Question

Epsom salts, a strong laxative used in veterinary medicine, is a hydrate, which means that a certain number of water molecules are included in the solid structure. The formula for Epsom salts can be written as \(\mathrm{MgSO}_{4} \cdot x \mathrm{H}_{2} \mathrm{O},\) where \(x\) indicates the number of moles of \(\mathrm{H}_{2} \mathrm{O}\) per mole of \(\mathrm{MgSO}_{4}\) When 5.061 \(\mathrm{g}\) of this hydrate is heated to \(250^{\circ} \mathrm{C},\) all the water of hydration is lost, leaving 2.472 g of MgSO \(_{4} .\) What is the value of \(x ?\)

Short answer

The value of \(x\) is approximately 7, and the formula for Epsom salts can be written as MgSO4·7H2O.

Step-by-step solution

Calculate the mass of water lost during heating

To find the mass of water lost after heating the hydrate, subtract the mass of anhydrous MgSO4 from the mass of the hydrate: Mass of H2O = mass of hydrate - mass of anhydrous MgSO4 Mass of H2O = 5.061g - 2.472g Mass of H2O = 2.589g

Find the number of moles of MgSO4 and H2O

We can find the moles of MgSO4 and H2O using the mass and molar mass of each substance. Molar mass of MgSO4 = (24.3 + 32.1 + 4×16) g/mol = 120.4 g/mol. And molar mass of H2O = (2×1 + 16) g/mol = 18 g/mol. Number of moles of MgSO4 = mass of MgSO4 / molar mass of MgSO4 Number of moles of MgSO4 = 2.472 g / 120.4 g/mol ≈ 0.0205 mol Number of moles of H2O = mass of H2O / molar mass of H2O Number of moles of H2O = 2.589 g / 18 g/mol ≈ 0.1438 mol

Determine the value of x

The value of x represents the number of moles of H2O per mole of MgSO4. Therefore, we will divide the number of moles of H2O by the number of moles of MgSO4 to find the value of x. x = moles of H2O / moles of MgSO4 x = 0.1438 mol / 0.0205 mol ≈ 7 So, the value of x is approximately 7, and the formula for Epsom salts can be written as MgSO4·7H2O.

Key concepts

Chemical Formulas

Understanding chemical formulas is fundamental in chemistry as they represent the composition of a compound. For example, in the case of Epsom salts, the compound's formula is \(\mathrm{MgSO}_{4}\cdot x \mathrm{H}_{2} \mathrm{O}\), indicating that magnesium sulfate (\(\mathrm{MgSO}_{4}\)) is combined with water molecules (\(\mathrm{H}_{2}\mathrm{O}\)) in a specific ratio.

The 'dot' in the formula signifies that it's a hydrate, which means water molecules are integrated into the crystal structure of the solid in a definite ratio. The variable 'x' represents the number of water molecules associated with each formula unit of magnesium sulfate. By finding the value of 'x', we can better understand the precise nature of the compound and its properties. In hydration or dehydration reactions, it's crucial to account for water molecules in the compound because they often affect its physical characteristics and chemical behavior.

To write the formula correctly after determining 'x', we simply append \(x \mathrm{H}_{2} \mathrm{O}\) to the compound, reflecting the composition of the hydrate.

Mole Concept

The mole concept is a bridge between the macroscopic world we can measure and the microscopic world of molecules and atoms. One mole is defined as the amount of substance that contains as many entities (atoms, molecules, or ions) as there are in 12 grams of carbon-12. This number is known as Avogadro's number, \(6.022 \times 10^{23}\).

In the provided problem, the mole concept allows us to quantify the amount of magnesium sulfate and water in Epsom salts. By calculating the number of moles, we can find the ratio of magnesium sulfate to water molecules, represented by 'x' in the chemical formula. This mole-to-mole comparison provides a clear view of the compound's stoichiometry, enabling us to determine that Epsom salts are comprised of one mole of magnesium sulfate to approximately seven moles of water.

Molar Mass Calculations

Molar mass is a crucial parameter for quantitative chemistry calculations, offering a link between the mass of a substance and the amount of substance (in moles). It is defined as the mass of one mole of a given substance and its unit is grams per mole (g/mol).

To calculate the molar mass, sum the atomic masses of all atoms in the molecule. For instance, the molar mass of magnesium sulfate, \(\mathrm{MgSO}_{4}\), is calculated by adding the atomic masses of magnesium (Mg), sulfur (S), and four oxygen (O) atoms. Once molar masses are known, as in this exercise with \(\mathrm{MgSO}_{4}\) and \(\mathrm{H}_{2}\mathrm{O}\), we can convert the mass of these substances lost or gained during a reaction into the number of moles. This data is pivotal in determining stoichiometric ratios in compounds, which in the original problem leads to identifying the hydrate formula accurately as \(\mathrm{MgSO}_{4}\cdot 7\mathrm{H}_{2}\mathrm{O}\).