Predict the shapes of (a) \(\mathrm{NH}_{2}^{-}\), (b) \(\mathrm{CO}_{3}^{2-}\), (c) \(\mathrm{IF}_{3},\) (d) \(\mathrm{Br}_{3}^{-},\) and (e) \(\mathrm{GaH}_{3}\).

The Valence Shell Electron Pair Repulsion (VSEPR) theory is fundamental for predicting the shape of molecules. At its core, VSEPR theory states that electron pairs surrounding an atom naturally repel each other, and they will arrange themselves as far apart as possible to minimize this repulsion. This behavior is due to the negative charge of the electrons which causes them to repel one another.

When we look at a molecule's structure, we focus primarily on the valence electrons of the central atom. These electrons, which may exist in pairs, dictate the spatial arrangement of the atoms within the molecule. For example, in the case of \(\mathrm{NH}_{2}^{-}\), the central nitrogen atom has four electron pairs that determine the molecular geometry.

The arrangement of electron pairs is significantly influenced by the number of pairs around a central atom. When we talk about electron pairs, we refer to both bonding pairs (electron pairs shared with other atoms to form bonds) and lone pairs (non-bonded electron pairs) around the central atom. The electron geometry is the three-dimensional arrangement of these electron pairs, which may be tetrahedral, trigonal planar, linear, octahedral, or various other configurations depending on the count and nature of electron pairs.

For instance, the carbonate ion \(\mathrm{CO}_{3}^{2-}\) exhibits a trigonal planar electron geometry because all its electron pairs are involved in bonding with no lone pairs on the central carbon atom.

A pivotal distinction in molecular geometry comes from understanding the difference between lone pairs and bonding pairs of electrons. Bonding pairs are shared between atoms, forming covalent bonds that create the skeleton of a molecule. In contrast, lone pairs are not shared and typically occupy more space than bonding pairs, leading to distortions in the geometry of a molecule.

For example, in the molecule iodine trifluoride \(\mathrm{IF}_{3}\), the presence of two lone pairs on the iodine atom causes a deviation from the expected trigonal bipyramidal electron geometry to a T-shaped molecular shape. This alteration is due to the extra space that lone pairs require.

The shape of a molecule is different from the electron geometry and is described as the molecular geometry. It factors in only the positions of the atomic nuclei, considering the spatial arrangements of atoms linked by chemical bonds regardless of the electron pair arrangement. This means that while the electron pair geometry considers both bonding and lone pairs, molecular shape reflects only the arrangement of atoms.

In the molecule gallium trihydride \(\mathrm{GaH}_{3}\), for instance, despite the electron pair geometry being trigonal planar, it's the same as the molecular shape since there are no lone pairs to alter the arrangement.