A hydrate of copper(II) chloride has the following formula: \(\mathrm{CuCl}_{2} \cdot x \mathrm{H}_{2} \mathrm{O} .\) The water in a \(3.41-\mathrm{g}\) sample of the hydrate is driven off by heating. The remaining sample has a mass of \(2.69 \mathrm{~g}\) Find the number of waters of hydration \((x)\) in the hydrate.

When studying chemistry, particularly when delving into reactions and chemical compounds, one fundamental principle that we often use is the mole concept. In simple terms, a mole is a unit that measures the amount of substance. Just like a dozen refers to 12 items, a mole refers to approximately 6.022 x 10²³ entities, whether they are atoms, molecules, ions, or electrons. This number is known as Avogadro's number.

To understand this better, let's take water as an example. The molar mass of water (H₂O) is roughly 18.015 g/mol, which means that 1 mole of water weighs approximately 18.015 grams, and contains Avogadro's number of molecules. In the exercise given, the mole concept is applied to determine the number of moles of water and CuCl₂ that were present in the sample, a crucial step towards finding the hydration number.

Now, stoichiometry is the section of chemistry that involves calculating the quantities of reactants and products in chemical reactions. It is deeply rooted in the balance of mass and the conservation of matter, embodying the idea that atoms are neither created nor destroyed in chemical reactions; they're simply rearranged. Stoichiometry allows us to predict how much product we'll end up with from certain amounts of reactants, or vice versa.

In practical terms, if we revisit our exercise where we are looking to find the hydration number (x) in the hydrate CuCl₂ · x H₂O, stoichiometry helps us calculate the precise ratio between CuCl₂ and the water molecules in the compound. By understanding the stoichiometric relationship between the components of our hydrate, the exact value of 'x' can be ascertained, reflecting the empirical formula of the compound.

The empirical formula is the simplest positive integer ratio of atoms present in a compound. Rather than indicating the number of atoms in a single molecule of a substance, the empirical formula gives the smallest whole number ratio of the elements within that compound. For instance, the empirical formula of hydrogen peroxide is HO, despite the actual molecular formula being H₂O₂, because it reflects the 1:1 ratio of hydrogen to oxygen atoms.

Applying this to our exercise, after determining the number of moles of CuCl₂ and water in the hydrate sample, the empirical formula can be found by expressing these moles as the smallest whole number ratio. This ratio corresponds to the variable 'x' in the hydrate's chemical formula. Hence, the empirical formula provides a clear and simplistic view of the compound's composition, which is indispensable for understanding the nature of the hydrate.