The half-life of (208)/(81) Tl is 3.1 minutes. A 84.6 -gram sample is studied in the laboratory.

The half-life of (208)/(81) Tl is 3.1 minutes. A 84.6 -gram...

The half-life of $\frac{208}{81} \mathrm{Tl}$ is 3.1 minutes. A 84.6 -gram sample is studied in the laboratory.
a) How many grams of the isotope will remain after 12 minutes?
b) In how many minutes will the mass of $^{208}$ TI be 3.48 grams?

Key Concepts

Exponential Decay

Exponential decay describes processes where a quantity decreases at a rate proportional to its current value. This mathematical model is used across various disciplines, including physics, biology, and finance, to represent systems that degrade, cool, or diminish over time. The function typically takes the form N(t) = N? e^(?kt), where N? is the initial amount and k is a constant.

Half-life

The half-life of a substance is the time required for it to reduce to half of its initial quantity. It is a specific application of exponential decay that provides a measure of the speed of decay processes and is widely used in fields such as nuclear physics, chemistry, and pharmacokinetics.

Logarithms

Logarithms are the inverses of exponential functions and are essential for solving for time or decay constants in exponential decay equations. By applying logarithms, one can isolate variables exponentiated in the decay formula, allowing the calculation of time or other parameters given certain performance measures.

Transcript

00:06 This question tells us that the half life of thallium 208 is 3 .1 minutes.

00:27 T half or the half life is equal to 3 .1 minutes.

00:36 And then we are asked to calculate the grams of remaining thallium 208 after 12 minutes if there was a mass of 84 .6 grams being studied in the laboratory.

00:59 So the initial mass here is, as we just heard, is 84 .6 grams.

01:16 And we are asked to calculate how much of this is going to be remaining after t equals 12 minutes in the first part of the question.

01:24 This is the data that's available.

01:29 So we can use this equation where we can calculate the mass of thallium remaining at time t, which is after 12 minutes, by taking the mass of thallium that was present at time 0, which is 84 .6 grams, and multiplying that by 0, which is raised to the phthalium, which is raised to the part.

01:59 T over t half and if we at the values in we get 84 .6 grams multiply that by 0 .5 t here is 12 minutes and the half life is 3 .1 minutes so these are both given in the minute units and they'll cancel out each other.

02:30 So the answer to this part of the question is going to be equal to 5 .78 grams.

02:44 This is the mass of talium 208 remaining after 12 minutes.

02:51 In the second part of the question, we are asked to calculate something else, and that is how many minutes will it take for the mass of talium 208 to, reduced to 3 .48 grams.

03:14 So let's take a second scenario.

03:16 Now we have different data.

03:20 In this case, t is equal to what we have to calculate...

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