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SAL: In the last video we saw
all sorts of different types

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of isotopes of atoms
experiencing radioactive decay

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and turning into other atoms or
releasing different types

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of particles.

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But the question is, when
does an atom or

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nucleus decide to decay?

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Let's say I have a bunch of,
let's say these are all atoms.

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I have a bunch of atoms here.

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And let's say we're talking
about the type of decay where

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an atom turns into
another atom.

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So your proton number
is going to change.

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Your atomic number is
going to change.

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So it could either be beta
decay, which would release

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electrons from the neutrons and
turn them into protons.

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Or maybe positron emission
turning protons into neutrons.

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But that's not what's
relevant here.

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Let's say we have a collection
of atoms. And normally when we

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have any small amount of any
element, we really have huge

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amounts of atoms of
that element.

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And we've talked about moles
and, you know, one gram of

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carbon-12-- I'm sorry, 12
grams-- 12 grams of carbon-12

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has one mole of carbon-12
in it.

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One mole of carbon-12.

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And what is one mole
of carbon-12?

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That's 6.02 times 10 to the 23rd
carbon-12 atoms. This is

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a ginormous number.

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This is more than we can, than
my head can really grasp

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around how large of
a number this is.

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And this is only when we have
12 grams. 12 grams is not a

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large mass.

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For example, one kilogram
is about two pounds.

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So this is about, what?

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I want to say [? 1/50 ?]

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of a pound if I'm
doing [? it. ?]

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But this is not a lot
of mass right here.

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And pounds is obviously force.

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You get the idea.

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On Earth, well anywhere,
mass is invariant.

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This is not a tremendous
amount.

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So with that said, let's go back
to the question of how do

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we know if one of these
guys are going to

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decay in some way.

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And maybe not carbon-12, maybe
we're talking about carbon-14

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or something.

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How do we know that they're
going to decay?

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And the answer is, you don't.

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They all have some probability
of the decaying.

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At any given moment, for a
certain type of element or a

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certain type of isotope of
an element, there's some

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probability that one
of them will decay.

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That, you know, maybe this guy
will decay this second.

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And then nothing happens for a
long time, a long time, and

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all of a sudden two
more guys decay.

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And so, like everything in
chemistry, and a lot of what

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we're starting to deal with in
physics and quantum mechanics,

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everything is probabilistic.

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I mean, maybe if we really
got in detail on the

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configurations of the nucleus,
maybe we could get a little

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bit better in terms of our
probabilities, but we don't

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know what's going on inside of
the nucleus, so all we can do

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is ascribe some probabilities
to something reacting.

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Now you could say, OK, what's
the probability of any given

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molecule reacting
in one second?

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Or you could define
it that way.

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But we're used to dealing with
things on the macro level, on

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dealing with, you know, huge
amounts of atoms. So what we

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do is we come up with terms
that help us get our head

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around this.

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And one of those terms is
the term half-life.

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And let me erase this
stuff down here.

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So I have a description, and
we're going to hopefully get

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an intuition of what
half-life means.

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So I wrote a decay reaction
right here,

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where you have carbon-14.

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It decays into nitrogen-14.

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And we could just do a
little bit of review.

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You go from six protons
to seven protons.

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Your mass changes the same.

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So one of the neutrons must have
turned into a proton and

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that is what happened.

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And it does that by releasing
an electron, which is also

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call a beta particle.

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We could have written this
as minus 1 charge.

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Relatively zero mass.

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It does have some mass,
but they write zero.

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This is kind of notation.

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So this is beta decay.

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Beta decay, this is
just a review.

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But the way we think about
half-life is, people have

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studied carbon and they said,
look, if I start off with 10

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grams-- if I have just a block
of carbon that's 10 grams. If

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I wait carbon-14's half-life--
this is a

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specific isotope of carbon.

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Remember, isotopes, if there's
carbon, can come in 12, with

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an atomic mass number of 12, or
with 14, or I mean, there's

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different isotopes of
different elements.

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And the atomic number
defines the carbon,

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because it has six protons.

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Carbon-12 has six protons.

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Carbon-14 has six protons.

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But they have a different
number of neutrons.

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So when you have the same
element with varying number of

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neutrons, that's an isotope.

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So the carbon-14 version, or
this isotope of carbon, let's

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say we start with 10 grams. If
they say that it's half-life

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is 5,740 years, that means that
if on day one we start

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off with 10 grams of pure
carbon-14, after 5,740 years,

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half of this will
have turned into

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nitrogen-14, by beta decay.

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And you might say, oh OK, so
maybe-- let's see, let me make

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nitrogen magenta, right there--
so you might say, OK,

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maybe that half turns
into nitrogen.

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And I've actually seen this
drawn this way in some

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chemistry classes or physics
classes, and my immediate

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question is how does this
half know that it

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must turn into nitrogen?

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And how does this half know that
it must stay as carbon?

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And the answer is
they don't know.

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And it really shouldn't
be drawn this way.

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So let me redraw it.

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So this is our original block
of our carbon-14.

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What happens over that
5,740 years is that,

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probabilistically, some of these
guys just start turning

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into nitrogen randomly,
at random points.

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And over 5,740 years, you
determine that there's a 50%

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chance that any one of these
carbon atoms will turn into a

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nitrogen atom.

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So that after 5,740 years, the
half-life of carbon, a 50%

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chance that any of the
guys that are carbon

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will turn to nitrogen.

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So if you go back after a
half-life, half of the atoms

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will now be nitrogen.

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So now you have, after
one half-life-- So

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let's ignore this.

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So we started with this.

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All 10 grams were carbon.

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10 grams of c-14.

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This is after one half-life.

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And now we have five
grams of c-14.

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And we have five grams
of nitrogen-14.

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Fair enough.

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Let's think about what happens
after another half-life.

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Well we said that during a
half-life, 5,740 years in the

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case of carbon-14-- all
different elements have a

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different half-life, if they're
radioactive-- over

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5,740 years there's a 50%-- and
if I just look at any one

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atom-- there's a 50%
chance it'll decay.

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So if we go to another
half-life, if we go another

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half-life from there, I had
five grams of carbon-14.

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So let me actually copy
and paste this one.

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This is what I started with.

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Now after another half-life--
you can ignore all my little,

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actually let me erase some
of this up here.

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Let me clean it up
a little bit.

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After one one half-life,
what happens?

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Well I now am left with five
grams of carbon-14.

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Those five grams of carbon-14,
every one of those atoms still

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has, over the next-- whatever
that number was, 5,740 years--

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after 5,740 years,
all of those once

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again have a 50% chance.

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And by the law of large numbers,
half of them will

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have converted into
nitrogen-14.

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So we'll have even more
conversion into nitrogen-14.

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So now half of that five grams.
So now we're only left

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with 2.5 grams of c-14.

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And how much nitrogen-14?

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Well we have another two and
a half went to nitrogen.

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So now we have seven and a half
grams of nitrogen-14.

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And we could keep going further
into the future, and

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after every half-life, 5,740
years, we will have half of

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the carbon that we started.

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But we'll always have an

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infinitesimal amount of carbon.

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But let me ask you a question.

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Let's say I'm just staring
at one carbon atom.

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Let's say I just have this
one carbon atom.

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You know, I've got its nucleus,
with its c-14.

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So it's got its six protons.

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1, 2, 3, 4, 5, 6.

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It's got its eight neutrons.

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It's got its six electrons.

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1, 2, 3, 4, 5, 6, whatever.

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What's going to happen?

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What's going to happen
after one second?

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Well, I don't know.

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It'll probably still be carbon,
but there's some

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probability that after one
second it will have already

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turned into nitrogen-14.

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What's going to happen after
one billion years?

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Well, after one billion years
I'll say, well you know, it'll

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probably have turned into
nitrogen-14 at that point, but

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I'm not sure.

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This might be the one
ultra-stable nucleus that just

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happened to, kind of,
go against the

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odds and stay carbon-14.

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So after one half-life, if
you're just looking at one

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atom after 5,740 years, you
don't know whether this turned

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into a nitrogen or not.

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This exact atom, you just know
that it had a 50% chance of

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turning into a nitrogen.

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Now, if you look at it over a
huge number of atoms. I mean,

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if you start approaching, you
know, Avogadro's number or

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anything larger--
I erased that.

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Then all of a sudden you can use
the law of large numbers

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and say, OK, on average, if each
of those atoms must have

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had a 50% chance, and if I have
gazillions of them, half

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of them will have turned
into nitrogen.

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I don't know which half,
but half of them

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will turn into it.

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So you might get a question
like, I start with, oh I don't

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know, let's say I start with
80 grams of something with,

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let's just call it x, and it has
a half-life of two years.

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I'm just making up
this compound.

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A two-year half-life.

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And then let's say we go into
a time machine and we look

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back at our sample, and let's
say we only have 10 grams of

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our sample left.

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And we want to know how much
time has passed by.

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So 10 grams left of x.

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How much time, you know, x is
decaying the whole time, how

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much time has passed?

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Well let's think about it.

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We're starting at time, 0 with
80 grams. After two years, how

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much are we going
to have left?

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We're going to have 40
grams. So t equals 2.

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But after two more years, how
many are we going to have?

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We're going to have 20 grams.
So this is t equals 3 I'm

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sorry, this is t
equals 4 years.

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And then after two more years,
I'll only have half of that

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left again.

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So now I'm only going to
have 10 grams left.

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And that's where I am.

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And this is t equals 6.

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So if you know you have
some compound.

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You're starting off with 80
grams. You know it has a

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two-year half-life.

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You get in a time machine.

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And then you didn't build
your time machine well.

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00:11:47,130 --> 00:11:49,080
You don't know how well it
calibrates against time.

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00:11:49,080 --> 00:11:50,900
You just look at your sample.

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00:11:50,900 --> 00:11:52,470
You say, oh, I only have
10 grams left.

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00:11:52,470 --> 00:11:59,470
You know that 1, 2, 3 half-lives
have gone by.

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00:11:59,470 --> 00:12:01,000
And you could also think
about it this way.

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00:12:01,000 --> 00:12:04,760
1/2 to the 3rd power, because
every time you have 1/2 of the

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00:12:04,760 --> 00:12:08,100
original sample-- that's the
number of half-lives-- after

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00:12:08,100 --> 00:12:11,340
three half-lives you'll have 1/8
of your original sample.

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00:12:11,340 --> 00:12:12,450
And that's what we have here.

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00:12:12,450 --> 00:12:16,230
We have 1/8 of 80 grams. And
this is just when you're doing

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00:12:16,230 --> 00:12:18,140
it with a discreet you know,
when you're right at the

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00:12:18,140 --> 00:12:19,270
half-life point.

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00:12:19,270 --> 00:12:21,200
In the next video we're going to
explore what if I asked you

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00:12:21,200 --> 00:12:24,000
a question, how many of the
particles, or how many grams

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00:12:24,000 --> 00:12:26,300
will you have in exactly
10 days?

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00:12:26,300 --> 00:12:28,420
Or at two and a half years?

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00:12:28,420 --> 00:00:00,000
And we'll do that in
the next video.