Give a valence bond description of the bonding in each of the following complexes. Include orbital diagrams for the free metal ion and the metal ion in the complex. Indicate which hybrid orbitals the metal ion uses for bonding, and specify the number of unpaired electrons. 
(b) [NiBr₄]^2- (tetrahedral) 

Verified step by step guidance

1. Step 1: Determine the oxidation state of the metal ion in the complex. For [NiBr_4]^{2-}, the oxidation state of Ni is +2.
2. Step 2: Write the electron configuration for the free Ni^{2+} ion. Nickel (Ni) has an atomic number of 28, so its electron configuration is [Ar] 3d^8 4s^2. For Ni^{2+}, remove two electrons from the 4s orbital, resulting in [Ar] 3d^8.
3. Step 3: Draw the orbital diagram for the free Ni^{2+} ion. The 3d orbitals will have 8 electrons, with two unpaired electrons in the 3d orbitals.
4. Step 4: Determine the hybridization of the metal ion in the complex. In a tetrahedral complex like [NiBr_4]^{2-}, the metal ion uses sp^3 hybrid orbitals for bonding.
5. Step 5: Indicate the number of unpaired electrons in the complex. Since the 3d orbitals of Ni^{2+} have two unpaired electrons, the complex will also have two unpaired electrons.>

Key Concepts

Valence Bond Theory

Valence Bond Theory (VBT) explains how atoms bond by overlapping their atomic orbitals to form covalent bonds. In this theory, the strength of the bond is determined by the extent of overlap between the orbitals. For transition metals, VBT also considers hybridization, where atomic orbitals mix to form new hybrid orbitals that can accommodate bonding with ligands.

Hybridization

Hybridization is the process of combining atomic orbitals to create new hybrid orbitals that are suitable for bonding. In the case of the tetrahedral complex [NiBr4]²⁻, nickel undergoes sp³ hybridization, resulting in four equivalent hybrid orbitals that can form bonds with the four bromide ligands. This concept is crucial for predicting the geometry and bonding properties of coordination complexes.

Unpaired Electrons and Magnetic Properties

Unpaired electrons in an atom or ion contribute to its magnetic properties. In transition metal complexes, the number of unpaired electrons can indicate whether the complex is paramagnetic (having unpaired electrons) or diamagnetic (all electrons paired). For [NiBr4]²⁻, analyzing the electron configuration of the nickel ion and the resulting hybridization helps determine the number of unpaired electrons and the complex's overall magnetic behavior.