The number of hydrogen bonded water molecule(s) associated with stoichiometry CuSO$$_4$$.5H$$_2$$O is _________.

We begin with the well-known crystalline compound copper(II) sulphate pentahydrate, whose formula is written as $$\mathrm{CuSO_4\cdot 5H_2O}.$$

It is experimentally established, through X-ray diffraction studies and many coordination-chemistry experiments, that these five water molecules are not all equivalent. Four of them are directly co-ordinated to the central copper(II) ion, while the remaining one is not bonded to the metal ion at all; instead, it is linked to the sulphate ion by hydrogen bonding.

To make that picture explicit, chemists often write the structural formula in the form

$$\bigl[\mathrm{Cu(H_2O)_4}\bigr]\mathrm{SO_4}\cdot\mathrm{H_2O}.$$

Here

$$\mathrm{Cu^{2+}} + 4\ \mathrm{H_2O}\; \longrightarrow\; \bigl[\mathrm{Cu(H_2O)_4}\bigr]^{2+}$$

represents the formation of a coordination complex. The sulphate anion $$\mathrm{SO_4^{2-}}$$ balances the charge of the $$\bigl[\mathrm{Cu(H_2O)_4}\bigr]^{2+}$$ unit, and the fifth water molecule is written separately with a dot to show that it is present in the crystal lattice but not bonded to the copper ion.

This fifth water molecule forms hydrogen bonds with the oxygen atoms of the $$\mathrm{SO_4^{2-}}$$ group. Because hydrogen bonding is a non-covalent interaction occurring between a hydrogen atom attached to one electronegative atom (such as oxygen) and another electronegative atom (again oxygen in sulphate), we classify this particular water molecule as “hydrogen-bonded water.”

Counting, therefore:

• Coordinated water molecules $$= 4.$$

• Hydrogen-bonded (lattice) water molecules $$= 1.$$

We are asked specifically for the number of hydrogen-bonded water molecules associated with the given stoichiometry. From the analysis above, that number is clearly

$$1.$$

So, the answer is $$1$$.