}{32}\,\,$
Thus, we have obtained the $\dfrac{1}{32}\,\,$of the original mass of carbon-14 in a total of 5 half lives.
Hence, 5 half lives will reduce the carbon-14 to $\dfrac{1}{32}\,\,$of the original mass.

Note:The half life of any element that undergoes decay is calculated using the formula, $A(t)={{A}_{0}}{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{{{t}_{{}^{1}/{}_{2}}}}}}$ , where A(t) is the amount of the remaining substance after decay, ${{A}_{0}}$is the initial amount of the substance, t is time taken for the decay, and ${{t}_{{}^{1}/{}_{2}}}$ is the half life.