Welcome to Mass and Energy!
In this chapter, we explore one of the most famous ideas in all of science: the fact that mass and energy are actually two sides of the same coin. This isn't just theoretical physics—it is the reason why the Sun shines and how nuclear power stations provide electricity to our homes. Don't worry if this feels a bit "mind-bending" at first; we will break it down step-by-step.
1. Einstein’s Famous Equation
The core of this chapter is the mass-energy equivalence formula: \(E = mc^2\).
This tells us that mass can be converted into energy, and energy can be converted into mass. In nuclear physics, we are usually looking at how a tiny change in mass (\(\Delta m\)) results in a huge release of energy (\(\Delta E\)).
The equation is written as:
\(\Delta E = \Delta m c^2\)
Where:
• \(\Delta E\) is the energy (measured in Joules, J).
• \(\Delta m\) is the mass difference (measured in kg).
• \(c\) is the speed of light (\(3.00 \times 10^8 \text{ m s}^{-1}\)).
Did you know? Because the speed of light squared (\(c^2\)) is such a massive number (\(9 \times 10^{16}\)), even a tiny speck of mass can turn into a terrifying amount of energy!
2. Atomic Mass Units (u) and MeV
Working in kilograms is fine for everyday objects, but for protons and neutrons, the numbers are too small to be practical. Instead, we use the atomic mass unit (u).
The AQA syllabus gives you a brilliant shortcut for your calculations:
\(1 \text{ u}\) is equivalent to \(931.5 \text{ MeV}\)
This is a "conversion factor" you will use constantly. If you calculate a mass change in \(u\), you just multiply it by \(931.5\) to get the energy in Mega-electronvolts (MeV). You don't even need to use \(E = mc^2\) if you use this shortcut!
3. Mass Defect and Binding Energy
Here is a strange fact: if you weigh a nucleus, it actually weighs less than the sum of the individual protons and neutrons that make it up. This "missing mass" is called the mass defect.
What is Mass Defect (\(\Delta m\))?
The mass defect is the difference between the mass of the completely separated nucleons (protons and neutrons) and the mass of the nucleus itself.
\(\text{Mass Defect} = (\text{Total mass of separate nucleons}) - (\text{Mass of the nucleus})\)
What is Binding Energy?
When protons and neutrons come together to form a nucleus, they release energy. To pull them apart again, you would need to put that exact same amount of energy back in. This is the binding energy.
Key Takeaway: The binding energy is the energy required to completely separate a nucleus into its individual protons and neutrons. Higher binding energy means the nucleus is more stable and "happier" staying together.
Analogy: Imagine trying to pull two strong magnets apart. You have to do "work" (use energy) to separate them. The nucleons in a nucleus are "stuck" together by the strong nuclear force, and the binding energy is the "work" needed to break them apart.
4. Binding Energy per Nucleon
To compare how stable different atoms are, we look at the binding energy per nucleon. This is simply the total binding energy of the nucleus divided by the number of nucleons (protons + neutrons) in that nucleus.
\(\text{Binding energy per nucleon} = \frac{\text{Total binding energy}}{\text{Nucleon number (A)}}\)
A higher binding energy per nucleon means the nucleus is more stable.
The Binding Energy Graph
If you plot a graph of binding energy per nucleon against nucleon number (\(A\)), you get a curve that is vital for your exam:
• The Peak: The most stable element is Iron-56 (\(^{56}\text{Fe}\)). It sits at the very top of the curve.
• Fusion: Small nuclei (like Hydrogen) are at the far left. They can join together to become more stable (moving up the curve towards Iron). This releases energy.
• Fission: Large, heavy nuclei (like Uranium) are at the far right. They can split into smaller pieces to become more stable (moving "backwards" up the curve towards Iron). This also releases energy.
5. Nuclear Fission and Fusion Calculations
In the exam, you might be asked to calculate how much energy is released in a nuclear reaction. You can do this in two ways depending on the data provided:
Method A: Using Mass
1. Find the total mass of the particles before the reaction.
2. Find the total mass of the particles after the reaction.
3. Calculate the difference (\(\Delta m\)).
4. Convert this mass into energy (using \(E=mc^2\) for kg, or the \(931.5\) factor for \(u\)).
Method B: Using Binding Energy
1. Find the total binding energy of the products (after).
2. Find the total binding energy of the reactants (before).
3. \(\text{Energy Released} = (\text{Total Binding Energy After}) - (\text{Total Binding Energy Before})\)
Common Mistake: Students often swap the "before" and "after" in binding energy calculations. Just remember: the products are more stable, so they have more binding energy. The "extra" energy is what gets released!
Summary Checklist
Quick Review:
• Do I know that \(1 \text{ u} = 931.5 \text{ MeV}\)?
• Can I define mass defect and binding energy?
• Can I sketch the binding energy per nucleon graph and label where Fusion and Fission happen?
• Do I remember that Iron-56 is the most stable nucleus?
• Can I calculate the energy released in a decay or reaction using mass or binding energy values?
For more details on how these reactions are used in power plants, see the chapter on "Induced fission and safety aspects". For details on how we measure the size of the nucleus, see "Rutherford scattering and nuclear radius".