To destroy a cancerous tumor, a dose of gamma radiation with...

To destroy a cancerous tumor, a dose of gamma radiation with a total energy of 2.12 J is to be delivered in 30.0 days from implanted sealed capsules containing palladium-103. Assume this isotope has a half-life of 17.0 d and emits gamma rays of energy 21.0 keV, which are entirely absorbed within the tumor. (a) Find the initial activity of the set of capsules. (b) Find the total mass of radioactive palladium these “seeds” should contain.

Key Concepts

Radioactive Decay

Radioactive decay is the process by which unstable nuclei lose energy by emitting radiation. This process follows an exponential law, meaning that the number of nuclei that remain undecayed decreases exponentially over time. This concept is foundational in calculating how the number of decays—and thus the amount of energy released—evolves in time, which is crucial for understanding both the dose delivery and the decay activity of radioactive sources.

Half-Life and Decay Constant

The half-life is the time required for half of the radioactive nuclei in a sample to decay. It is intimately related to the decay constant (?) through the equation ? = ln(2)/half-life. This relationship is essential in modeling the decay process over time, allowing one to calculate the rate at which a radioactive substance diminishes and how this impacts the activity and cumulative energy release.

Activity

Activity is defined as the number of radioactive decays per unit time, typically measured in becquerels or curies. It is directly proportional to the number of radioactive nuclei present and the decay constant, as given by A = ?N. Understanding activity is critical for determining the rate at which energy is deposited via decays, which in turn is used for dosimetry in medical applications such as tumor treatment.

Energy per Decay and Dosimetry

Each radioactive decay event releases a specific amount of energy, and the total energy delivered by a radioactive source is the product of the energy per decay and the total number of decays over a given period. This concept is at the heart of dosimetry, where the goal is to deliver a precise amount of radiation energy to a target, such as a cancerous tumor, ensuring both effectiveness and safety in medical treatments.

Mass Calculation Using Nuclear Decay Data

Once the total number of decays required to deliver a specific energy dose is known, it is possible to back-calculate the number of radioactive atoms that must be present initially. By knowing the atomic mass of the isotope and using Avogadro's number to relate the number of atoms to a mole, one can determine the mass of the radioactive material needed. This calculation bridges nuclear physics and chemistry, and is essential for accurately preparing radioactive sources for medical applications.

Transcript

00:01 So the initial activity, a0, is defined as the decay rate multiplied by the initial number of nucleates.

00:11 We know the decay rate by taking lon2 over t half, and we are given t half, to be 17 days.

00:21 Right? but what is ni? the initial number of nucleides? well, that will depend on the number of nucleides that will depend on the number of nucleides that we need in order to actually produce the sufficient amount of energy that is absorbed by the cancerous tumor.

00:51 So the amount of energy that has been, or the number of nucleides that have decayed to produce this energy will be n -i minus nf, where nf is the number of nucleides after 30 days.

01:09 And so this will tell us how much nucleides has decayed.

01:18 And from this we can get the amount of energy that is being absorbed by multiplying by the energy released per nucleide.

01:33 Now nf is related to n i by the decay law, so we can very nicely rearrange this equation over here into n i 1 minus exponential.

01:51 So the number of nuclei that has decayed in 30 days is constrained by the energy requirement for the tumor.

02:09 The energy required by the tumor is 2 .12 joules.

02:17 And from this we can actually calculate what is the required number of nuclei that needs to be decayed, will be 2 .12 divided by the energy for each decay which is given as 21 kilo electron volts and we will need to convert that into juice right since the 2 .12 is given in juice.

02:59 Now from this required number of nucleus we equate that to the equation over here which is given as n .i minus nf right? and so we can actually find out what is our n -i...

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