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- [Voiceover] Phosphorus-32 is radioactive

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and undergoes beta decay.

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So we talked about beta
decay in the last video.

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Here's our beta particle, and the

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phosphorus is going to turn into sulfur.

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Let's say we started with four
milligrams of phosphorus-32.

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And we wait 14.3 days,

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and we see how much of
our phosphorus is left.

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You're going to find two
milligrams of your phosphorus left.

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The rest has turned into sulfur.

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And this is the idea of half-life.

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Let's look at the definition
for half-life here.

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It's the time it takes for 1/2 of

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your radioactive nuclei to decay.

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So, if we start with four milligrams,

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and we lose 1/2 of that, right,

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then we're left with two milligrams.

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And it took 14.3 days for this to happen.

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So 14.3 days is the
half-life of phosphorus-32.

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And this is the symbol for half-life.

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So, 14.3 days is the
half-life for phosphorus-32.

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The half-life depends on
what you're talking about.

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So if you're talking about
something like uranium-238,

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the half-life is different,
it's approximately

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4.47 times 10 to the ninth, in years.

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That's obviously much
longer than phosphorus-32.

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We're going to stick with
phosphorus-32 in this video,

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and we're going to actually start with

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four milligrams every time in this video

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just to help us understand
what half-life is.

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Next, let's graph the rate
of decay of phosphorus-32.

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Let's look at our graph here.

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On the Y-axis, let's do the amount

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of phosphorus-32,

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and we're working in milligrams here,

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so this will be in milligrams.

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On the X-axis, let's do time,
and since the half-life is in

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days, it just makes it
easier to do this in days.

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Alright, we're going to start with

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four milligrams of our sample.

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Let's go ahead and mark
this off so this would be

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one milligram, two
milligrams, three, and four.

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So we're going to start
with four milligrams.

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So when time is equal to
zero, we have four milligrams.

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Let's go ahead and mark this off.

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So one, two, three, and four.

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We wait 14.3 days, so this is 14.3 days,

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and half of our sample should be left.

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So what's half of four,
it's of course, two.

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And so, we can go ahead and
graph our next data point.

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There should be two milligrams left

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after 14.3 days so that's our point.

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Alright, we wait another 14.3 days,

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so we wait another half-life, so after

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two half-lives, that should be 28.6 days.

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So we know that after 28.6
days, it's another half-life,

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so what's 1/2 of two, it's one, of course.

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So that's our next point.

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So after 28.6 days, we should have

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one milligram of our sample.

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Let's wait another half-life.

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28.6 plus 14.3,

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should be 42.9.

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So that's our next point.

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And what's half of one?

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It's 0.5, of course, so, in here,

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that's about 0.5, and so that gives us

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an idea about where
our next data point is.

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And we could keep going, but this is

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enough to give you an idea
of what the graph looks like.

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Right, so if I think about this graph,

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this is exponential decay.

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That's what we're talking about when

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we're talking about
radioactive decay here.

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We'll talk a little bit more about

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exponential decay in the next video.

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But this just helps you
understand what's happening.

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So as you increase the
number of half-lives,

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you can see the amount of radioactive

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material is decreasing.

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Alright, let's do a very
simple problem here.

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If you start with four
milligrams of phosphorus-32,

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how much is left after 57.2 days?

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So if you're waiting 57.2 days,

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well, the half-life of
phosphorus-32 is 14.3 days.

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So, how many half-lives is that?

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57.2 days divided by 14.3 days

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would give us how many
half-lives, and that's four.

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So there are four half-lives,
so four half-lives here.

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We're starting with four milligrams,

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so one very simple way of doing this is to

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think about what happens
after each half-life.

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So four milligrams, if
we wait one half-life,

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goes to two milligrams.

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Wait another half-life,
goes to one milligram.

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Wait another half life,
goes to 0.5 milligrams.

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And, if we wait one more half-life,

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then that would go to 0.25 milligrams.

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So that would be our answer,
because that's four half-lives.

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Here's one half-life,
two, three, and four,

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which is how many we
needed to account for.

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That's one way to do the math.

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Another way, would be
starting with four milligrams,

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we need to multiply that by 1/2,

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and that would give us two, and then

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multiply by 1/2 again,

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and 1/2 again, and 1/2 again.

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So that's four half-lives, right?

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So this represents our four half-lives.

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And that's the same thing as going

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four, times 1/2 to the fourth power,

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which mathematically, is

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four times one over 16,

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so that's 4/16, so that's
the same thing as 1/4,

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and so that's 0.25 milligrams.

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So it doesn't really matter how you

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do the math, there are
lots of ways to do it.

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You should get the same answer.

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You could also get this on the graph

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if you had a decent graph.

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After four half-lives, you would be

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you would be over here somewhere.

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And so you could just find where that is.

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So let me use red, so you could

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find where that is on your graph,

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and then go over to here,

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so that would be approximately right here,

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and then read that off your graph.

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And that looks like about
0.25 milligrams as well.

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We'll talk more about
graphing in the next video.