Problem 54 - Chemistry: Principles and Reactions

Question

Epsom salts are hydrates of magnesium sulfate. The formula for Epsom salts is \(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}\). A \(7.834\) -g sample is heated until a constant mass is obtained indicating that all the water has been evaporated off. What is the mass of the anhydrous magnesium sulfate? What percentage of the hydrate is water?

Step-by-Step Solution

Answer

Answer: The mass of anhydrous magnesium sulfate after heating the Epsom salt is approximately 3.822 g, and the percentage of the hydrate that is water is approximately 51.22 %.

Step 1: Calculate the molar mass of the Epsom salt and its anhydrous component

First, we need to calculate the molar mass of the Epsom salt, \(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}\), and its anhydrous component, \(\mathrm{MgSO}_{4}\). The molar mass of the Epsom salt is given by the sum of the molar masses of its components: Molar mass of \(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}\) = Molar mass of \(\mathrm{Mg} + \mathrm{S} + 4(\mathrm{O}) + 7(2(\mathrm{H}) + \mathrm{O})\) Use the periodic table to determine the molar masses of each element: Mg (magnesium) = 24.3 g/mol S (sulfur) = 32.06 g/mol O (oxygen) = 16 g/mol H (hydrogen) = 1.01 g/mol Now, calculate the molar mass of \(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{MgSO}_{4}\): Molar mass of \(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}\) = 24.3 + 32.06 + 4(16) + 7(2(1.01) + 16) = 246.47 g/mol Molar mass of \(\mathrm{MgSO}_{4}\) = 24.3 + 32.06 + 4(16) = 120.37 g/mol

Step 2: Calculate the mass of the water and anhydrous magnesium sulfate

The initial mass of the sample is given as 7.834 g. To find the mass of the water and anhydrous magnesium sulfate, we need to find the ratio of their molar masses: Ratio = Molar mass of \(\mathrm{MgSO}_{4}\)/Molar mass of \(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}\) = 120.37/246.47 = 0.488 Now, multiply the initial mass by the ratio to find the mass of anhydrous magnesium sulfate after heating: Mass of anhydrous \(\mathrm{MgSO}_{4}\) = Initial mass × ratio = 7.834 × 0.488 = 3.822 g To find the mass of the water, subtract the mass of anhydrous \(\mathrm{MgSO}_{4}\) from the initial mass: Mass of water = Initial mass - Mass of anhydrous \(\mathrm{MgSO}_{4}\) = 7.834 - 3.822 = 4.012 g

Step 3: Calculate the percentage of the hydrate that is water

To calculate the percentage of the hydrate that is water, divide the mass of water by the initial mass of the sample and multiply by 100: Percentage of water = (Mass of water / Initial mass) × 100 = (4.012 / 7.834) × 100 ≈ 51.22 % The mass of the anhydrous magnesium sulfate is approximately 3.822 g, and the percentage of the hydrate that is water is approximately 51.22 %.

Key Concepts

Epsom Salts

Epsom salts, scientifically known as magnesium sulfate heptahydrate, are a fascinating compound. They are frequently used in therapeutic baths to soothe muscles and reduce inflammation. The chemical formula for Epsom salts is \( \mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O} \). This formula indicates that each molecule of magnesium sulfate is associated with seven molecules of water. Intriguingly, this makes Epsom salts a type of hydrate—a compound that includes water molecules integrated into its crystal structure.

When Epsom salts are heated, they lose their water content and convert into an anhydrous form, meaning they no longer contain water. This process of losing water is known as dehydration, and it allows us to determine the anhydrous magnesium sulfate's mass from the original hydrate sample, which can have important applications in understanding chemical compositions and reactions.

Molar Mass Calculation

Molar mass is crucial when dealing with chemical compounds because it helps us understand the amount of substance present. The molar mass of a compound is the sum of the molar masses of its individual atoms. Using the periodic table, each atom's molar mass can be found in grams per mole (g/mol).

For Epsom salts \( \mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O} \), we add the molar masses:

• \(\mathrm{Mg} = 24.3\, \text{g/mol}\)
• \(\mathrm{S} = 32.06\, \text{g/mol}\)
• \(\mathrm{O} = 16\, \text{g/mol}\) * 4
• Water \((\mathrm{H}_{2} \mathrm{O}) = 18.02\, \text{g/mol}\) * 7

When calculated, this results in a total molar mass for Epsom salts of approximately 246.47 g/mol and for anhydrous magnesium sulfate (\( \mathrm{MgSO}_{4} \)) 120.37 g/mol.

Hydration and Anhydrous Compounds

A pivotal concept in chemistry is hydration, which refers to compounds that incorporate water molecules into their structure. These are called hydrates. Epsom salts are a hydrate, meaning they have a specific stoichiometric amount of water linked to their structure, as seen in \( \mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O} \). This feature significantly influences the substance's properties, such as solubility and melting point.

In contrast, anhydrous compounds are those from which water has been removed, either through heating or other methods of dehydration. For magnesium sulfate, once it has been dehydrated, it is termed anhydrous magnesium sulfate (\( \mathrm{MgSO}_{4} \)). The transition from hydrated to anhydrous form typically involves weight loss, dedicated solely to water evaporation, which can be measured to determine the proportion of water in the hydrate.

Percentage Composition

Percentage composition is a simple yet powerful tool in chemistry that explains the relative amount of each component in a compound, expressed in percentage terms. Here, it is calculated to determine what proportion of Epsom salts is made up of water.

To find the water percentage in Epsom salts, you start with the total mass of water lost during heating and divide it by the initial mass of the Epsom salt sample. This gives the water's fraction in the salt. Multiplying this fraction by 100 converts it to a percentage.

In this exercise, the result demonstrated that approximately 51.22% of the original hydrate sample is water. This means that over half of the Epsom salt's mass is due to its water content, which is intrinsic to its formula, \( \mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O} \). Understanding this percentage helps in grasping how hydrates behave physically and chemically.