Hi. It’s Mr. Andersen and this is AP Physics
essentials 136. It is on half-life and radioactive
decay. To model that my daughter and I started
rolling some dice. We have 32 dice in the
cup. And what she would do is she would roll
the dice and then she would pull all of a
certain number out. So in this first simulation
she is going to pull all of the 1s out. And
so we put those to the side. That is generation
one. And now generation two. And generation
three. And generation four. Now when you get
to the fourth generation, she has hit her
first half life. What does that mean? Well
if there were 32 to start, that is the amount
of time it takes for half of those to decay.
So that would be 16. So 16 have decayed right
here. So she is going to keep doing that.
And so if we watch what happens in the rest
of the simulation you are not pulling as many
dice because there are not as many left. And
eventually they all have decayed after 19
generations. But what you can see is that
the half life is consistent. So this was to
16 and then to 8 and then to 4 and then to
2 and then to 1. And so radioactive nuclei
will decay. What does that mean? They are
going to quick off some kind of a particle,
it could be alpha, beta or gamma. And then
they are going to form a new, usually more
stable nuclei. And so in this radioactive
decay, mass and energy are conserved. We have
talked about that in previous videos. But
what we are going to talk about here is what
is the probability of that decay occurring?
Well it is chance. So we never know when the
next one is going to decay. But what we can
use is the law of large numbers to calculate
the half-life. The half-life is the amount
of time it takes for half of the radioactive
nuclei to decay. In other words if this represents
all of the radioactive nuclei that would be
the time it takes for half of them to decay.
Now we have all of these ones left. And so
we would have another half life like that.
We could just keep going and going and going
like that. And so there is an equation we
can use to figure out how many are going to
decay at each generation. And so delta N,
which is the change in the radioactive nuclei,
so N is going to be our radioactive nuclei.
So the change in N is minus, because we are
losing those nuclei, so negative decay constant,
I will get to that in a second, times N which
is the number of radioactive nuclei times
time. And so if we kind of work backwards
for that, in this simulation here our time
was advancing 1 generation after another.
And so that is going to be 1 each time. What
was our decay constant? Well it is 1 in 6.
So it is a one-sixth probability that you
are going to roll a 1 and that that one is
going to decay. And so we could model that.
I will just use a quick little spreadsheet
to do that. So at time 0 how many radioactive
nuclei did we have? Well our N value was 32.
So let me walk you through this formula. What
is our time going to be? It is going to be
1. What is going to be our N value? It is
32. So 1 times 32. And then what is our decay
constant? It is one-sixth. And so it is going
to be negative one-sixth times 32. So what
is that value? It is going to be negative
5.33. Now if we look, how many actually decayed?
It is 6, but it is close, -5.33. This is what
it would predict to be and this is what we
actually got with a really small number of
dice. Now how do you do the next one? Well
what we have to do is we have to take that
32, these ones decayed, so we are going to
take 32 minus 5.33 and that is my N value
for the next generation. And so now what do
I do? I go back to this formula again. So
it is going to be T, which is 1 times my new
N which is going to be 26.67 times my decay
constant, which remains constant, and so what
I am going to get is negative 4.45. Now how
does that match up? It is pretty close to
this. So then we subtract that value like
that and we could just do this over and over
again. So right here on the right is going
to be what we would predict to occur. And
this is what actually occurred in this little
simulation. And the number of dice are so
small compared to the number of nuclei in
a sample. But if you look at my data, the
green line represents the actual data that
we find. The red line represents the predicted.
And so you can see that it matches up pretty
quickly. What would happen if we changed the
decay constant. What if we changed it from
one-sixth to one-half. So how would you do
that in the modeling? Instead of pulling 1s
out, she is going to pull the 1s, 3s and 5s
out. So what is going to happen? Well it is
going to happen more quickly. So more of them
are going to decay at each point. And so it
is going to take less time for all of them
to decay. In other words our half-life has
gotten much much shorter. And so you should
be able to calculate half-life. So how long
would it be for half of them decay? You can
see it is going to be somewhere around 4 generations.
And so when you look at a curve, the first
thing you want to do is figure out how long
does it take to go from 100 radioactive nuclei
to 50 percent of that. And so if I look across
here it took 1 year for 50 percent of them
to decay. So let’s watch the next generation.
So now we should go to 25. You can see it
is consistent. 1 year. 1 year. 1 year. And
so you could say for this perfect model it
is going to be a half-life of 1 year. But
on a test you are more likely to get something
like this. Calculate the half-life decay of
carbon 14. So if you are given this curve
right here, we go from 100 to 50, so you could
count across like that. So this is 50 percent
of the carbon 14. And then you just read the
time on the bottom. So if this right here
is 10,000 years, what is our radioactive half-life?
It is around 6,000 years. But let’s keep
going. So now let’s go down to 25 percent
and you can see it is around 12,000. And so
it is pretty consistent over time. Now each
form of radioactive decay is going to kick
off a different particle. Let’s start with
alpha particles. An alpha particle is 2 protons
and 2 neutrons. So if we look at the alpha
decay of uranium 238, let’s make sure that
the mass is conserved. And so mass on the
left side is 238, mass on the right side is
238. So we are fine there. Let’s make sure
charge is conserved. 92 positive charge on
the left. 92 positive charge on the right.
So that is conserved as well. Now what is
the half-life of uranium 238 decay? It is
4.47 billion years. It takes a huge amount
of time for just half of the uranium 238 to
decay. But there are so many nuclei that we
can actually measure this. And this is how
we determine the age of the earth. Let’s
look at beta decay. Beta decay we converted
a neutron to a proton. We also give off an
electron and an electron antineutrino. So
if we look across the top, mass is conserved,
14 and 14. Charge is conserved. On the left
side we have 6 positive charges. On the right
side we have 7 positive, 1 negative. And so
we have 6 positive charges on the right as
well. What is the half-life of carbon 14 decay?
It is going to be 5730 years. So we had shown
that just a few slides ago. It was around
6000 for half-life. And we can use this to
date living material. And then we could look
at gamma decay. Remember in gamma decay we
are just giving off these gamma rays. We are
going from barium 137 that is charged to barium
137 that is not charged. And so we are conserving
charge and conserving mass. What is the half-life?
2.6 minutes. So it is really really short.
And so half-life is going to change depending
on that decay constant. But you should be
able to take a graph like this. Figure out
the half-life. And I hope that was helpful.
