The color of transition metals and their ions is due to the presence of partially filled d orbitals. These orbitals are located in the inner shell of the atom and can hold a maximum of 10 electrons. When a transition metal atom loses electrons to form an ion, the d orbitals may not be completely filled. This means that there are energy levels that are available for electrons to occupy.

When light strikes a transition metal ion, electrons in the d orbitals can absorb energy and move to higher energy levels. This is called an "electronic transition." The energy of the light that is absorbed is related to the energy difference between the two energy levels.

When the electron falls back down to its original energy level, it releases the energy that it absorbed in the form of light. The color of the light that is emitted is related to the energy difference between the two energy levels.

For example, if the energy difference between the two energy levels is small, the light that is emitted will be red. If the energy difference between the two energy levels is large, the light that is emitted will be blue.

The color of a transition metal ion can also be affected by the environment around the ion. For example, the color of a transition metal ion can be different in a solid state than it is in a solution. This is because the environment around the ion can affect the energy levels of the d orbitals.

Zinc ions, \((Zn^{2+})\), have a fully filled d orbital configuration. This means that there are no energy levels that are available for electrons to occupy. Therefore, zinc ions cannot absorb light and therefore do not emit any color. This is why zinc ions are colorless.

The other ions listed, \(Ti^{3+}, Mn^{3+}\), and \(Cr^{3+}\), all have partially filled d orbitals. This means that there are energy levels that are available for electrons to occupy. Therefore, these ions can absorb light and emit color.