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 \(2.472 \mathrm{~g}\) of \(\mathrm{MgSO}_{4}\). What is the value of \(x ?\)

Short answer

The value of x in the formula for Epsom salt is found using the steps: 1. Determine the moles of MgSO4 in the 2.472 g sample as \(\frac{2.472 \mathrm{~g}}{120.367 \mathrm{~g~mol}^{-1}}\). 2. Calculate the moles of H2O removed by heating from the mass of water lost \(2.589 \mathrm{~g}\) as \(\frac{2.589 \mathrm{~g}}{18.015 \mathrm{~g~mol}^{-1}}\). 3. Divide the moles of H2O by the moles of MgSO4 to find the value of x. This will result in the formula \(\mathrm{MgSO}_{4} \cdot x \mathrm{H}_{2} \mathrm{O}\), where x represents the moles of water molecules per mole of magnesium sulfate in Epsom salt.

Step-by-step solution

Calculate the moles of MgSO4

First, we need to determine the moles of MgSO4 present in the 2.472 g sample. To do this, we can use the molar mass of MgSO4: Mg: \(24.3050 \mathrm{~g~mol}^{-1}\) S: \(32.0660 \mathrm{~g~mol}^{-1}\) O: \(15.9990 \mathrm{~g~mol}^{-1}\) The molar mass of MgSO4 is: \[ M_{MgSO4} = 24.305 + 32.066 + 4 \times 15.999 = 120.367 \mathrm{~g~mol}^{-1} \] Now we can calculate the moles of MgSO4: \[ \mathrm{moles ~of ~MgSO4} = \frac{2.472 \mathrm{~g}}{120.367 \mathrm{~g~mol}^{-1}} \]

Calculate the moles of H2O removed by heating the Epsom salt hydrate

To find the moles of H2O removed by heating, first find the mass of the water lost: \[ \mathrm{mass ~of ~H2O~lost} = 5.061 \mathrm{~g} - 2.472 \mathrm{~g} = 2.589 \mathrm{~g} \] Next, we calculate the moles of H2O using its molar mass: H: \(1.00784 \mathrm{~g~mol}^{-1}\) O: \(15.9990 \mathrm{~g~mol}^{-1}\) The molar mass of H2O is: \[ M_{H2O} = 2 \times 1.00784 + 15.999 = 18.015 \mathrm{~g~mol}^{-1} \] Now we can calculate the moles of H2O: \[ \mathrm{moles ~of ~H2O} = \frac{2.589 \mathrm{~g}}{18.015 \mathrm{~g~mol}^{-1}} \]

Calculate the value of x, the moles of H2O per mole of MgSO4 in the Epsom salt hydrate

Finally, we can find the value of x by dividing the moles of H2O by the moles of MgSO4: \[ x = \frac{\mathrm{moles ~of ~H2O}}{\mathrm{moles ~of ~MgSO4}} \] Using the values we've previously calculated, we can determine the value of x and conclude that the formula of Epsom salt is \(\mathrm{MgSO}_{4} \cdot x \mathrm{H}_{2} \mathrm{O}\).

Key concepts

Molar Mass Calculation

Understanding molar mass is crucial when engaging with stoichiometry, the field of chemistry that quantifies the relationships between reactants and products in chemical reactions. The molar mass is the weight of one mole of a substance, usually expressed in grams per mole (\text{g/mol}). One mole contains Avogadro's number of particles, which is approximately 6.022 \(\times\) 10²³.

To calculate the molar mass, one must sum the atomic masses of all the atoms in the molecule. For example, to find the molar mass of \(MgSO_4\), you would add the atomic masses of magnesium (Mg), sulfur (S), and four oxygen (O) atoms:

\[M_{MgSO4} = 24.305 + 32.066 + 4 \times 15.999 = 120.367 \text{ g/mol}\]
This calculated molar mass serves as a conversion factor between grams and moles, enabling us to solve various stoichiometric problems.

Chemical Hydrates

In chemistry, a hydrate is a compound that contains water molecules integrated into its structure. These water molecules are not loosely attached; they are an integral part of the crystalline framework. This feature is seen in compounds like Epsom salts (\(MgSO_4 \cdot xH_2O\)), where \(x\) represents the moles of water per mole of the salt.

When a hydrate is heated, the water molecules are released. The remaining anhydrous compound typically has a lower mass. In stoichiometric problems involving hydrates, it's often necessary to calculate the number of moles of water per mole of the anhydrous compound. This is essential for identifying the empirical formula of the hydrate.

Real-life Application

Knowing the composition of hydrates is not just academic; it's practical. For example, in pharmaceuticals, the stability and solubility of hydrates can affect drug efficacy and shelf-life.

Mole Concept

The mole is a fundamental concept in chemistry that defines the quantity of a substance. One mole corresponds to Avogadro's number (6.022 \(\times\) 10²³) of particles, whether they are atoms, ions, or molecules. The mole provides a bridge between the microscopic world (atoms and molecules) and the macroscopic (gram-quantities we can measure).

For instance, if we consider the case study of Epsom salts, the mole concept allows us to deduce the number of water molecules associated with each magnesium sulfate unit. By comparing the moles of \(H_2O\) lost to the moles of \(MgSO_4\) remaining, we establish their stoichiometric relationship in the hydrate's empirical formula.

Understanding the mole concept is central to mastering stoichiometry, as it underpins the calculations for chemical reactions and the creation of solutions in the laboratory.