17-24= Radioactive Decay These exercises use the radioactive...

$17-24=$ Radioactive Decay These exercises use the radioactive decay model. Radioactive Cesium The half-life of cesium- 137 is 30 years. Suppose we have a $10-$ g sample. $$ \begin{array}{l}{\text { (a) Find a function } m(t)=m_{0} 2^{-t / h} \text { that models the mass }} \\ {\text { remaining after } t \text { years. }} \\ {\text { (b) Find a function } m(t)=m_{0} e^{-r t} \text { that models the mass }} \\ {\text { remaining after } t \text { years. }} \\ {\text { (c) How much of the sample will remain after } 80 \text { years? }} \\ {\text { (d) After how many years will remain after } 80 \text { years? }} \\ {\text { remain? }}\end{array} $$

$17-24=$ Radioactive Decay These exercises use the radioactive
decay model.

Radioactive Cesium The half-life of cesium- 137 is
30 years. Suppose we have a $10-$ g sample.

$$
\begin{array}{l}{\text { (a) Find a function } m(t)=m_{0} 2^{-t / h} \text { that models the mass }} \\ {\text { remaining after } t \text { years. }} \\ {\text { (b) Find a function } m(t)=m_{0} e^{-r t} \text { that models the mass }} \\ {\text { remaining after } t \text { years. }} \\ {\text { (c) How much of the sample will remain after } 80 \text { years? }} \\ {\text { (d) After how many years will remain after } 80 \text { years? }} \\ {\text { remain? }}\end{array}
$$

Key Concepts

Radioactive Decay

Radioactive decay refers to the process by which an unstable atomic nucleus loses energy by emitting radiation. This process occurs randomly at the level of individual atoms but follows predictable statistical rules across large collections of atoms. The overall phenomenon is fundamental to nuclear physics and is modeled mathematically to predict how the quantity of a radioactive substance decreases over time.

Half-life

Half-life is the time required for half of the atoms in a radioactive sample to decay. It is a key characteristic of each radioactive substance and provides a convenient measure for quantifying the rate of decay. Knowing the half-life allows one to calculate the remaining amount of a substance after a given period, using mathematical models of exponential decay.

Exponential Decay Functions

Exponential decay functions are mathematical models used to describe processes where a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, these functions provide an effective way to represent how the mass of a radioactive sample diminishes over time. Two common forms of these functions use different bases: one involving the half-life formulation (using a base of 2) and another using the natural base e.

Decay Constant

The decay constant is a parameter in exponential decay formulas that quantifies the rate at which a substance undergoes decay. It is directly related to the half-life of the material; a larger decay constant indicates a quicker decay process. Converting between the half-life and the decay constant is essential for using different forms of exponential decay equations and for understanding the speed of the decay process in a quantitative manner.

Transcript

00:01 In this problem, we're dealing with the radioactive decay of radium 226.

00:06 For part a, we have to utilize the half -life provided a function that models the remaining mass after t years.

00:17 So the equation we're going to use for this is m of t is equal to m initial, which is the initial mass, times 2 to the negative t.

00:38 Over h power, where in this case, m initial is the initial mass, t is the number of years, and h is the half -life.

00:50 The sample size, right now, the mass of it is 10 grams.

00:55 So we know that m initial is 10 grams.

00:58 It's also provided that the half -life is 30 years.

01:02 Therefore, for this part, we can simply write the equation as m of t.

01:11 Is equal to 10 times 2 to the negative t over 30.

01:23 Let's get that a little lower.

01:28 T over 30.

01:32 Now for part b, we just have to use the basic radioactive decay function.

01:38 So that equation would be m of t is equal to m initial times e to the negative rt, where m initial is our initial mass, r is the decay, i guess, factor, like how much it decays by, and t is the number of years.

01:59 So m of t is equal to the initial mass, which we are told is 10, and multiply that by e to the negative r of t.

02:21 But this isn't done yet.

02:22 We have to find r.

02:24 So what we're going to do is solve for r by equating this equation with this equation.

02:35 And that's going to look like m initial times 2 to the negative t over h is equal to, or sorry, it's going to look like 10 times 2 to the negative t over 30 is equal to 10 times 2 to the negative t over 30 is equal to 10 times.

03:09 Times e to the negative rt.

03:13 And the reason we have to equate these two functions is because we have to get rid of t somehow so we can make r the only unknown variable.

03:23 So right off the bat, the two tens cancel and our equation becomes 2 to the negative t over 30 is equal to e to the negative rt...

Please give Ace some feedback