Electricity is generated in a nuclear power station. Fission is the process by which energy is released in the nuclear reactor. Figure one shows a uranium-235 nucleus undergoing nuclear fission. How many free neutrons are present after a single uranium-235 nucleus undergoes this
Okay, so let’s take a quick look at figure one, which shows us a neutron being
absorbed by a uranium-235 nucleus to produce a barium nucleus, a krypton nucleus,
and some more neutrons. In other words, these are the reactants — the neutron and the uranium nucleus — and
these are the products — the barium, the krypton, and the more neutrons.
We’ve been asked to figure out how many free neutrons there are present after a
single uranium-235 nucleus undergoes this reaction. In other words, we know that there are more neutrons, but how many neutrons,
specifically after this one single uranium nucleus undergoes the reaction?
Now to answer this, we need to remember that, in a nuclear reaction, the total mass
number is conserved. So what does this mean? Well, the total mass number of the reactants has to be the same as the total mass
number of the products. In other words, the total number of protons and neutrons before the reaction has
occurred is equal to the total number of protons and neutrons after the reaction has
Now this statement of mass number conservation is basically just a statement talking
about the conservation of mass. In other words, the amount of stuff that we had before the reaction is the same as
the amount of stuff that we have after the reaction, because stuff can’t just
disappear, right? So what’s the total mass number before the reaction?
Well, before the reaction, we’ve got the neutron and the uranium nucleus. Now the uranium clearly has a mass number of 235. We can see this. And of course, a neutron has a mass number of one, because of course the mass number
of an element is a measure of the number of protons and neutrons in the nucleus of
the atom. Now of course, a neutron by itself is either a proton or a neutron. It’s a neutron. And therefore, it has a mass number of one.
So before the reaction, we have a total mass number of one plus 235, and so the total
mass number of the reactants is 236. This means that the products must also have a total mass number of 236. So what’s the total mass number of the products?
Well, the total mass number of the products is the mass number of barium — that’s 144
— plus the mass number of Krypton — that’s 89 — plus the number of neutrons that
we’ve got in these more neutrons, because each neutron has a mass number of one as
we’ve already seen. In other words then, 236 — that’s the mass number of the reactants — is equal to 144
— that’s the mass number of the barium atom that we have — plus 89 — that’s the mass
number of the krypton that we have — plus we’ll call this 𝑥 — that’s the number of
neutrons that we have. And now all we have to do is to rearrange so that we solve for 𝑥.
We can subtract 144 and 89 from both sides of the equation so that, on the right-hand
side, we just have 𝑥 left. And then on the left, we’ve got 236 minus 144 minus 89. This leaves us with three is equal to 𝑥. Or in other words, three free neutrons are present after a single uranium-235 nucleus
undergoes this reaction. So let’s replace more neutrons with three neutrons.
Now interestingly, these three neutrons can go on to interact with other uranium
nuclei just like this neutron did. Let’s see what happens in that situation. What is the maximum number of uranium-235 nuclei that the neutrons produced in this
reaction could cause to undergo nuclear fission?
So we’ve seen that three neutrons are produced in this reaction, so here they are:
one, two, three. Now each one of these neutrons could behave, as we said earlier, just like this
neutron and interact with a uranium nucleus each. So in other words, the maximum number of uranium nuclei that these three neutrons
could cause to decay is if each one of them interacted with a uranium nucleus. So our three neutrons could cause three uranium nuclei to undergo nuclear
fission. But if that were to happen, what would be the consequences? Well, let’s find out.
If each neutron produced in this reaction causes another uranium-235 nucleus to
undergo nuclear fission, producing the same fission products, what is the total
number of neutrons that will be produced?
Okay, so what this question is saying is that the three neutrons that we’ve produced
here do end up interacting with three other uranium nuclei. And this results in each one of these uranium nuclei decaying exactly like this one
did. In other words, this nucleus decays to produce a barium nucleus, a krypton nucleus,
and three neutrons, as does this one and as does this one. So in this case, what are the total number of neutrons that have been produced?
Well, this nucleus produces three neutrons, this nucleus produces three more
neutrons, and this one produces three neutrons as well. Therefore, the total number of neutrons produced is three plus three plus three, or
nine. So if each neutron produced in this reaction causes another uranium-235 nucleus to
undergo nuclear fission, then, in total, nine neutrons are produced.
Okay, so up until now, we’ve been looking at this one particular reaction at the
atomic scale. But let’s see what happens when we multiply this reaction by a few million and
harness it in a nuclear reactor.
Figure two shows the inside of a nuclear fission reactor in a nuclear power
station. Describe the action of the control rods in a nuclear fission reactor.
So that’s these rods here, the control rods. Now to answer this question, we first need to look back at the previous part of the
question. We saw that one neutron interacting with one uranium nucleus could produce something
like three neutrons. And then each one of those neutrons could interact with other uranium-235 nuclei,
producing nine neutrons. And then those nine neutrons could go on to interact with other uranium nuclei, and
so on and so forth.
Now each time this reaction happens, each time a neutron interacts with a uranium
nucleus, some energy is released. That’s how energy is generated in a nuclear fission reactor. However, as we can see, the number of neutrons that are produced can very quickly get
out of control. And therefore, we could very quickly have all of the uranium that’s present in fuel
rods undergoing nuclear fission because of the massive number of neutrons that can
be very, very quickly produced, because the problem is, all of the energy that gets
generated in one of these reactions gets transferred to the cold water that’s
entering the reactor. And this energy ends up heating the water. The hot water then leaves the system with all of this energy. And then all of that energy gets converted into electricity later on at the power
However, it’s not safe to have ridiculous numbers of uranium nuclei decaying as
quickly as possible, because essentially this can cause an explosion and, along with
that, the distribution of radioactive material all over the place basically. Or we could transfer energy so quickly to the water that it becomes steam very, very
fast. And once again, this could lead to an explosion.
So at this point, we get the idea. We need to be able to control how many uranium nuclei are decaying every second or
every minute. And the way we do that is with control rods, because what control rods do is to
absorb some of the neutrons that are flying about in the reactor.
So at this point in the reaction, we had nine neutrons. However, the control rods might absorb a few. So let’s say they absorb this one and this one and this one and this one. So now instead of having nine neutrons that could potentially interact with nine
uranium nuclei, we’ve only got five, a much safer number. And we can actually control how many neutrons get absorbed. The way to do this is to either lower the control rods into the reactor or raise them
up out of the reactor.
Now the lower we push them in, the more neutrons they will absorb, because remember,
the nuclear reactions are going on in this region around the fuel rods, which means
lots of neutrons are being released all over the place but especially near the fuel
rods. Now if we push the control rods in, then they can block the flow of some of these
neutrons and actually absorb them. This means that fewer neutrons get across to other fuel rods and so we can control
how quickly our supply of uranium-235 in each one of these fuel rods gets
depleted. Or in other words, we can control the number of reactions happening every minute or
every second. And so we can control the amount of energy being transferred to the cold water every
minute or second.
Conversely, if our reaction is not happening fast enough, then we pull the control
rods up out of the reactor. This way, fewer neutrons are absorbed, so more of them can interact with the uranium
nuclei in the fuel rods. Thus, we have more reactions every minute or second and more power being transferred
to the cold water.
So to summarize then, we can firstly say that control rods absorb neutrons. Therefore, lowering the rods into the nuclear reactor reduces the number of neutrons
available for fission. This decreases the amount of energy transferred to the water running through the
nuclear reactor. Conversely, raising the rods out of the reactor increases the number of neutrons
available for fission. This increases the amount of energy transferred to the water. And so this is the action of the control rods in a nuclear fission reactor.
Let’s look at this in more quantitative detail though. Figure three shows how the power output of a nuclear fission reactor would change if
control rods were lowered further into the reactor core. Calculate the rate of decrease of power output at four minutes. And from the answer blank, we can see that our answer needs to be in megawatts per
So first of all, we’ve been given a graph that shows the power output of the nuclear
reactor against time. Now we’ve also been told that as time goes on, we lower the control rods further into
the reactor core. In other words, at the beginning, the rods aren’t very low down into the reactor core
at all. And as time progresses, we steadily lower them down further and further.
Now the shape of this graph actually aligns well with what we discussed earlier. We said that as we lower the control rods into the reactor, fewer neutrons are
available for fission. Therefore, fewer nuclear reactions happen every second or every minute. And so the power output of the nuclear reactor is lower. We can see that as the rods get lowered, the power output goes from very high to
Now also we’ve been given the power output in megawatts, and the time on the
horizontal axis has been given to us in minutes. We’re also trying to calculate the rate of decrease of power output. In other words, what is the rate of change of power output?
Now we can recall that the rate of change of something, such as, for example, the
power output, is equal to the change of that thing divided by the time taken for
that change to occur. Now because in our graph we’ve got power output on the vertical axis and time on the
horizontal axis, essentially what we’re trying to do here is to calculate the
gradient of this curve. But what does that even mean? Doesn’t the gradient of the curve change, like it’s very steep here and very shallow
Well, yes it does, but we’ve been asked to calculate the gradient of this curve at a
very specific point, when the time is four minutes. In other words, we need to calculate the rate of change of the power output at four
minutes. So what’s the gradient of the curve at this point here?
Well, to answer that, we actually need to draw a tangent to the curve, because this
way we can work out the gradient of that tangent, which tells us the gradient of the
curve at that very specific point, because essentially a tangent is just a straight
line that has the same gradient as the curve at the point at which the tangent is
drawn. So here is our tangent drawn by eye.
Now it’s worth noting that a tangent also only touches the curve at one specific
point. This is the only way that the tangent can have the same gradient as the curve at that
So anyway, to find the gradient of the curve at four minutes, let’s find the gradient
of this tangent. Let’s say we’re working with this point over here and this point over here to find
the gradient. Now we can recall that, in order to find the gradient of a straight line, we need to
find the difference in the power output of the two points that we’ve already
mentioned, or in other words the difference in the vertical coordinates of the two
points. And we need to divide this by the difference in the horizontal coordinates of the two
points. And our graphical way to see this is that we need to divide this distance — that’s
the difference in the vertical coordinates — by this distance — that’s the
difference in the horizontal coordinates.
So now when finding the difference between the vertical and horizontal coordinates,
we’ll do the vertical coordinate of this one minus the vertical coordinate of this
one, and same with the horizontal coordinates. We can see then that this bottom-right point has a vertical coordinate of 300
megawatts. And from this, we can subtract the vertical coordinate here, which is 1950
megawatts. And then we do the same thing for the denominator. The horizontal coordinate of this point goes down to eight minutes. And from this, we subtract the horizontal coordinate of the top-left point, which is
Now before we simplify the right-hand side of this equation, we can see that the
numerator is some number of megawatts minus some number of megawatts and the
denominator is some number of minutes minus some number of minutes.
And so what we end up getting is negative 1650 megawatts in the numerator. Let’s ignore the negative just for a second. And in the denominator, we’ve got eight minutes. In other words, we’ve got a unit of megawatt per minute, which is exactly what we
want our answer to be.
Now let’s actually evaluate negative 1650 divided by eight. When we do this, we end up getting negative 206.25 megawatts per minute. So why is the answer negative? Well, we just calculated the gradient of this curve at four minutes. In other words, we’ve calculated the rate of change of the power output. And the rate of change of the power output is clearly negative. We can see that the power output is decreasing as time goes by, because it starts up
very high and then it decreases to a lower value. And so this is why the gradient is negative.
However, this is not actually our answer because we’ve been asked to calculate the
rate of decrease, not just the rate of change. And so the rate of decrease is simply 206.25 megawatts per minute, because the fact
that we’ve been asked to calculate the rate of decrease already means that we’ve
taken this negative sign into account. The rate of change, which is what we calculated as the gradient, is a very general
term. And it will tell you whether the power output is increasing or decreasing as it is in
However, as we’ve already said, we’ve been asked to calculate the rate of decrease,
and so we only need to look at how much it’s decreasing by and the unit. Hence, our final answer is that the rate of decrease of power output is 206.25
megawatts per minute.