For each of the following complexes, draw a crystal field energy-level diagram, assign the electrons to orbitals, and predict the number of unpaired electrons. 
(a) [CrF₆]^3-
(b) [V(H₂O)₆]³⁺
(c) [Fe(CN)₆]^3-

Verified step by step guidance

1. Identify the oxidation state of the metal in each complex and determine the electron configuration of the metal ion.
2. Determine the geometry of each complex (octahedral, tetrahedral, etc.) and use it to draw the crystal field splitting diagram.
3. Assign the electrons to the d-orbitals according to the crystal field splitting diagram, considering the electron configuration and the nature of the ligands (strong or weak field).
4. Count the number of unpaired electrons in the d-orbitals for each complex.
5. Use the number of unpaired electrons to predict the magnetic properties of each complex (paramagnetic or diamagnetic).

Key Concepts

Crystal Field Theory

Crystal Field Theory (CFT) explains how the arrangement of ligands around a central metal ion affects the energy levels of the d-orbitals. In an octahedral complex, for example, the d-orbitals split into two energy levels: the lower-energy t2g and the higher-energy eg orbitals. This splitting is crucial for determining the electronic configuration of the metal ion and predicting properties such as color and magnetism.

Electron Configuration and Orbital Filling

The electron configuration of a transition metal ion is determined by the number of electrons it has after accounting for its oxidation state. Electrons fill the d-orbitals according to the Aufbau principle, Hund's rule, and the Pauli exclusion principle. Understanding how to assign electrons to the split d-orbitals in a crystal field diagram is essential for predicting the number of unpaired electrons, which influences the magnetic properties of the complex.

Ligand Field Strength and Crystal Field Splitting

The strength of the ligands surrounding a metal ion affects the degree of d-orbital splitting, known as crystal field splitting energy (Δ). Strong field ligands, like CN⁻, cause a larger splitting, leading to lower energy configurations and potentially fewer unpaired electrons. Conversely, weak field ligands, like F⁻, result in smaller splitting, which can lead to more unpaired electrons. This concept is vital for predicting the magnetic behavior of the complexes.