Radioactive iodine, ^131 I, with a half-life of 8.0252 d, is used in some forms of medical diagn

Radioactive iodine, ^131 I, with a half-life of 8.0252 d, is...

Radioactive iodine, $^{131} \mathrm{I},$ with a half-life of $8.0252 \mathrm{d},$ is used in some forms of medical diagnostics. (a) If the initial activity of a sample is $64.5 \mathrm{mCi},$ what is the mass of $^{131} \mathrm{I}$ in the sample? (b) What will the activity be 4.5 d later?

Key Concepts

Radioactive Decay

Radioactive decay is the spontaneous process by which unstable atomic nuclei lose energy by emitting radiation. This process is stochastic, meaning it cannot be predicted exactly for each nucleus but is described statistically for large numbers of atoms, and it forms the basis for understanding nuclear reactions and decay processes.

Half-Life

The half-life is the time required for half of a given sample of radioactive nuclei to decay. It is a characteristic property of each radioactive isotope and provides a measure of the rate at which the material decays over time.

Activity

Activity is the measure of the number of radioactive disintegrations occurring per unit time in a sample. It is often expressed in units such as curies or becquerels, serving as an indicator of the radioactivity of the material.

Exponential Decay Law

The exponential decay law describes how the number of radioactive nuclei or the activity of a sample decreases over time. It is expressed as N(t) = N? exp(–?t) or A(t) = A? exp(–?t), where N? and A? represent the initial number of nuclei and activity, respectively.

Conversion from Activity to Mass

Converting activity to mass involves several steps: first, relating the observed activity to the number of decaying nuclei using the decay constant; then, calculating the amount in moles with Avogadro’s number; and finally, using the molar mass of the radioactive substance to find its mass. This conversion is essential in connecting the measurable activity to the physical quantity of material present.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time that an individual nucleus will undergo radioactive decay. It is mathematically related to the half-life by the equation ? = ln(2)/T?/? and is crucial for formulating the exponential decay law.

Transcript

00:01 So for this question, we want to find what is the mass of the initial sample of iodin -131.

00:10 We are given that the activity is actually 64 .5 miliquries.

00:19 Alright, to find out what is the mass, we need to find out first what is the number of molecules there are, or what is the number of radioactive nuclei that are.

00:31 And it is related to the activity with the rate of decay.

00:39 So the rate of decay multiplied by the number of nucleus gives you the activity.

00:49 So first off, we will need to find what is this rate of decay.

00:53 Now rate of decay, we are given half -life, so we can calculate it from the half -life.

00:59 T -half.

01:01 So the loan 2 over t -half will give you.

01:03 Us the decay rate and we want that in terms of per seconds right because that will be the si units so long to over but we are given days so we got to convert that into seconds 252 days 24 times 3 ,600 and this will give us in terms of seconds per second next we got to convert our activity to back rows as well for si unit we simply multiply that by 10, sorry, 3 .4, no, 3 .7, sorry.

01:48 10 of 10, backers.

01:52 This is the amount of backers per qe.

01:55 Right, and we got also multiplied by minus 3.

01:59 Since this we are dealing with midi qeys.

02:09 Now that we got all of this, we should be able to find the number of iodin molecules.

02:15 Which we can just rearrange.

02:22 We just put the values in over here.

02:27 64 .5 times 10 to the minus 3.

02:32 3 .7 times 10 to power of 10.

02:37 Right? and for our decay rate, there will be 0 2...