Question

Whether a neutral whole atom behaves as bosons or a fermion is independent of $\mathit{Z}\mathbf{,}$ instead depending entirely on the number of the neutrons in its nucleus. Why? What is it about this number that determines whether the atom is a boson or a fermion?

Short answer

If the neutrons are in odd number then the atom will have a net half-integral spin and will behave as Fermion, whereas if it is even then it will behave as boson.

Step-by-step solution

For Fermion.

Particles come in two types: particles that make up matter, known as "fermions," and particles that carry forces, known as "bosons." The difference between the two is that fermions take up space, while bosons can pile on top of each other.

The sum of the number of electrons and number of protons will be an even number in a neutral atom. This will result in a net integral spin of the atom (excluding neutron). In this case, one can determine that the net spin of the system only if one knows the number of neutrons whether it is odd or even. That is why the behaviour of the neutral atom depends on the number of neutrons instead of $Z.$If the number of neutrons is odd, the atom will behave as Fermion. If the number of neutrons is even, the atom will behave as Boson. The number that determines this behaviour is called intrinsic angular momentum (spin) of the particle.

For Boson.

In a neutral atom, the number of electrons and number of protons are equal (and the sum of the number of electrons and protons is a even number). As these two particles have odd half-integral spin $\left(\frac{1}{2},\frac{3}{2},\frac{5}{2},.....\right)$the total even number of particles results in a net integral spin.

Hence, the behaviour of a neutral atom completely depends on the number of neutrons whether the neutrons are in odd number or even number. If the neutrons are in odd number, the atom will have a net half-integral spin and will behave as Fermion. If the neutrons are in even number, the atom will have a net integral spin and the atom behaves as Boson.