# Lesson 13: Binding energy and mass defect

## Introduction

In this lesson we’ll look in more detail at the amount of energy released when nuclei become more stable.

## Pushing protons together increases potential energy

Imagine pusing two protons closer and closer to each other. They repel each other because they are both positively charged so you have to do more and more work as they get closer. Energy and work are equivalent ideas.

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If you let the protons go then they fly apart again, so you get the energy back. Because you can get the energy back we say that you store potential energy as you push protons together.

The maximum potential energy is when they are quite close together and it's zero when they're a long way apart. Low energy means more stable. If the protons don’t touch then the stable, low-energy state is for them to be a long way apart.

## Like falling into a well

When two protons are close enough the strong force binds them tightly together. It’s as if they’ve fallen down a deep well and the potential energy has suddenly become very negative.

In other words once nucleons are bound in a nucleus then the stable, low energy state is for them to stay bound.

## Negative potential energy is energy you didn't have to put in

Potential energy is normally defined to be negative for attractive forces and positive for repulsive ones. Only attractive forces, like gravity and the strong force, bind systems together. All bound systems have negative potential energy.

When the protons aren’t bound in the nucleus you store potential energy by pushing the protons together. When the protons are bound in a nucleus you store potential energy by pulling the protons apart, provided they don't unbind.

You have to do more work pulling the nucleus apart than you had to put in squeezing the protons together to make it. Energy is released when the protons bind together because the total potential energy of the system is reduced.

The energy seems to come from nowhere because the strong force suddenly attracts the protons when they are very close.

## Binding energy is energy you have to put in to break a nucleus up

The energy needed to drag the protons a long way apart is called the binding energy of the nucleus. The binding energy is always positive because it’s work you have to put in.

The binding energy is equal in size to the potential energy of the nucleus. The potential energy of a nucleus is negative because the stability comes from an attractive force (the strong force).

We tend to talk about binding energy rather than potential energy because positive numbers cause less confusion than negative ones. It also means that you have the relationship of big binding energy meaning stable.

Binding energy can also be thought of as the energy you observe when a nucleus forms.

## Total binding energy always increases as you make a bigger nucleus

As you make a bigger and bigger nucleus the potential energy gets more and more negative. So the total binding energy gets bigger and bigger.

## Average binding energy per nucleon goes up steeply then down gently

Individual nucleons don’t ‘have’ binding energy. A nucleus has a total binding energy, which changes as nucleons are added or subtracted. But we can IMAGINE sharing this total binding energy equally among all the nucleons.

This is called the average binding energy per nucleon.

The stability of a nucleus depends on the average binding energy per nucleon not the total binding energy. In other words it depends on the energy needed to remove a single nucleon rather than all the nucleons.

The average binding energy per nucleon goes up and then goes down as you make bigger and bigger nuclei. More binding energy per nucleon means more stability for the whole nucleus.

So as you make bigger and bigger nuclei the stability starts low, rises rapidly, then drops off slowly.

## How the average binding energy per nucleon can rise and fall

But if the total binding energy always increases how can the average binding energy per nucleon rise then fall?

Here is an analogy. Imagine a group of people playing a game. The objective is for each person to acquire as many of their own points as possible. More points means more stability. The total number of points that the whole group has is irrelevant.

When a new member joins he or she bring points with them. All the points are pooled and the total shared equally. At the moment there are five members. They each have 100 points. What happens when a sixth member joins?

Regardless of whether they bring 90, 100 or 110 points, the total points for the group always increases. This is like adding nucleons to a nucleus. The total binding energy always increases.

But what happens to the points of each member when the total points are shared out equally? If the new member brings only 90 points the total still increases. But each player ends up with fewer points. Each is less ‘stable’.

In the same way adding an extra nucleon always increases the total binding energy but the average binding energy PER NUCLEON can go up or down.

Remember nucleons don’t bring their own binding energy but we can IMAGINE sharing out the total as if they did.

## Energy released in fusion

Let’s imagine starting with a single nucleon and then building nuclei of increasing size around it.

Initially every extra nucleon increases the average binding energy per nucleon. Energy is released if the binding energy per nucleon increases. This is because the nucleus is becoming more stable.

Nickel-62 is the most stable nuclide. But iron-56 is the most commonly used example of a very stable nuclide.

Fusion releases energy up to about iron-56.

## Energy released in fission

Now let's imagine starting with a very large nucleus and then gradually taking it to pieces nucleon by nucleon.

Again, every nucleon you remove increases the average binding energy per nucleon a little so energy is always released. And again, the nucleus gets more and more stable as you decrease it's size.

Fission releases energy until the nucleus is as small as iron-56.

## Estimating energy released using the average BE per nucleon graph

We can use the average binding energy per nucleon graph to estimate the energy released when fusion or fission take place.

For example how much energy is released if 4 protons fuse to form a helium-4 nucleus?

The binding energy of a single proton or a single neutron is zero. The binding energy per nucleon for helium-4 is 7.2 MeV. So each of the four nucleons increases its binding energy on average by 7.2 MeV, making a total of 28.8 MeV.

Higher binding energy means more stable. Energy is released when systems become more stable. So 28.8 MeV of energy is released every time a helium nucleus is synthesised. How does this energy appear?

You might think it appears as the energy of the electrons and neutrinos created when the neutrons turn into protons. But this is extra energy due to weak force interactions. Binding energy involves the strong force and appears as gamma photons.

## Net energy released is independent of path of reactions

Now it’s almost impossible for four protons to come together at the same time to form a helium-4 nucleus. We’ve seen that one way helium synthesis takes place in stars is by the proton-proton cycle.

But it doesn’t matter how complex the steps involved are, the net energy released is always 28.8 MeV per helium nucleus formed.

## Another fusion example

What about if four helium-4 nuclei fuse by some method to form one oxygen-16 nucleus?

Each helium nucleon increases its binding energy by about 0.8 MeV when it becomes part of an oxygen nucleus. There are 16 nucleons, making the total energy released about 12.8 MeV.

## Calculating energy released during fission

We can do the same for fission.

A uranium-235 nucleus will fission if it absorbs a neutron. A roughly 40/60 split is most likely. We can use the average binding energy per nucleon graph to estimate some values for typical fission products barium-144 and krypton-90.

## Comparing chemical energy with nuclear energy

It turns out that each uranium-235 nucleus releases energy of the order of hundreds of millions of electronvolts. Chemical reactions, like those involved when coal burns, release a few tens of electronvolts of energy per atom.

Why do nuclear reactions release millions of times more energy per atom than chemical reactions?

Chemical reactions are all about electrons. Electrons are governed by the electromagnetic force.

Nuclear reactions are all about the nucleus. Nucleons are governed by the strong force. The strong force is much bigger than the electromagnetic force and the nucleons are much closer to each other than electrons are. That’s why the energy changes are so much bigger.

## Comparing energy yield from fission and fusion

Now let’s compare fission and fusion. Fissioning a single uranium-235 nucleus yields about 180 MeV. Synthesising a single helium-4 nucleus by fusion yields about 30 MeV.

Fission tends to cause a fairly small change in binding energy per nucleon. But the nucleus is bigger so there are lots of nucleons.

Fusion tends to cause a much larger change in binding energy per nucleon but the nucleus formed is smaller so there are fewer nucleons.

Fissioning a single uranium-235 nucleus yields far more energy than synthesising a single helium-4 nucleus by fusion. But a helium-4 nucleus is only about 1/60th of the mass of a uranium-235 nucleus. Synthesising 60 helium-4 nuclei by fusion yields about 1800 MeV. In other words 10 times more than the same mass of uranium-235.

In the future it may be possible to harness fusion as a source of clean energy.

## The nucleus has potential energy not binding energy

One final concept we need to look at is ‘mass defect’.

A nucleus doesn’t ‘have’ binding energy. Binding energy is the energy you put in if you happen to want to pull the nucleus to pieces. The energy that a nucleus has, separate from anything we do to it, is its potential energy.

When we form a nucleus by fusion from its nucleons the potential energy gets smaller – i.e. more negative.

## Energy has mass

Now one of the consequences of Einstein’s special theory of relativity is that energy has mass. A reduction in energy also appears as a reduction in mass.

When protons bind together their energy is reduced so their mass is also reduced.

This reduction in mass is called the ‘mass defect’. It’s the mass that disappears when nuclei form from their individual nucleons. Einstein’s famous equation E=mc^{2} tells us how much mass each joule of energy has. It’s less than a trillionth of a gram.

But with an atomic nucleus the mass of the energy is quite big compared with the mass of the nucleus. For a large nucleus like uranium-235 the mass defect can be equivalent to almost two nucleons.

Mass defect is important because you can measure mass very accurately using a ‘mass spectrometer’. If you know masses then you can calculate energy changes using Einstein’s relationship.

## The atomic mass unit, u

The masses involved are still of the order of trillionths of trillionths of a gram so we use a very small unit of mass. This is called an ‘atomic mass unit’. It’s arbitrarily defined as one twelfth of the mass of one atom of carbon-12.

## Using mass defect to see whether a nuclear reaction can happen spontaneously

We can use the idea of mass defect to work out whether a nuclear reaction can happen.

Nuclear reactions can only happen spontaneously if the products are more stable than the reactants. More stable means lower energy. Lower energy means lower mass.

So if the mass goes down then a nuclear reaction can happen spontaneously. If it goes up then it can’t.

For example you can tell that oxygen-16 can't emit an alpha particle to form carbon-12 because the mass of the alpha plus carbon-12 nucleus is bigger than the original oxygen-16 nucleus.

## Using mass defect to work out energy liberated

Equally we can predict how much energy a nuclear reaction will liberate, for example when deuterium and tritium fuse to form helium-4 plus a free neutron.

First we calculate the decrease in mass. Then we work out how much energy has this mass. We know if the mass has decreased then this energy must be released, in this case about 17.6 MeV.

So in this lesson we’ve found out how to do some simple calculations involving potential energy, binding energy and mass defect. Next lesson we’ll see how the energy liberated in fission can be used to provide us with useful electrical energy.

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