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 \(x\) in the formula for Epsom salts, \(MgSO_{4} \cdot xH_{2}O\), is 7. Consequently, the formula for Epsom salts is \(MgSO_{4} \cdot 7H_{2}O\).

Step-by-step solution

Calculate moles of MgSO4

To find the moles of MgSO4, we will first find the molar mass of MgSO4 and then use the formula: Moles = Mass / Molar Mass. Molar mass of MgSO4 can be calculated using the formula: \(Molar \, Mass = Mass_{Mg} + Mass_{S} + 4 \times Mass_{O} = 24.305 \mathrm{~g/mol} + 32.07 \mathrm{~g/mol} + 4 \times 16.00 \mathrm{~g/mol} = 120.37 \mathrm{~g/mol}\) Next, we will find the moles of MgSO4 using the formula: \(Moles = \frac{Mass}{Molar \, Mass} = \frac{2.472 \mathrm{~g}}{120.37 \mathrm{~g/mol}} = 0.02053 \mathrm{~mol}\)

Calculate moles of water

To find the moles of water, we will first find the mass of the water lost during the dehydration process. Then we will find the moles of water using the formula: Moles = Mass / Molar Mass. Mass of water lost = Mass of hydrate - Mass of anhydrous salt \(Mass \, of \, water \, lost = 5.061 \mathrm{~g} - 2.472 \mathrm{~g} = 2.589 \mathrm{~g}\) Now we will find the moles of water using the formula: \(Moles = \frac{Mass}{Molar \, Mass} = \frac{2.589 \mathrm{~g}}{18.015 \mathrm{~g/mol}} = 0.1437 \mathrm{~mol}\)

Calculate the value of x

With the moles of MgSO4 and water calculated, we can find the value of x using the formula: \(x = \frac{Moles \, of \, H_{2}O}{Moles \, of \, MgSO_{4}}\) \(x = \frac{0.1437 \mathrm{~mol}}{0.02053 \mathrm{~mol}} = 7.00\) Therefore, the formula for Epsom salts is: \(MgSO_{4} \cdot 7H_{2}O\).