Foundations — Nuclear reactions — Q-value calculation

This page builds every piece of notation the parent Q-value note uses, starting from a reader who has seen none of it. Read top to bottom: each symbol is earned before the next one leans on it.

1. The nuclide symbol ${}_Z^{A}X$ — a nucleus's ID card

Before we can weigh anything, we must name it. A nucleus is written ${}_Z^{A}X$.

Look at the figure: the nucleus is a bag of balls. Blue balls = protons (there are $Z$ of them), pale balls = neutrons (there are $A-Z$). The total count of balls is $A$. That is all $A$ and $Z$ ever mean.

2. Mass $m$ and the atomic mass unit $\text{u}$ — how we weigh

3. The speed of light $c$ and $E=m{c}^2$ — the exchange rate

Here is the crux of the whole topic: mass and energy are the same thing in different currencies, and $c$ is the exchange rate.

See Mass-energy equivalence E=mc^2 for the full story. For us, the only job of $c^2$ is: multiply a missing mass to get the energy that appeared.

4. The 931.5 conversion — one shortcut, derived

Combining the last two sections gives the single most-used number on the parent page.

Where does $931.5$ come from? Just feed $1\text{u}$ through $E=m{c}^2$ and convert joules to MeV:

5. Binding energy and the "missing mass" $\Delta m$

Now: why should any mass go missing? Because a bound nucleus weighs less than its loose parts.

In the figure the loose balls sit high (heavy, unbound); once snapped together they drop into a well (lighter, bound). The depth of the well is the binding energy.

6. Kinetic energy $K$ and momentum $p$ — where the energy shows up

The freed energy doesn't vanish into thin air; it becomes motion of the products.

Look at the figure. A parent at rest has zero total momentum. After it splits, the two fragments must fly apart with equal-and-opposite momenta so the arrows still cancel to zero. This is why the heavy daughter must recoil — and why, later, no fragment can be left perfectly at rest.

7. Putting it together — the symbols of $Q$

Now every symbol in the parent's headline formula is defined:

Everything the reactions (Alpha decay, Beta decay, Nuclear fission, Nuclear fusion) do is a special case of this one bookkeeping line.

8. How the foundations feed the topic

Equipment checklist

Cover the right side and test yourself — you are ready for the parent page only if you can answer all of these.