The homoleptic and octahedral complex of Co$$^{2+}$$ and H$$_2$$O has _____ unpaired electron(s) in the t$$_{2g}$$ set of orbitals.

The homoleptic octahedral complex of Co²⁺ with H₂O is [Co(H₂O)₆]²⁺.

Co²⁺ has electronic configuration: [Ar] 3d⁷.

H₂O is a weak field ligand, so the complex is high spin.

In a high-spin octahedral d⁷ complex:

$$t_{2g}^5 \, e_g^2$$

The t₂g orbitals (dxy, dxz, dyz) have 5 electrons distributed as: ↑↓, ↑↓, ↑

Number of unpaired electrons in t₂g = 1 (two orbitals are fully paired, one has 1 unpaired).

The number of unpaired electrons in t₂g is $$\mathbf{1}$$.