What percentage of cesium chloride made from cesium- 137 (beta emitter, t1 / 2=30.1 y) remains a

What percentage of cesium chloride made from cesium- 137...

Key Concepts

Radioactive Decay and Half-Life

This concept describes the process whereby an unstable nucleus loses energy over time by emitting radiation. The half-life is a measure of the time it takes for half of a sample of a radioactive substance to decay. Using the exponential decay law, one can calculate the remaining fraction of the substance over a period, which is key for determining percentages over extended durations.

Beta Decay and Nuclear Transmutation

In beta decay, a nucleus emits a beta particle (an electron), resulting in an increase in the atomic number and thereby transforming the element into another. This nuclear transmutation alters the element’s chemical behavior. When the original radioactive isotope is part of a compound, such as a monovalent halide salt, the decay product may form a new compound with different stoichiometry, reflecting changes in the ionic charges and overall chemical structure.

Transcript

00:01 All right, we're dealing with cesium -137, which is a beta emitter, which has a half -life of 30 .1 years.

00:11 And they want to know how much remains after 150 years.

00:16 All right.

00:17 What percentage, i should say.

00:20 Okay, so first off, anything that is radioactive in nature is first order kinetics.

00:29 That dictates what equations we get to use here, which means the half -life equation is that the half -life equals the natural log of 2 divided by k, which is called the rate constant, or the decay constant.

00:45 The second equation we get to use is the natural log of n equals the natural log of n -not.

00:58 Sorry, i have to sneeze.

01:01 It's not going to come now.

01:02 It'll come eventually.

01:03 I apologize.

01:04 Okay.

01:05 Minus kt, where t is the time.

01:08 All right.

01:09 So the ns can be many different things.

01:13 They can be the concentrations.

01:14 They can be how many atoms you've got.

01:17 They can be what percentage remains, which is actually the format we're going to use.

01:22 Because we're interested in our percent, what percentage remains.

01:26 So we would assume that at time zero, we have 100%.

01:30 So that's going to be the natural log of n.

01:33 The n not.

01:33 The n not part is going to be 100.

01:35 And we're looking for what percentage remains.

01:39 Now we've got our time.

01:41 Our time is 150 years.

01:44 We need to get k.

01:46 And that's where the half -life comes in.

01:48 We're going to solve that first equation for k, and then plug that into this one to then get n.

01:55 All right, so for starters, the half -life equation is that the half -life equals natural log of 2 over k.

02:04 To solve that, you would multiply both sides by k.

02:07 So you've got to get k out of the denominator, which would give us k times 30 .1 years equals natural log of 2.

02:19 Then you can divide by 30 .1 years to get a value for k, natural log over 30 .1 gives you 0 .0230 years.

02:37 And it's actually inverse years, 1 over years.

02:41 Okay, so now we know the number we can plug in for k over here.

02:44 0 .0230 years.

02:47 Inverse years.

02:49 That is a minus one.

02:50 Okay, so now we can go ahead and do that part.

02:54 So we have got natural log of, i'm going to call it, no, we'll leave it as n.

02:58 We're solving for n.

03:01 Equals natural log of 100 minus 0 .0230 inverse years times 150 years.

03:14 Doing the math on the right hand side, natural log of 100 minus 0 .230 times 150 gives you 1 .1 .140 gives you 1 .115 gives you 1 .115 .1 .5.

03:24 So the natural log of n equals 1 .155.

03:31 All right.

03:32 So now we need to get rid of the natural log because we just want the n part.

03:36 So to get rid of these, you have to remember that ln is the same thing as log base e, where e is the number e, not the letter.

03:46 Similar to how pi is 3 .14.

03:49 E is 2.

03:52 Something.

03:52 It should be on your calculator.

03:54 What is e equal 2 .718? okay.

03:59 So the way that you undo these, in other words, the way that you get the n out is you do e to the, and then you raise natural log of n as the exponent.

04:10 So e to the exponent, natural log of n.

04:13 This cancels out the natural log and the e both, and you're left with n.

04:19 So this is going to be n equals.

04:21 Now you can't just do something to one side of the equation or you mess up the whole thing.

04:25 So we also have to do that to the right hand side, e to the 1 .15.

04:30 Power.

04:32 So e to the 1 .155 gave me 3 .17...