SAL: In the last video we saw

all sorts of different types of isotopes of atoms

experiencing radioactive decay and turning into other atoms or

releasing different types of particles. But the question is, when

does an atom or nucleus decide to decay? Let’s say I have a bunch of,

let’s say these are all atoms. I have a bunch of atoms here. And let’s say we’re talking

about the type of decay where an atom turns into

another atom. So your proton number

is going to change. Your atomic number is

going to change. So it could either be beta

decay, which would release electrons from the neutrons and

turn them into protons. Or maybe positron emission

turning protons into neutrons. But that’s not what’s

relevant here. Let’s say we have a collection

of atoms. And normally when we have any small amount of any

element, we really have huge amounts of atoms of

that element. And we’ve talked about moles

and, you know, one gram of carbon-12– I’m sorry, 12

grams– 12 grams of carbon-12 has one mole of carbon-12

in it. One mole of carbon-12. And what is one mole

of carbon-12? That’s 6.02 times 10 to the 23rd

carbon-12 atoms. This is a ginormous number. This is more than we can, than

my head can really grasp around how large of

a number this is. And this is only when we have

12 grams. 12 grams is not a large mass. For example, one kilogram

is about two pounds. So this is about, what? I want to say [? 1/50 ?] of a pound if I’m

doing [? it. ?] But this is not a lot

of mass right here. And pounds is obviously force. You get the idea. On Earth, well anywhere,

mass is invariant. This is not a tremendous

amount. So with that said, let’s go back

to the question of how do we know if one of these

guys are going to decay in some way. And maybe not carbon-12, maybe

we’re talking about carbon-14 or something. How do we know that they’re

going to decay? And the answer is, you don’t. They all have some probability

of the decaying. At any given moment, for a

certain type of element or a certain type of isotope of

an element, there’s some probability that one

of them will decay. That, you know, maybe this guy

will decay this second. And then nothing happens for a

long time, a long time, and all of a sudden two

more guys decay. And so, like everything in

chemistry, and a lot of what we’re starting to deal with in

physics and quantum mechanics, everything is probabilistic. I mean, maybe if we really

got in detail on the configurations of the nucleus,

maybe we could get a little bit better in terms of our

probabilities, but we don’t know what’s going on inside of

the nucleus, so all we can do is ascribe some probabilities

to something reacting. Now you could say, OK, what’s

the probability of any given molecule reacting

in one second? Or you could define

it that way. But we’re used to dealing with

things on the macro level, on dealing with, you know, huge

amounts of atoms. So what we do is we come up with terms

that help us get our head around this. And one of those terms is

the term half-life. And let me erase this

stuff down here. So I have a description, and

we’re going to hopefully get an intuition of what

half-life means. So I wrote a decay reaction

right here, where you have carbon-14. It decays into nitrogen-14. And we could just do a

little bit of review. You go from six protons

to seven protons. Your mass changes the same. So one of the neutrons must have

turned into a proton and that is what happened. And it does that by releasing

an electron, which is also call a beta particle. We could have written this

as minus 1 charge. Relatively zero mass. It does have some mass,

but they write zero. This is kind of notation. So this is beta decay. Beta decay, this is

just a review. But the way we think about

half-life is, people have studied carbon and they said,

look, if I start off with 10 grams– if I have just a block

of carbon that’s 10 grams. If I wait carbon-14’s half-life–

this is a specific isotope of carbon. Remember, isotopes, if there’s

carbon, can come in 12, with an atomic mass number of 12, or

with 14, or I mean, there’s different isotopes of

different elements. And the atomic number

defines the carbon, because it has six protons. Carbon-12 has six protons. Carbon-14 has six protons. But they have a different

number of neutrons. So when you have the same

element with varying number of neutrons, that’s an isotope. So the carbon-14 version, or

this isotope of carbon, let’s say we start with 10 grams. If

they say that it’s half-life is 5,740 years, that means that

if on day one we start off with 10 grams of pure

carbon-14, after 5,740 years, half of this will

have turned into nitrogen-14, by beta decay. And you might say, oh OK, so

maybe– let’s see, let me make nitrogen magenta, right there–

so you might say, OK, maybe that half turns

into nitrogen. And I’ve actually seen this

drawn this way in some chemistry classes or physics

classes, and my immediate question is how does this

half know that it must turn into nitrogen? And how does this half know that

it must stay as carbon? And the answer is

they don’t know. And it really shouldn’t

be drawn this way. So let me redraw it. So this is our original block

of our carbon-14. What happens over that

5,740 years is that, probabilistically, some of these

guys just start turning into nitrogen randomly,

at random points. And over 5,740 years, you

determine that there’s a 50% chance that any one of these

carbon atoms will turn into a nitrogen atom. So that after 5,740 years, the

half-life of carbon, a 50% chance that any of the

guys that are carbon will turn to nitrogen. So if you go back after a

half-life, half of the atoms will now be nitrogen. So now you have, after

one half-life– So let’s ignore this. So we started with this. All 10 grams were carbon. 10 grams of c-14. This is after one half-life. And now we have five

grams of c-14. And we have five grams

of nitrogen-14. Fair enough. Let’s think about what happens

after another half-life. Well we said that during a

half-life, 5,740 years in the case of carbon-14– all

different elements have a different half-life, if they’re

radioactive– over 5,740 years there’s a 50%– and

if I just look at any one atom– there’s a 50%

chance it’ll decay. So if we go to another

half-life, if we go another half-life from there, I had

five grams of carbon-14. So let me actually copy

and paste this one. This is what I started with. Now after another half-life–

you can ignore all my little, actually let me erase some

of this up here. Let me clean it up

a little bit. After one one half-life,

what happens? Well I now am left with five

grams of carbon-14. Those five grams of carbon-14,

every one of those atoms still has, over the next– whatever

that number was, 5,740 years– after 5,740 years,

all of those once again have a 50% chance. And by the law of large numbers,

half of them will have converted into

nitrogen-14. So we’ll have even more

conversion into nitrogen-14. So now half of that five grams.

So now we’re only left with 2.5 grams of c-14. And how much nitrogen-14? Well we have another two and

a half went to nitrogen. So now we have seven and a half

grams of nitrogen-14. And we could keep going further

into the future, and after every half-life, 5,740

years, we will have half of the carbon that we started. But we’ll always have an infinitesimal amount of carbon. But let me ask you a question. Let’s say I’m just staring

at one carbon atom. Let’s say I just have this

one carbon atom. You know, I’ve got its nucleus,

with its c-14. So it’s got its six protons. 1, 2, 3, 4, 5, 6. It’s got its eight neutrons. It’s got its six electrons. 1, 2, 3, 4, 5, 6, whatever. What’s going to happen? What’s going to happen

after one second? Well, I don’t know. It’ll probably still be carbon,

but there’s some probability that after one

second it will have already turned into nitrogen-14. What’s going to happen after

one billion years? Well, after one billion years

I’ll say, well you know, it’ll probably have turned into

nitrogen-14 at that point, but I’m not sure. This might be the one

ultra-stable nucleus that just happened to, kind of,

go against the odds and stay carbon-14. So after one half-life, if

you’re just looking at one atom after 5,740 years, you

don’t know whether this turned into a nitrogen or not. This exact atom, you just know

that it had a 50% chance of turning into a nitrogen. Now, if you look at it over a

huge number of atoms. I mean, if you start approaching, you

know, Avogadro’s number or anything larger–

I erased that. Then all of a sudden you can use

the law of large numbers and say, OK, on average, if each

of those atoms must have had a 50% chance, and if I have

gazillions of them, half of them will have turned

into nitrogen. I don’t know which half,

but half of them will turn into it. So you might get a question

like, I start with, oh I don’t know, let’s say I start with

80 grams of something with, let’s just call it x, and it has

a half-life of two years. I’m just making up

this compound. A two-year half-life. And then let’s say we go into

a time machine and we look back at our sample, and let’s

say we only have 10 grams of our sample left. And we want to know how much

time has passed by. So 10 grams left of x. How much time, you know, x is

decaying the whole time, how much time has passed? Well let’s think about it. We’re starting at time, 0 with

80 grams. After two years, how much are we going

to have left? We’re going to have 40

grams. So t equals 2. But after two more years, how

many are we going to have? We’re going to have 20 grams.

So this is t equals 3 I’m sorry, this is t

equals 4 years. And then after two more years,

I’ll only have half of that left again. So now I’m only going to

have 10 grams left. And that’s where I am. And this is t equals 6. So if you know you have

some compound. You’re starting off with 80

grams. You know it has a two-year half-life. You get in a time machine. And then you didn’t build

your time machine well. You don’t know how well it

calibrates against time. You just look at your sample. You say, oh, I only have

10 grams left. You know that 1, 2, 3 half-lives

have gone by. And you could also think

about it this way. 1/2 to the 3rd power, because

every time you have 1/2 of the original sample– that’s the

number of half-lives– after three half-lives you’ll have 1/8

of your original sample. And that’s what we have here. We have 1/8 of 80 grams. And

this is just when you’re doing it with a discreet you know,

when you’re right at the half-life point. In the next video we’re going to

explore what if I asked you a question, how many of the

particles, or how many grams will you have in exactly

10 days? Or at two and a half years? And we’ll do that in

the next video.

12:08 Confirmed