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' is approximately 7, which means there are 7 moles of \(\mathrm{H}_{2} \mathrm{O}\) per mole of \(\mathrm{MgSO}_{4}\). Therefore, the formula for Epsom salt hydrate is $\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}$.

Step-by-step solution

Calculate the mass of water in the hydrated Epsom salt

First, we need to find the mass of water in the hydrated Epsom salt. We can do this by subtracting the mass of anhydrous Epsom salt from the mass of hydrated Epsom salt: Mass of water = Mass of hydrated Epsom salt - Mass of anhydrous Epsom salt Mass of water = \(5.061 \mathrm{~g}\) - \(2.472 \mathrm{~g}\) Mass of water = \(2.589 \mathrm{~g}\)

Calculate the moles of Epsom salt (\(\mathrm{MgSO}_{4}\)) and water (\(\mathrm{H}_{2} \mathrm{O}\))

Next, we need to find the number of moles for both Epsom salt and water. To do this, we will use the molar mass of each compound: Moles of Epsom salt = Mass of anhydrous Epsom salt / Molar mass of \(\mathrm{MgSO}_{4}\) Moles of Epsom salt = \(2.472 \mathrm{~g}\) / \(120.37 \mathrm{~g/mol}\) Moles of Epsom salt = \(0.0205 \mathrm{~mol}\) Moles of water = Mass of water / Molar mass of \(\mathrm{H}_{2} \mathrm{O}\) Moles of water = \(2.589 \mathrm{~g}\) / \(18.015 \mathrm{~g/mol}\) Moles of water = \(0.1437 \mathrm{~mol}\)

Find the value of 'x'

Now, we need to find the number of moles of \(\mathrm{H}_{2} \mathrm{O}\) per mole of \(\mathrm{MgSO}_{4}\), which is represented by 'x'. We can do this by dividing the moles of water by the moles of Epsom salt: Value of x = Moles of water / Moles of Epsom salt Value of x = \(0.1437 \mathrm{~mol}\) / \(0.0205 \mathrm{~mol}\) Value of x ≈ 7 The value of 'x' is approximately 7, which means there are 7 moles of \(\mathrm{H}_{2} \mathrm{O}\) per mole of \(\mathrm{MgSO}_{4}\). Therefore, the formula for Epsom salt hydrate is $\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}$.

Key concepts

Hydrate Formula Determination

Understanding the composition of hydrates is crucial in chemistry, as it reveals the number of water molecules associated with a compound. Hydrates are substances that incorporate water into their molecular structure, often in a fixed ratio. To determine the hydrate formula, like in the case of Epsom salts (also known as magnesium sulfate hydrate), you need to identify the value of 'x', which represents the amount of water molecules per formula unit of the compound.

To find 'x', you first calculate the mass of the water lost upon heating, which is the difference between the hydrated and anhydrous compound. Next, you convert this mass to moles using the mole concept and molar masses of water and the anhydrous compound. By comparing the mole ratio of water to the anhydrous substance, you can determine the correct stoichiometric number to describe the hydrate formula. For Epsom salts, this step-by-step approach reveals that the correct hydrate formula is \(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}\).

Mole Concept

The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we observe. A mole represents Avogadro's number, which is approximately \(6.022 \times 10^{23}\) entities. This number is similar to using 'dozen' for counting, except it applies to atoms, ions, or molecules in chemistry.

To utilize the mole concept in practical situations, like finding the number of water molecules in a hydrate, we relate the mass of a substance (in grams) to the amount of substance (in moles). This comparison requires the use of molar mass and opens a pathway to understanding chemical ratios and formulas. In solving our exercise, we converted grams of \(\mathrm{MgSO}_{4}\) and \(\mathrm{H}_{2}\mathrm{O}\) to moles to find out how many moles of water were present per mole of \(\mathrm{MgSO}_{4}\).

Molar Mass Calculation

Molar mass is the weight of one mole of a substance and it is expressed in grams per mole (g/mol). Understanding how to calculate molar mass is fundamental because it allows you to convert between the mass of a substance and the number of moles—a key step in many stoichiometric problems.

To calculate the molar mass, you sum up the atomic masses of all the atoms in the molecular formula. For \(\mathrm{MgSO}_{4}\), you would add the atomic masses of magnesium, sulfur, and four oxygen atoms. Molar mass is typically found on the periodic table and used in calculations to find the number of moles, as seen in our Epsom salt exercise where we used the molar masses of \(\mathrm{H}_{2}\mathrm{O}\) and \(\mathrm{MgSO}_{4}\) to determine the moles of each substance. This information was then used to establish the hydrate formula.