Draw a crystal field energy-level diagram, and predict the number of unpaired electrons for each of the following: 
(a) [Mn(H2O)6]2+

Verified step by step guidance

1. Identify the metal ion and its oxidation state. For [Mn(H2O)_6]^{2+}, manganese (Mn) is in the +2 oxidation state.
2. Determine the electron configuration of the Mn^{2+} ion. Manganese in its elemental form is [Ar] 3d^5 4s^2, so Mn^{2+} is [Ar] 3d^5.
3. Recognize that [Mn(H2O)_6]^{2+} is an octahedral complex, which means the 3d orbitals will split into two sets: t_{2g} (lower energy) and e_g (higher energy).
4. Distribute the 5 d-electrons of Mn^{2+} among the split d-orbitals according to Hund's rule, filling the lower energy t_{2g} orbitals first before pairing electrons in the higher energy e_g orbitals.
5. Count the number of unpaired electrons in the d-orbitals after distribution.

Key Concepts

Crystal Field Theory

Crystal Field Theory (CFT) explains how the arrangement of ligands around a central metal ion affects its electronic structure and energy levels. It describes the splitting of d-orbitals in transition metal complexes due to the electrostatic interactions between the metal ion and surrounding ligands. This theory is crucial for predicting the magnetic properties and color of the complexes.

d-Orbital Splitting

In an octahedral field, the five d-orbitals split into two energy levels: the lower-energy t2g (dxy, dyz, dzx) and the higher-energy eg (dx2-y2, dz2) orbitals. The extent of this splitting depends on the nature of the ligands and their field strength. Understanding this splitting is essential for determining the electron configuration and the number of unpaired electrons in a complex.

Unpaired Electrons and Magnetism

Unpaired electrons in an atom or ion contribute to its magnetic properties. A species with unpaired electrons exhibits paramagnetism, while those with all paired electrons are diamagnetic. In the context of transition metal complexes, the number of unpaired electrons can be predicted from the electron configuration after considering d-orbital splitting, which is vital for understanding the complex's behavior in a magnetic field.