use the half-life information to complete each table. (The formula 𝒩=𝒩0 e^k t is not required.)

Use the half-life information to complete each table. (The...

use the half-life information to complete each table. (The formula $\mathcal{N}=\mathcal{N}_{0} e^{k t}$ is not required.)
(a) Polonium- 210 : half-life $=138.4$ days
(b) Polonium- 214 : half-life $=1.63 \times 10^{-4}$ second

Key Concepts

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. This mathematical model is used to explain the decay of unstable isotopes, where the decrease in the amount of substance over time follows a consistent exponential law.

Decay Table Completion

Decay table completion involves applying the concept of half-life to systematically calculate the remaining amount of a substance after successive intervals. By understanding that each half-life period results in a halving of the substance's quantity, one can fill in tables that organize the decay process over multiple periods.

Half-life

Half-life is the time period required for a substance undergoing exponential decay to reduce to half of its initial amount. It is a fundamental concept in nuclear physics and chemistry that allows the prediction of how quickly a radioactive material decays over time.

Transcript

00:02 Here, given that the half -life of an isotope is 138 .4 days and that the initial amount was 0 .4 grams.

00:11 And we want to know how much is left after several half -lives have elapsed.

00:16 So after one half -life, that's 138 .4 days, there's going to be exactly half of that 0 .4 grams, right? what did we start with? times one -half is going to leave us with two grams.

00:32 Now, how about another half -life later? so after the next 138 .4 days, this is going to be 276 .8 days.

00:47 We're going to, that's exactly one half -life later.

00:50 So we had two, and then after half a life, point two, times a half, we're going to have 0 .1.

01:04 And then after another 138 .4 days, our total time elapsed will have been 415 .2 days.

01:18 And that's 138 .4 is another half life.

01:21 So that point one that we had is going to be reduced in half again.

01:26 So 0 .1 times one half is going to equal to 0 .05 .0 .05.

01:37 And then one more half -life later.

01:41 So if we add another 138 .4 days, that's going to add up to 553 .6 total days.

01:55 And a half -life has elapsed since we were at 0 .05.

02:00 So that's going to get cut in half.

02:03 So 0 .05 times 1⁄2.

02:07 Is going to equal to 0 .02.

02:16 Now, we were also, we were given a second isotope that has a half -life of 1 .63 times 10 to the minus 4 seconds.

02:25 And we were given that the initial amount was 0 .1 grams.

02:31 So if we get cut in half, so if we take our 0 .1 and cut it in half, 0 .1 times a half, we're going to get .05 grams.

02:47 So if there are 0 .05 grams, that was one half of what we started with...

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