A student determines the half-life of a radioactive isotope. The student uses a detector over five minutes and plots a graph showing how the count rate shown on the detector varies with time. The count rate due to background radiation is 30 counts per minute. What is the half-life of this isotope?
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Answer:
We know that radiation is more than just the spooky, silent threat that we see in movies. Healthcare providers can actually harness the unique properties of radiation to look inside the human body and diagnose diseases in new ways. We also know that all radiation occurs when an unstable nucleus releases energy to become more stable. This happens when the nucleus changes into a different nucleus This happens in three different ways:
Alpha decay: The nucleus splits into two chunks, a little chunk called an “alpha particle” (which is just two protons and two neutrons) and a daughter nucleus with a lower atomic number than the initial nucleus. The “radiation” here is the small chunk, which generally moves away from the nucleus at a pretty high speed.
Beta decay: There are two types of beta decay: In beta-minus decay, a neutron in an atom changes into a proton, an electron, and an antineutrino, creating and releasing an electron along the way (since the total charge has to stay the same!). The radiation here is the super fast-moving electron released. In beta-positive decay, a proton turns into a neutron, causing the nucleus to shoot out an exotic positive particle called a “positron” or “anti-electron.”
Gamma decay: The number of protons, neutrons, and electrons stays the same, but they rearrange themselves within the atom, giving off energy in the form of high-energy photons (gamma radiation), in order to have lower overall energy.
For all of these cases, the total amount of the radioactive element decreases over time. So if a scientist takes a chunk of carbon-10 (which undergoes beta decay), counts the number of carbon-10 atoms inside it, goes to make coffee, and then comes back to count the number of atoms again, she’ll find that the total number of atoms of carbon-10 is now smaller! In their place she’ll find the beta decay product of carbon-10, which is the element boron.
Although the decay of individual nuclei happens randomly, it turns out that large numbers of nuclei can be modelled by a mathematical function that predicts the amount of radioactive nuclei remaining at a given time:
N(t) = N_0
0
start subscript, 0, end subscripte^{-kt}
−kt
start superscript, minus, k, t, end superscript
This states that the number of carbon-10 nuclei (N(t)) left in a sample that started out with N0 atoms decreases exponentially in time. The constant k is called the decay constant, which controls how quickly the total number of nuclei decreases. The value of the decay constant is specific to the type of decay (alpha, beta, gamma) and isotope being studied, and so unknown
isotopes can be identified based on how quickly they decay.