### Video Transcript

A radioactive isotope with a
half-life of two hours contains 100 billion unstable nuclei. How many unstable nuclei would
remain after 10 hours?

Isotopes are atoms that have the
same number of protons but a different number of neutrons. A radioactive isotope can undergo
radioactive decay, where radioactive decay is the spontaneous emission of radiation
by an unstable nucleus. The unstable nucleus emits
radiation in an attempt to become more stable. The emitted radiation could be in
the form of 𝛾 rays, 𝛼 particles, or 𝛽 particles. The radiation can be detected, and
the amount of time required for one-half of the radioactive nuclei in the sample to
decay can be measured. This is called the half-life.

So, for example, if a sample
contained 100 radioactive nuclei, then the time taken for half of those to decay
leaving 50 radioactive nuclei remaining is the half-life. The isotope in the question starts
off with 100 billion unstable nuclei. These are represented by the orange
dots. The isotope has a half-life of two
hours. So after two hours, half of the
undecayed unstable nuclei would have undergone radioactive decay, forming decayed
stable nuclei represented by the pink dots. To calculate how many undecayed
nuclei we have remaining, we need to divide the original number of undecayed nuclei,
which is 100 billion, by two. This gives us a value of 50 billion
undecayed unstable nuclei. After a further two hours, the
number of undecayed unstable nuclei would halve again.

So to find out how many undecayed
unstable nuclei we have remaining, we need to divide 50 billion by two. This gives us a value of 25
billion. So after two half-lives, totaling
four hours, there are 25 billion unstable nuclei remaining. But the question asks us how many
unstable nuclei would remain after 10 hours. So we need to repeat the process
until we get to a total of 10 hours. After another half-life, bringing
the total to six hours, the number of unstable nuclei would halve again. So this time, we need to divide 25
billion by two. This gives a value of 12.5 billion,
but we are still yet to reach 10 hours, so we need to repeat the process again.

After a further two hours, the
number of unstable nuclei would halve again. So we need to divide 12.5 billion
by two. This gives a value of 6.25
billion. So after a total of eight hours,
there would be a total of 6.25 billion undecayed unstable nuclei remaining. After another half-life, the number
of unstable nuclei remaining would halve again. If we divide 6.25 billion by two,
we get a value of 3.125 billion. So after a total of five
half-lives, which amounts to a total of 10 hours, there would be 3.125 billion
undecayed unstable nuclei remaining. So the answer to the question “How
many unstable nuclei would remain after 10 hours?” is 3.125 billion.