Step 1: What the nucleus must account for.
Any correct nuclear model must reproduce three measured quantities of an atom \(^{A}_{Z}X\): its charge \(+Ze\), its mass number \(A\), and its spin/magnetic behaviour. The proton-neutron hypothesis is judged by how well it does this.
Step 2: The two building blocks.
Only two particles are placed inside the nucleus:
Proton, charge \(+e\), mass \(\approx 1.67\times10^{-27}\) kg.
Neutron, charge \(0\), mass \(\approx 1.67\times10^{-27}\) kg.
Both are called nucleons. No electron is placed inside the nucleus.
Step 3: Counting the particles.
To get a nuclear charge of \(+Ze\) we need exactly \(Z\) protons. To make up the remaining mass we add neutrons. If the mass number is \(A\), then
Neutron number \(N = A - Z\).
So the nucleus holds \(Z\) protons and \(A-Z\) neutrons.
Step 4: Checking the model.
Charge check: \(Z(+e) + (A-Z)(0) = +Ze\), correct.
Mass check: \(A\) nucleons each of mass \(\approx m_p\) give mass number \(A\), correct.
Isotope check: atoms with the same \(Z\) but different \(N\) (hence different \(A\)) are naturally explained as isotopes, e.g. \(^{1}_{1}H\), \(^{2}_{1}H\), \(^{3}_{1}H\).
Step 5: Why the electron model was dropped.
Placing an electron in a \(10^{-15}\) m nucleus violates the uncertainty principle (it would need enormous energy) and gives wrong nuclear spin. The proton-neutron model has no such trouble, so it is accepted.
\[\boxed{^{A}_{Z}X:\ Z\ \text{protons},\ (A-Z)\ \text{neutrons}}\]