Heating a sample of \(\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot \mathrm{xH}_{2} \mathrm{O}\) weighing 4.640 g until the removal of the water of hydration leaves 1.720 g of anhydrous \(\mathrm{Na}_{2} \mathrm{CO}_{3} .\) What is the formula of the hydrated compound?

Understanding the stoichiometry of chemical hydrates is essential for grasping the relationship between the anhydrous compound and the water molecules attached to it. Hydrates are compounds that include a specific number of water molecules bound in their crystal structure. The process to determine a hydrate's formula involves calculating the ratio of the moles of water to the moles of the anhydrous substance.

For hydrates, the stoichiometric coefficients tell us the number of water molecules per formula unit of the compound. If you have a hydrate represented by \( \mathrm{A} \cdot x\mathrm{H}_{2}\mathrm{O} \), where \( \mathrm{A} \) is the anhydrous compound and x is the number of water molecules, then x is what we're solving for. We can find x by dividing the number of moles of water by the number of moles of \( \mathrm{A} \). This procedure exemplifies hydrate stoichiometry and how it plays a crucial role in determining the formula of a hydrate.

Calculating the molar mass, which is the mass of one mole of a substance, is a fundamental skill in chemistry. The molar mass of a compound can be found by summing the atomic masses of each element in the compound, as given in the periodic table, multiplied by the number of atoms of that element in the formula.

To give a clearer illustration, let's consider the molar mass of water (\(\mathrm{H}_2\mathrm{O}\)). Oxygen has a molar mass of approximately 16.00 g/mol, and hydrogen is about 1.008 g/mol. Since water has two hydrogen atoms and one oxygen atom, its molar mass is calculated as follows: \(2 \times 1.008 \mathrm{g/mol} + 16.00 \mathrm{g/mol} = 18.016 \mathrm{g/mol}\). In the case of hydrates, calculating the molar mass of both the anhydrous compound and the attached water molecules is crucial for further stoichiometric calculations.

When venturing through chemical calculations, the conversion from moles to mass and vice versa is ubiquitous. This conversion utilizes the molar mass as a conversion factor and follows the formula: \( \text{Number of moles} (n) = \frac{\text{Mass} (m)}{\text{Molar mass} (M)} \).

Let's take a practical example: If we have a given mass of a substance and we wish to find out how many moles it represents, we simply divide the mass by the molar mass. Conversely, to find the mass when given moles, we multiply the number of moles by the molar mass. For example, to convert 2 moles of water to grams, you would calculate \(2 \text{ moles} \times 18.016 \text{ g/mol} = 36.032 \text{ g}\). This moles to mass conversion is significant when determining the x value in a hydrate's chemical formula, since it allows to correlate the mass of water lost upon heating to the quantity of water molecules that were originally present.