Problem 60 - Chemistry: The Central Science

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 \(x\) ?

Step-by-Step Solution

Answer

The value of \(x\) in the formula for Epsom salts \(\mathrm{MgSO}_{4} \cdot x \mathrm{H}_{2}\mathrm{O}\) is 7, meaning there are 7 moles of water molecules associated with each mole of magnesium sulfate in the hydrate.

Step 1: Calculate the moles of anhydrous magnesium sulfate

To calculate the moles of \(\mathrm{MgSO}_{4}\), we need to use the mass given and its molar mass. The molar mass of \(\mathrm{MgSO}_{4}\) is approximately \(120.4 \ \mathrm{g/mol}\). Using the mass given: moles of \(\mathrm{MgSO}_{4} = \frac{mass}{molar\ mass} = \frac{2.472 \ \mathrm{g}}{120.4 \ \mathrm{g/mol}} = 0.02053 \mathrm{mol}\)

Step 2: Find the mass of water

To find the mass of water, we can subtract the mass of anhydrous \(\mathrm{MgSO}_{4}\) from the mass of the initial hydrate: mass of \(\mathrm{H}_{2}\mathrm{O} = mass\ of\ hydrate - mass\ of\ anhydrous\ \mathrm{MgSO}_{4} = 5.061 \ \mathrm{g} - 2.472 \ \mathrm{g} = 2.589 \ \mathrm{g}\)

Step 3: Calculate the moles of water

To calculate the moles of \(\mathrm{H}_{2}\mathrm{O}\), we need to use the mass found in the previous step and its molar mass. The molar mass of \(\mathrm{H}_{2}\mathrm{O}\) is approximately \(18.02 \ \mathrm{g/mol}\). Using the mass of water: moles of \(\mathrm{H}_{2}\mathrm{O} = \frac{mass}{molar\ mass} = \frac{2.589 \ \mathrm{g}}{18.02 \ \mathrm{g/mol}} = 0.1437 \mathrm{mol}\)

Step 4: Find the ratio of moles of water to moles of magnesium sulfate

Now we can find the value of x by dividing the moles of \(\mathrm{H}_{2}\mathrm{O}\) by the moles of \(\mathrm{MgSO}_{4}\): x = \(\frac{moles\ of\ H_{2}O}{moles\ of\ MgSO_{4}} = \frac{0.1437 \ \mathrm{mol}}{0.02053 \ \mathrm{mol}} = 7\)

Step 5: Write the formula with the value of x

Now that we know the value of x, we can write the formula for the Epsom salts: Epsom salts formula = \(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2}\mathrm{O}\) The value of x is 7, which means that there are 7 moles of water molecules associated with each mole of magnesium sulfate in the Epsom salts hydrate.

Key Concepts

Magnesium Sulfate

Magnesium sulfate is a chemical compound consisting of magnesium, sulfur, and oxygen, with the chemical formula \( \mathrm{MgSO}_4 \). It is commonly found in a hydrated form, known as Epsom salts, which are used in various applications including as a laxative in veterinary medicine.
When magnesium sulfate is in its anhydrous form, meaning without water, it appears as a white crystalline solid. However, it naturally integrates water molecules into its structure, forming a hydrate.
This ability to incorporate water makes magnesium sulfate an excellent example to study hydrates in chemistry, especially when examining the stoichiometry of hydrated compounds.

Moles Calculation

In chemistry, calculating moles is crucial to understand and quantify substances in a reaction or formula. The mole is a unit that expresses the amount of a chemical substance, and it is based on the number of atoms found in 12 grams of carbon-12. One mole typically corresponds to \( 6.022 \times 10^{23} \) particles (Avogadro's number).
To calculate the moles of any substance, you'll need two things: the mass of the substance and its molar mass. The molar mass is the mass of one mole of a substance and is usually expressed in grams per mole (g/mol).
For magnesium sulfate in its anhydrous form (\( \mathrm{MgSO}_4 \)), the molar mass is approximately \( 120.4 \, \mathrm{g/mol} \). Therefore, to find the number of moles of \( \mathrm{MgSO}_4 \), you divide the mass by the molar mass. This straightforward equation helps pinpoint the amount present in a given mass of compound: \( \text{moles} = \frac{\text{mass}}{\text{molar\ mass}} \).

Water of Hydration

Hydrates are compounds that include water molecules within their crystalline structure. These water molecules are not just physically trapped, but chemically bound and can often impact the properties of the material.
The term "water of hydration" refers to the number of water molecules associated with each formula unit of the compound. This is often represented in the chemical formula by a dot, like \( \mathrm{MgSO}_4 \cdot x \mathrm{H}_2\mathrm{O} \), where \( x \) denotes the number of water molecules.
For instance, in Epsom salts, experiments reveal that there are 7 moles of water (\( \mathrm{H}_2\mathrm{O} \)) per mole of magnesium sulfate (\( \mathrm{MgSO}_4 \)), resulting in the formula \( \mathrm{MgSO}_4 \cdot 7 \mathrm{H}_2\mathrm{O} \). This specific hydration level is crucial for the properties of Epsom salts and is determined through experimental measurements, like heating the hydrate and measuring the water released.