Number of Water Molecules Directly Bonded to Copper in Blue Vitriol

In the crystalline structure of copper(II) sulfate pentahydrate, $\text{CuSO}_4\cdot 5\text{H}_2\text{O}$ (blue vitriol), the compound is formulated as:
$[\text{Cu}(\text{H}_2\text{O})_4]\text{SO}_4\cdot\text{H}_2\text{O}$

In this coordination complex:

1. Four water molecules act as ligands and are directly coordinated (bonded via coordinate covalent bonds) to the central $\text{Cu}^{2+}$ ion, typically forming a square-planar geometry around the copper center in the equatorial positions (with axial positions occupied by oxygen atoms from the sulfate groups in a distorted octahedral environment).
2. The fifth water molecule is not coordinated to the metal center; instead, it is held in the crystal lattice through hydrogen bonding between the coordinated water molecules and the sulfate ions ($\text{SO}_4^{2-}$).

Therefore, the number of water molecules directly bonded to the metal center is $4$.