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### Course: Precalculus (Eureka Math/EngageNY) > Unit 3

Lesson 15: Topic C: Lessons 20-21: Inverse relationship of exponentials and logarithms- Relationship between exponentials & logarithms
- Relationship between exponentials & logarithms: graphs
- Relationship between exponentials & logarithms: tables
- Relationship between exponentials & logarithms
- Half-life and carbon dating
- Exponential model word problem: medication dissolve
- Exponential model word problem: bacteria growth
- Exponential model word problems

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# Half-life and carbon dating

Carbon dating is a real-life example of a first-order reaction. This video explains half-life in the context of radioactive decay. Created by Sal Khan.

## Want to join the conversation?

- If all C-14 will eventually become nitrogen, then why is there C-14 at the first place? and why does it still exist?(43 votes)
- Because more C-14 is being created all the time.(6 votes)

- Can hydrogen with an atomic mass of 1 decay?(22 votes)
- No, it is stable. But a radioactive isotope of hydrogen called Tritium with an atomic mass of 3 can give a Helium, an electron and an anti-neutrino.(37 votes)

- I have read several estimates for the "half-life" of a proton. The longest I read was at least 10 to the 109th years. I would like to verify an "at least this long" and a competent source for the half-life of a proton, Thank you!(12 votes)
- To the best of my knowledge, proton decay (not meaning β⁺) is entirely hypothetical and has never been observed despite multiple efforts to do so. I would not put much credence in any such conjecture. It might be the case that the 13.8 billion or so years the universe has existed in its current form is just far too short of a time for protons to decay. Or, maybe they never decay. Who knows. But, if it happens at all, it would appear to be an exceedingly rare event -- perhaps even more rare than Big Bang events.(18 votes)

- I am learning about half--lifes and this video explains pretty well but I am still confused on the overall picture, could someone please explain this to me in a easier sense? It would really help me on the half-life quiz we're about to have.(4 votes)
- At roughly8:20or so, Sal mentions that after the half-life is up, we're left with half and half (half old element/half new element). He then implies that in another set number of years (the half-life), the remaining concentration of the original element will have a probability of "changing" via beta decay, ending in 75% of the original concentration being the "new" element by the end of this 2nd half-life.

My question: If this is all based on probabilities, why do we press the figurative "reset" button once the half-life number of years has been achieved? In other words, don't the remaining original elements have an even more likely chance of decaying sooner than later, because they've already been waiting for their turn to decay (for the original amount of years it took to achieve a half-life)?? I'm envisioning a bell-curve here, where the chance at decaying becomes exponential as one (an element) misses the "average" time-line of a decay...

Hope this makes sense...thanks!(1 vote)- It is possible to determine the probability that a single atomic nucleus will "survive" during a given interval. This probability amounts to 50% for one half-life. In an interval twice as long (2 T) the nucleus survives only with a 25% probability (half of 50%), in an interval of three half-life periods (3 T) only with 12.5% (half of 25%), and so on.

You can't, however, predict the time at which a given atomic nucleus will decay. For example, even if the probability of a decay within the next second is 99%, it is nevertheless possible (but improbable) that the nucleus will decay only after millions of years.

It’s like flipping a coin. If you flip “heads” ten times in a row, what are the chances that the next flip will be “heads”. You might say, “I’ve flipped ten heads in a row. The next one is much more likely to come up “tails”. Nevertheless, your chances of flipping heads are the same as before: 1 out of 2.

Some nuclei are much luckier at flipping the coin than others. They keep flipping until they get “tails”; then they decay.(4 votes)

- How is the half-life of Carbon-14 calculated at 5,740 years when no one has lived that long to actually measure the amount left? What is the technique that actually establishes the half-life time duration?(4 votes)
- The half-life of a radioactive element is the time it takes before half of the atoms in a sample of the element have decayed.

If you know how many atoms you have in a sample, and you measure how many of them decay per second, it is easy to figure out how long you would have to wait before half of all the atoms have decayed. You do not have to measure until half the atoms actually have decayed, but the tradition is that we use the half-life as a measure of how quickly a radioactive element decays.(4 votes)

- After a half-life, half of 10g of C-14 turns into N-14 like Sal shown above and we have 5g of C-14 and 5g of N-14. So after another half-life, half of 5g of C-14 turns into N-14, but what will the 5g of N-14 turns into?(3 votes)
- It does not turn into anything else. It remains the same. It is stable.

Hope this helps :)(4 votes)

- How do they turn into nitrogen?(3 votes)
- When an atom loses or gains protons in its' nucleus, it changes what type of element it is.

In this case in particular of beta decay, a neutron becomes a proton in the carbon atom and ejects an electron. The new atom has 1 proton more (the number of neutrons does not determine the type of atom) and thus become the element with one more proton than carbon --> which is nitrogen.

So in a similar way, for alpha decay, the nucleus ejects two proton and two neutrons as an alpha particle, then the nucleus has changed its' composition and due to the loss of protons, the atom will be the element with two fewer protons.(4 votes)

- At5:17the video states that indiviudal C14 atoms don't know when to change, and that it's ultimately up to random chance when a certain atom decays. How then are we able to accurately determine a specific rate of decay for a large mass of atoms if each atom's chance of decay is random?(2 votes)
- The laws of chance work very precisely when you are dealing with billions and billions of atoms.

In the same way, if you flipped a coin a billion billion times, it would come up heads almost exactly half the time.(4 votes)

- How long does it take them to actually change the nucleus composition?

Is it a gradual change over 5740 years or an instantaneous change after each half-life?(3 votes)- It does not work like that at all. Each particular atom decays more or less instantly at a random, unpredictable time. However, when you have a large number of particles, since we know the odds of any atom decaying in a given amount of time, we can determine how long it will take for about half of the atoms to decay.

It is much like a roll of the dice. You have no idea what number each particular roll will produce, but you can predict the outcome of millions of rolls based on statistics. You can never predict which rolls will produce a six and which will produce a three, but you can predict how many sixes and how many threes will be thrown if there are a very large number of rolls.

It is the same thing with radioactive decay. You cannot in anyway predict just when any particular atom will decay. But, you can predict how many out of a large number of particles will decay in a given time -- but you cannot predict which of those particles will decay, just how many will decay in total.(1 vote)

## Video transcript

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.