Question

Epsom salts is a hydrated ionic compound with the following formula: \(\mathrm{MgSO}_{4} \cdot x \mathrm{H}_{2} \mathrm{O} . \mathrm{A} 4.93-\mathrm{g}\) sample of Epsom salts was heated to drive of the water of hydration. The mass of the sample after complete dehydration was 2.41 \(\mathrm{g} .\) Find the number of waters of hydration \((x)\) in Epsom salts.

Short answer

The number of waters of hydration \(x\) in Epsom salts is 7.

Step-by-step solution

Determine Mass of Water

Calculate the mass of the water lost by subtracting the mass of the anhydrous compound (after dehydration) from the mass of the hydrated compound (before dehydration). The mass of the water is given by: Mass of hydrated compound - Mass of anhydrous compound = 4.93 g - 2.41 g = 2.52 g.

Determine Moles of Anhydrous Compound

Calculate the moles of the anhydrous compound, magnesium sulfate \(MgSO_4\), using its molar mass (120.36 g/mol). The number of moles is given by: Number of moles = Mass of anhydrous compound / Molar mass of anhydrous compound = 2.41 g / 120.36 g/mol.

Determine Moles of Water

Calculate the moles of the water using its molar mass (18.015 g/mol). The number of moles is given by: Number of moles = Mass of water / Molar mass of water = 2.52 g / 18.015 g/mol.

Determine the Molar Ratio and Solve for x

Find the molar ratio of water to anhydrous compound to determine x, the number of waters of hydration. This is given by: Molar ratio = Moles of water / Moles of anhydrous compound. Solve for x to find the number of water molecules per formula unit of \(MgSO_4\).

Key concepts

Understanding Hydrated Ionic Compounds

When you picture salt, you might think of the dry grains you sprinkle on your food. However, salts don't always come dry. Many, like the Epsom salts in our exercise, exist as hydrated ionic compounds. These are salts that have water molecules incorporated into their crystal structure. This water isn't just mixed in; it's chemically bonded to the salt. These water molecules are known as waters of hydration, and they can vastly change the properties of the salt, such as its color, solubility, and even its mass.

When Epsom salts are heated, this bound water is driven off, leaving behind an anhydrous (water-free) compound. The process of heating to remove this water is known as dehydration. Understanding the concept of hydrated salts is essential for analyzing the exercise about finding the number of waters of hydration in a given sample of Epsom salt.

Diving into Molar Mass

The molar mass is an incredibly important concept in chemistry. It's essentially the mass of one mole of a substance, usually expressed in grams per mole (g/mol). You can think of it as the chemical equivalent of counting eggs by the dozen – it’s a way to measure entities at the atomic or molecular scale. For Epsom salts (\( MgSO_4 \times xH_2O \times \) ), and for water (\( H_2O \times \) ), the molar mass tells us how much one mole of each compound weighs.

In our exercise, the molar mass of anhydrous Epsom salts (\( MgSO_4 \times \) ) and the molar mass of water (\( H_2O \times \) ) are used to convert the mass of substances lost upon heating and the mass remaining into moles, an essential step towards solving the problem.

Stoichiometry: The Art of Chemical Ratios

Stoichiometry may sound intimidating, but it's really just dealing with the relationships between the quantities of reactants and products in a chemical reaction. It plays a starring role in the exercise at hand. Through stoichiometry, we use the concept of the mole—the standard unit for 'amount of substance' in chemistry—to relate the masses of the Epsom salts before and after dehydration with their respective molar masses.

Using stoichiometry, we find the ratio of moles of water lost to the moles of the remaining anhydrous compound. This reveals, in a precise way, how many water molecules were originally associated with each formula unit of the Epsom salts. This ratio is key to solving the exercise and finding the value of x, which represents the number of water molecules per formula unit of Epsom salts.

Empirical Formula Calculation: The Blueprint of a Compound

When chemists talk about the empirical formula, they're referring to the simplest whole-number ratio of elements in a compound. It’s like the compound’s basic blueprint, showing the fundamental composition without diving into how it’s arranged or its actual molecular size. For our exercise, we need to know the empirical formula of the hydrated Epsom salts to determine the number of waters of hydration, x.

In practice, the calculation involves dividing the number of moles of each element by the smallest number of moles present among all the elements. The result gives us the proportions of each element in the compound. However, in the case of hydrated salts, we are finding the ratio of water molecules to the salt formula units through the steps of dehydration and stoichiometric calculations. Once we determine the number of moles of water compared to anhydrous compound, we've solved for x and found our empirical formula.