Question

Epsom salts are hydrated crystals of magnesium sulfate, \(\mathrm{MgSO}_{4} \cdot x \mathrm{H}_{2} \mathrm{O} .\) When \(6.499 \mathrm{~g}\) of Epsom salts are dehydrated by heating, 3.173 g remain. What is the mass percent of water in the hydrated sample? What is \(x\) ?

Short answer

Answer: The mass percent of water in the hydrated magnesium sulfate sample is 51.2%, and the stoichiometric coefficient \(x\) is 7. The formula for hydrated Epsom salts is \(\mathrm{MgSO}_4 \cdot 7 \mathrm{H}_2\mathrm{O}\).

Step-by-step solution

Find the mass of water lost during dehydration

First, we need to determine the mass of water that was lost during the dehydration process. To do so, subtract the mass after dehydration from the initial mass of the Epsom salts. Initial mass of Epsom salts: \(6.499 g\) Mass after dehydration: \(3.173 g\) Mass of water lost: \(6.499 g - 3.173 g = 3.326 g\)

Calculate the mass percent of water in the hydrated sample

Now that we know the mass of water lost, we can calculate the mass percent of water in the initial hydrated sample. To do this, divide the mass of water lost by the initial mass of the Epsom salts and multiply by 100. Mass percent of water: \(\frac{3.326\ \mathrm{g}}{6.499\ \mathrm{g}} \times 100\% = 51.2\%\)

Determine the molar mass of magnesium sulfate and water

In order to find the value of \(x\), we need to know the molar masses of magnesium sulfate (\(\mathrm{MgSO}_4\)) and water (\(\mathrm{H}_2\mathrm{O}\)). Molar mass of Mg: \(24.31\ \mathrm{g/mol}\) Molar mass of S: \(32.07\ \mathrm{g/mol}\) Molar mass of O: \(16.00\ \mathrm{g/mol}\) So, Molar mass of \(\mathrm{MgSO}_4 = 24.31 + 32.07 + 4(16.00) = 120.37\ \mathrm{g/mol}\) Molar mass of H: \(1.01\ \mathrm{g/mol}\) Then, Molar mass of \(\mathrm{H}_2\mathrm{O} = 2(1.01) + 16.00 = 18.02\ \mathrm{g/mol}\)

Calculate the moles of magnesium sulfate and water

Now that we have the molar masses of magnesium sulfate and water, we can calculate the moles of each substance using the masses obtained in Step 1 and the molar masses obtained in Step 3. Moles of magnesium sulfate: \(\frac{3.173\ \mathrm{g}}{120.37\ \mathrm{g/mol}} = 0.0264\ \mathrm{mol}\) Moles of water: \(\frac{3.326\ \mathrm{g}}{18.02\ \mathrm{g/mol}} = 0.1846\ \mathrm{mol}\)

Determine the value of \(x\)

With both the moles of magnesium sulfate and water calculated, we can now find the stoichiometric coefficient \(x\). Divide the moles of water by the moles of magnesium sulfate and round to the nearest whole number. \(x = \frac{\text{moles of water}}{\text{moles of magnesium sulfate}} = \frac{0.1846\ \mathrm{mol}}{0.0264\ \mathrm{mol}} = 6.99 \approx 7\) Therefore, the value of \(x\) is 7, and the formula for hydrated Epsom salts is \(\mathrm{MgSO}_4 \cdot 7 \mathrm{H}_2\mathrm{O}\). The mass percent of water in the hydrated sample is 51.2%.

Key concepts

Mass Percent of Water

The mass percent of water in a substance is a measure of the amount of water, by mass, contained in a compound compared to the total mass of the compound. In the context of hydrated salts, this is the mass of water that has been chemically bonded to the salt crystals.

Calculating the mass percent is a straightforward process. First, determine the mass of water by subtracting the mass of the anhydrous salt (after water is lost) from the initial mass of the hydrated salt. Then, calculate the mass percent by dividing the mass of water by the initial mass of the hydrated compound and multiplying by 100. For example, with the dehydrated Epsom salts exercise, the mass percent of water is found to be 51.2%.

Molar Mass Calculation

Understanding the concept of molar mass is crucial in chemistry. It represents the mass of one mole of a substance and is expressed in grams per mole (g/mol). The molar mass can be calculated by summing the atomic masses of all atoms present in a chemical formula.

For instance, to find the molar mass of magnesium sulfate \( \mathrm{MgSO}_4 \), you add the molar masses of magnesium, sulfur, and oxygen. These calculations are fundamental in stoichiometry as they allow chemists to convert between mass and moles of a substance, providing a bridge between the macroscopic and molecular world.

Stoichiometry

Stoichiometry is like the recipe for a chemical reaction, dictating the proportional relationship between reactants and products. In stoichiometry, the mole concept is critical as it allows comparison of quantities of different substances involved in a reaction.

In the Epsom salts example, stoichiometry is used to find the ratio of moles of water to moles of anhydrous magnesium sulfate. This ratio helps determine the number of water molecules associated with each formula unit of magnesium sulfate, known as the hydration number. The process of identifying this ratio is a fundamental application of stoichiometry in determining the formulas of hydrated ionic compounds.

Chemical Formula Determination

Determining the chemical formula of a compound, especially a hydrated one, involves using stoichiometric calculations to find the mole ratio of its components. With hydrates, this process unveils how many water molecules are included per formula unit of the salt.

In the case of hydrated Epsom salts, this means identifying the 'x' in \( \mathrm{MgSO}_4 \cdot x \mathrm{H}_2\mathrm{O} \). The steps involve finding moles of both the anhydrous compound and the water, and then determining their mole ratio. In our exercise, the calculation concluded that 'x' equals 7, leading to the formula \( \mathrm{MgSO}_4 \cdot 7 \mathrm{H}_2\mathrm{O} \). Understanding this process of finding the chemical formula is essential in chemistry, as it reveals the exact nature of a substance.