\(\mathrm{PCl}_{3}\) and \(\mathrm{PCl}_{5}\) both exists; \(\mathrm{NCl}_{3}\) exists but \(\mathrm{NCl}_{5}\) does not exist. It is due to (a) Lower electronegativity of \(\mathrm{P}\) than \(\mathrm{N}\) (b) Lower tendency of \(\mathrm{N}\) to form covalent bond (c) Availability of vacant d-orbital in \(\mathrm{P}\) but not in \(\mathrm{N}\) (d) Statement is itself incorrect.

Covalent bonding is one of the fundamental types of chemical bonds where atoms share pairs of electrons to achieve greater stability. The sharing enables each atom to fill its valence shell, reaching a noble gas configuration. This type of bond often forms between nonmetal atoms with similar electronegativities. For example, water (H_{2}O) consists of oxygen sharing electrons with two hydrogen atoms, creating a molecule with covalent bonds.

When studying molecules like PCl_{3} and PCl_{5}, it becomes evident that the capacity of an atom to form covalent bonds is not solely dependent on the number of unpaired electrons in its outermost shell but also on the availability of orbitals for electron sharing. This will become clearer as we take a closer look at d-orbitals in chemical bonding and octet rule exceptions.

The valence shell electron configuration of an atom dictates how it interacts and bonds with other atoms. The outermost electrons play a pivotal role in chemical reactions and bond formation. Atoms aim to achieve a full valence shell, typically by obtaining eight electrons to fulfill the octet rule, which is reminiscent of the electron configuration of noble gases.

Nitrogen, with the configuration [He]2s^2 2p^3, has five valence electrons and seeks three more to complete its octet, which explains why NCl_{3} is stable. But without available d-orbitals, it cannot expand its octet to form NCl_{5}. Phosphorus, on the other hand, has the configuration [Ne]3s^2 3p^3 and can make use of the nearby d-orbitals to accommodate more than eight electrons, making the formation of PCl_{5} possible.

While the s and p orbitals in the valence shell are sufficient for many elements to form typical bonds, the d-orbitals can become involved in chemical bonding in elements in the third period and below. This involvement is crucial as it allows for bond formation that extends beyond the octet rule. Phosphorus can utilize its 3d-orbitals, which are vacant but accessible, to accept more electrons and form bonds like those in PCl_{5}.

This flexibility is not available to elements like nitrogen in the second period due to the absence of vacant d-orbitals in their valence shells. Therefore, they cannot expand their octet to form hypervalent compounds, which is why NCl_{5} does not exist.

The octet rule states that atoms are generally most stable when they have eight electrons in their valence shell, emulating noble gas configurations. However, there are exceptions to this rule. Hypercoordinated molecules like PCl_{5} showcase an expanded octet, made possible by the use of d-orbitals. Atoms such as phosphorus have the needed d-orbitals to harbor more than eight valence electrons, resulting in compounds that defy the conventional octet rule.

Another exception involves molecules with less than a complete octet, such as boron in BF_{3}, which is stable with only six valence electrons. The approach to chemical bonding is deeply enriched by understanding these exceptions, which highlight the versatility of electron configurations and the predictive challenges they pose in understanding molecular geometry and reactivity.