More Exponential Decay Examples A few more examples of exponential decay.
More Exponential Decay Examples
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- SAL: Let's do a couple more of these exponential decay
- problems, because a lot of this really is just practice
- and being very comfortable with the general formula, and
- I'll write it again.
- Where the amount of the element that's decaying, that
- we have at any period in time, is equal to the amount that we
- started with, times e to the minus kt.
- Where the k value is specific to any certain element with a
- certain half-life, and sometimes they don't even give
- you the half-life.
- So let's try this interesting situation.
- Let's say that I have an element.
- Let me just give you a formula.
- Let's say that I have some magic element here, where its
- formula is, its k value I give to you, k is equal to minus,
- let me think of a-- [coughs]
- Excuse me, I just had a lot of walnuts and my throat is dry.
- Let's say that k is equal to, well k, we're putting a minus
- in front of it, so I'll say the k value
- is a positive 0.05.
- So its exponential decay formula would be the amount
- that you start off with, times e to the minus 0.05t.
- My question to you is, given this, what is the half-life of
- the compound that we're talking about?
- What is the half-life?
- So to figure that out, we need to figure out what t value can
- we put here, so that if we start off with whatever value
- here, we end up with 1/2 of that value there.
- So let's do that.
- So we're starting off with N sub 0 This is just some value,
- our initial starting point.
- We could put 100 there.
- Actually, let's do that, just to keep things less abstract.
- So let's say we start with 100.
- I'm just picking 100 out of air.
- I could have left it abstract with N.
- Let's say I'm starting with 100.
- And I take the 100 times e to the minus 0.05, times t.
- t is whatever our half-life.
- So after our half-life we're going to have 1/2 of this
- stuff left.
- So this should be equal to 50.
- We just solved for t.
- Divide both sides by 100.
- You get e to the minus 0.05t, is equal to 1/2.
- You take the natural log of both sides of this.
- The natural log of this, the natural log of that.
- And then you get-- the natural log of e to anything, I've
- said it before, is just the anything.
- So it is minus 0.05t is equal to the natural log of 1/2.
- And then you get t is equal to the natural log of 1/2,
- divided by minus 0.05.
- So let's figure out what that is.
- Actually, someone just made a comment, and I
- might as well do that.
- I could just put this minus up here.
- I could make this a plus, and this a minus, if I just
- multiply the numerator and the denominator by negative 1.
- And if I want to, just to make the calculator math a little
- easier, if you put a minus in front of a natural log, or any
- logarithm, that's the same thing as the log of the
- inverse of 2 over 0.05.
- It makes the calculator math a little bit easier.
- The same thing.
- So if I do 2 natural log, divided by 0.05,
- it is equal to 13.86.
- So when t is equal to 13.86.
- And I'm assuming that we're dealing with time in years.
- That tends to be the convention, although sometimes
- it could be something else and you'd always have
- to convert to years.
- But assuming that this original formula, where they
- gave this k value 0.05, that was with the assumption that t
- is in years, and I've just solved its half-life.
- I just solved that after 13.86 years, you can expect to have
- 1/2 of the substance left.
- We started with 100, we ended up with 50.
- I could have started with x and ended up with x over 2.
- Let's do one more of these problems, just so that we're
- really comfortable with the formula.
- Let's say that I have something with a half-life of,
- I don't know, let's say I have it as one month.
- Half-life of one month.
- And after, well let's say that I-- well let me just for the
- sake of time, let me make it a little bit simpler.
- Let's say I just have my k value is equal to-- I mean you
- can go from half-life to a k value, we did that in the
- previous video.
- Let's say my k value is equal to 0.001.
- So my general formula is the amount of product I have, is
- equal to the amount that I started with times e to the
- minus 0.001 times t.
- And I gave you this, if you have to figure it out from
- half-life, I did that in the previous video with carbon-14.
- But let's say this is the formula.
- And let's say that after, I don't know, let's say after
- 1000 years I have 500 grams of whatever element is described.
- The decay formula for whatever element is
- described by this formula.
- How much did I start off with?
- So essentially I need to figure out N sub 0, right?
- I'm saying that after 1000 years, so N of 1000, which is
- equal to N sub naught times e to the minus
- 0.001, times 1000.
- That's the N of 1000.
- And I'm saying that that's equal to 500 grams. That
- equals 500 grams. So I just have to
- solve for N sub naught.
- So what's the e value?
- So if I have 0.0001 times 1000, so this is N sub naught.
- This is 1/1000 of a 1000-- so times e to the minus 1 is
- equal to 500 grams. Or I could multiply both sides by e, and
- I have N sub naught is equal to 500e, which is about 2.71.
- So it's 500 times 2.71.
- I don't actually have e on this calculator or at least I
- don't see it.
- So we'll have 1,355 grams. So it's equal to 1,355 grams, or
- 1.355 kilograms. That's what I started with.
- So hopefully you see now.
- I mean, I think we've approached this pretty much at
- almost any direction that a chemistry test or teacher
- could throw the problem at you.
- But you really just need to remember this formula.
- And this applies to a lot of things.
- Later you'll learn, you know, when you do compound interest
- in finance, the k will just be a positive value, but it's
- essentially the same formula.
- And there's a lot of things that this formula actually
- describes well beyond just radioactive decay.
- But the simple idea is, use information they give you to
- solve for as many of these constants as you can.
- And then whatever they're asking for, solve for
- whatever's left over.
- And hopefully I've given you enough examples of that.
- But let me know, I'm happy to do more.
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