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?

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

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

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 %.