Although the number of bond pair of electrons in the valence shell of phosphorus in a \(\mathrm{PCl}_{3}\) molecule is 3 , its shape is not trigonal planar. Why?

Valence electrons are the electrons in the outermost shell of an atom that are involved in chemical bonding. They are critical in determining an atom's reactivity and the types of bonds it can form with other atoms. In the case of phosphorus (\textbf{P}) in \textbf{PCl}\(_3\), phosphorus is in group 15 of the periodic table, which means it has 5 valence electrons. These electrons play a crucial role in determining the structure of the molecule. When bonding occurs, phosphorus will share some of its valence electrons with chlorine atoms to form chemical bonds, called covalent bonds.

In the phosphorus trichloride (\textbf{PCl}\(_3\)) example, three of the phosphorus valence electrons are shared with three chlorine atoms, forming three bond pairs. A key point to note is that the total count of valence electrons isn't just involved in bonding. This leads to the existence of a lone pair of electrons after the bond formation, significantly affecting the molecular geometry, which helps explain why the shape isn't trigonal planar but instead trigonal pyramidal.

The electron domain geometry of a molecule is determined based on the regions of electron density that surround the central atom. These regions include both bond pairs and lone pairs of electrons. VSEPR theory outlines that electron domains will arrange themselves as far apart as possible to minimize repulsions.

In \textbf{PCl}\(_3\), there are a total of 4 electron domains around the central phosphorus atom: three bond pairs formed with chlorine atoms and one lone pair of electrons. Even though the number of bonding pairs is three, the lone pair contributes to the total count of electron domains. According to VSEPR theory, these four domains will arrange themselves in a tetrahedral electron domain geometry to minimize electron pair repulsions. Despite the same number of electron domains as would be in a molecule with four bond pairs (also arranging in a tetrahedral pattern), the presence of a lone pair alters the resulting molecular geometry.

Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It's influenced by the electron domains but focuses specifically on the position of the nuclei, ignoring the lone pair locations once the electron domain geometry has been determined. This geometrical structure can have significant implications for the molecule's physical and chemical properties, such as polarity and reactivity.

For \textbf{PCl}\(_3\), although the electron domain geometry is tetrahedral due to four total electron domains (three bonding pairs plus one lone pair), the molecular geometry is identified by considering only the positions of the bond pairs. Thus, \textbf{PCl}\(_3\) has a trigonal pyramidal molecular geometry. Each chlorine atom takes a vertex of a triangular base, with the phosphorus atom above the base, forming a pyramid-like shape. The lone pair occupies more space than the bonding pairs, which pushes the bond pairs closer together, creating the trigonal pyramidal shape rather than a flat trigonal planar structure.