Question #39370

1 Answer
Jun 20, 2017

Answer:

#1/256#

Explanation:

You know that the nuclear half-life of a radioactive nuclide, #t_"1/2"#, tells you the time that must pass in order for half of an initial sample of said to nuclide to undergo radioactive decay.

This means that with every passing half-life, the mass of the sample gets reduced by half.

If you take #A_0# to be the initial mass of the sample, you can say that you will be left with

#A_t = A_0 * (1/2)^color(red)(n)#

Here

  • #A_t# is the mass of the sample that remains undecayed after a period of time #t#
  • #color(red)(n)# represents the number of half-lives that pass in a given time period #t#

In your case, you know that

#t_"1.2" = "3 hours"#

You also know that the total time that passes is equal to

#"1 day = 24 hours"#

You can thus say that you have

#color(red)(n) = (24 color(red)(cancel(color(black)("hours"))))/(3color(red)(cancel(color(black)("hours")))) = color(red)(8) ->#this means that eight half-lives pass in a #"24-hour"# period.

Therefore, the mass of the sample that remains undecayed is equal to

#A_t = A_0 * (1/2)^color(red)(8)#

#A_t = A_0 * 1/256#

To find the fraction that remains undecayed, simply divide the amount that remains undecayed by the initial amount

#A_t/A_0 = (color(red)(cancel(color(black)(A_0))) * 1/256)/color(red)(cancel(color(black)(A_0))) = 1/256#