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SAL: Let's do a couple more
of these exponential decay

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problems, because a lot of this
really is just practice

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and being very comfortable with
the general formula, and

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I'll write it again.

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Where the amount of the element
that's decaying, that

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we have at any period in time,
is equal to the amount that we

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started with, times
e to the minus kt.

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Where the k value is specific to
any certain element with a

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certain half-life, and sometimes
they don't even give

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you the half-life.

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So let's try this interesting
situation.

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Let's say that I have
an element.

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Let me just give
you a formula.

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Let's say that I have some magic
element here, where its

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formula is, its k value I give
to you, k is equal to minus,

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let me think of a-- [coughs]

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Excuse me, I just had a lot of
walnuts and my throat is dry.

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Let's say that k is equal to,
well k, we're putting a minus

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in front of it, so I'll
say the k value

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is a positive 0.05.

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So its exponential decay formula
would be the amount

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that you start off with, times
e to the minus 0.05t.

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My question to you is, given
this, what is the half-life of

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the compound that we're
talking about?

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What is the half-life?

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So to figure that out, we need
to figure out what t value can

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we put here, so that if we start
off with whatever value

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here, we end up with 1/2
of that value there.

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So let's do that.

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So we're starting off with N sub
0 This is just some value,

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our initial starting point.

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We could put 100 there.

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Actually, let's do that, just to
keep things less abstract.

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So let's say we start
with 100.

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I'm just picking
100 out of air.

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I could have left it
abstract with N.

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Let's say I'm starting
with 100.

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And I take the 100 times e to
the minus 0.05, times t.

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t is whatever our half-life.

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So after our half-life we're
going to have 1/2 of this

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stuff left.

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So this should be equal to 50.

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We just solved for t.

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Divide both sides by 100.

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You get e to the minus 0.05t,
is equal to 1/2.

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You take the natural log
of both sides of this.

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The natural log of this, the
natural log of that.

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And then you get-- the natural
log of e to anything, I've

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said it before, is just
the anything.

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So it is minus 0.05t is equal
to the natural log of 1/2.

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And then you get t is equal
to the natural log of 1/2,

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divided by minus 0.05.

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So let's figure out
what that is.

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Actually, someone just
made a comment, and I

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might as well do that.

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I could just put this
minus up here.

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I could make this a plus, and
this a minus, if I just

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multiply the numerator and the
denominator by negative 1.

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And if I want to, just to make
the calculator math a little

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easier, if you put a minus in
front of a natural log, or any

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logarithm, that's the same
thing as the log of the

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inverse of 2 over 0.05.

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It makes the calculator math
a little bit easier.

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The same thing.

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So if I do 2 natural log,
divided by 0.05,

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it is equal to 13.86.

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So when t is equal to 13.86.

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And I'm assuming that we're
dealing with time in years.

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That tends to be the convention,
although sometimes

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it could be something else
and you'd always have

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to convert to years.

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But assuming that this original
formula, where they

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gave this k value 0.05, that was
with the assumption that t

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is in years, and I've just
solved its half-life.

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I just solved that after 13.86
years, you can expect to have

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1/2 of the substance left.

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We started with 100, we
ended up with 50.

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I could have started with x and
ended up with x over 2.

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Let's do one more of these
problems, just so that we're

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really comfortable
with the formula.

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Let's say that I have something
with a half-life of,

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I don't know, let's say I
have it as one month.

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Half-life of one month.

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And after, well let's say that
I-- well let me just for the

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sake of time, let me make
it a little bit simpler.

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Let's say I just have my k value
is equal to-- I mean you

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can go from half-life to a k
value, we did that in the

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previous video.

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Let's say my k value
is equal to 0.001.

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So my general formula is the
amount of product I have, is

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equal to the amount that I
started with times e to the

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minus 0.001 times t.

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And I gave you this, if you
have to figure it out from

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half-life, I did that in the
previous video with carbon-14.

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But let's say this
is the formula.

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And let's say that after, I
don't know, let's say after

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1000 years I have 500 grams of
whatever element is described.

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The decay formula for
whatever element is

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described by this formula.

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How much did I start off with?

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So essentially I need to figure
out N sub 0, right?

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I'm saying that after 1000
years, so N of 1000, which is

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equal to N sub naught
times e to the minus

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0.001, times 1000.

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Right?

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That's the N of 1000.

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And I'm saying that that's
equal to 500 grams. That

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equals 500 grams. So
I just have to

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solve for N sub naught.

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So what's the e value?

115
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So if I have 0.0001 times 1000,
so this is N sub naught.

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This is 1/1000 of a 1000-- so
times e to the minus 1 is

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equal to 500 grams. Or I could
multiply both sides by e, and

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I have N sub naught is equal to
500e, which is about 2.71.

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So it's 500 times 2.71.

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I don't actually have e on this
calculator or at least I

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don't see it.

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So we'll have 1,355 grams. So
it's equal to 1,355 grams, or

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1.355 kilograms. That's
what I started with.

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So hopefully you see now.

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I mean, I think we've approached
this pretty much at

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almost any direction that a
chemistry test or teacher

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could throw the problem
at you.

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But you really just need to
remember this formula.

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And this applies to
a lot of things.

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Later you'll learn, you know,
when you do compound interest

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in finance, the k will just be
a positive value, but it's

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essentially the same formula.

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And there's a lot of things
that this formula actually

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describes well beyond just
radioactive decay.

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But the simple idea is, use
information they give you to

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solve for as many of these
constants as you can.

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And then whatever they're
asking for, solve for

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whatever's left over.

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And hopefully I've given you
enough examples of that.

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But let me know, I'm
happy to do more.