Question Video: Determining the Amount of Unstable Nuclei Present after a Given Amount of Time Chemistry

A radioactive isotope with a half-life of 2 hours contains 100 billion unstable nuclei. How many unstable nuclei would remain after 10 hours?

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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.

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