The activity of a sample of radioactive material is measured,and found to be 880 Bq. After 160 minutes the activity has fallen to 55 Bq. What is the half-life of Bq?

1 Answer
Jan 13, 2016

Answer:

#"40 minutes"#

Explanation:

Before doing anything else, make sure that you understand what's going on here.

You're dealing with a sample of radioactive material that has an activity equal to #"880 Bq"#. Now, what does that mean?

A becquerel, #"Bq"#, is a unit used to measure of the radioactivity of an isotope in which one nucleus decays per second.

#color(blue)("1 becquerel" = "1 decay"/"1 s")#

This means that you have

#"1 Bq" = "1 s"^(-1)#

So, an activity of #"880 Bq"# means that #880# nuclei are decaying every second.

Now, after #160# minutes pass, the activity falls to #"55 Bq"#, which is equivalent to saying that at this point only #55# nuclei are decaying per second.

As you know, nuclear half-life is simply the time needed for a sample of radioactive material to decay to half of its initial size.

The equation that establishes a relationship between the amount left undecayed, #A#, the initial amount, #A_0#, and the number of half-lives that pass in a period of time #t# looks like this

#color(blue)(A = A_0 * 1/2^n)" "#, where

#n# - the number of half-lives

Plug in your values to get

#55 color(red)(cancel(color(black)("Bq"))) = 880 color(red)(cancel(color(black)("Bq"))) * 1/2^n#

This is equivalent to

#55/880 = 1/2^n#

#1/16 = 1/2^n implies 2^n = 16#

You will thus have

#2^n = 16 implies n = 4#

Therefore, four half-lives must pass in order for the activity of the sample to go from #"880 Bq"# to #"55 Bq"#.

Since the number of half-lives can be thought of as

#color(blue)(n = "given period of time"/"half-life" = t/t_"1/2")#

you can say that

#t_"1/2" = t/n#

This will get you

#t_"1/2" = "160 minutes"/4 = color(green)("40. minutes")#