Which element exists as a monoatomic molecule?

The correct option is Argon.

To understand why argon exists as a monoatomic molecule, let us look at the nature of the elements listed in the options:
1. Oxygen (O): Oxygen belongs to Group 16 of the periodic table. It needs two electrons to complete its valence shell. Therefore, it shares electrons with another oxygen atom to form a diatomic molecule ($O_2$).
2. Chlorine (Cl): Chlorine is a halogen (Group 17) and needs one electron to achieve a stable octet. It shares an electron pair with another chlorine atom to exist as a diatomic molecule (${\mathrm{Cl}}_2$).
3. Nitrogen (N): Nitrogen belongs to Group 15 and needs three electrons to complete its valence shell. It shares three pairs of electrons with another nitrogen atom, forming a strong triple covalent bond to exist as a diatomic molecule ($N_2$).
4. Argon (Ar): Argon is a noble gas (Group 18). It has a completely filled valence shell (an octet: $3s^{2}3p^6$). Because of this stable, low-energy electron configuration, argon atoms do not need to share, gain, or lose electrons to attain stability. Consequently, they do not bond with other atoms under standard conditions and exist stably as single, isolated atoms, which are called monoatomic molecules.