(I) Fe → Fe³⁺ + 3e^- 

Fe (Z = 26)

Fe³⁺ = number of electrons = 23

1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁶ 4s² for Fe atom.

1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁵ for Fe³⁺ ion.

So, it contain 5 unpaired electrons. 

Mn (Z = 25) Electronic configuration is

1s² 2s² 2p⁵ 3s² 3p⁶ 4s² 3d⁵

Mn → Mn²⁺ + 2e^-

Number of unpaired electrons in Mn²⁺ = 5

Ar (Z = 18). Electronic configuration is

1s² 2s² 2p⁶ 3s² 3p⁶

All orbitals are completely filled. So, no unpaired electrons in it.

(II) Significance of de Broglie equation:

1. λ = \(\frac{h}{mv}\) This equation implies that a moving particle can be considered as a wave and a wave can exhibit the properties of a particle. 

2. For a particle with high linear momentum (mv) the wavelength will be so small and cannot be observed.

3. For a microscopic particle such as an electron, the mass is of the order of 10^-31 kg, hence the wavelength is much larger than the size of atom and it becomes significant. 

4. For the electron, the de Broglie wavelength is significant and measurable while for the iron ball it is too small to measure, hence it becomes insignificant.