Question

Epsom salt is used in tanning leather and in medicine. It is hydrated magnesium sulfate, \(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O} .\) The water of hydration is lost on heating, with the number lost depending on the temperature. Suppose you heat a \(1.394-\mathrm{g}\) sample at \(100^{\circ} \mathrm{C}\) and obtain \(0.885 \mathrm{g}\) of a partially hydrated sample, \(\mathrm{MgSO}_{4} \cdot x \mathrm{H}_{2} \mathrm{O} .\) What is the value of \(x ?\)

Short answer

The value of \(x\) is 4.

Step-by-step solution

Determine Mass of Water Lost

Start by calculating the mass of water that is lost during heating. The initial mass of the hydrated sample is given as 1.394 g and the mass after heating is 0.885 g. Subtract the final mass from the initial mass to find the mass of the water lost.\[\text{Mass of water lost} = 1.394 \text{ g} - 0.885 \text{ g} = 0.509 \text{ g}\]

Calculate Moles of Water Lost

Next, calculate the moles of water lost. Use the molar mass of water, which is approximately 18.015 g/mol, to convert the mass of water lost to moles.\[\text{Moles of } \mathrm{H_2O} = \frac{0.509 \text{ g}}{18.015 \text{ g/mol}} \approx 0.0283 \text{ mol}\]

Determine Moles of MgSO4

Calculate the moles of anhydrous MgSO4 in the sample after heating. The mass of this anhydrous part is 0.885 g. The molar mass of MgSO4 is approximately 120.366 g/mol.\[\text{Moles of } \mathrm{MgSO_4} = \frac{0.885 \text{ g}}{120.366 \text{ g/mol}} \approx 0.00735 \text{ mol}\]

Calculate the Number of Water Molecules

Determine the value of \(x\), which indicates the number of water molecules per formula unit of MgSO4 in the partially hydrated sample. This is done using the ratio of moles of water to moles of MgSO4.\[x = \frac{\text{Moles of Water}}{\text{Moles of \(\mathrm{MgSO_4}\)}} = \frac{0.0283}{0.00735} \approx 3.85\]

Round to the Nearest Whole Number

Since \(x\) represents the number of water molecules, it should be a whole number. Round the calculated value of 3.85 to the nearest whole number to find that \(x = 4\).

Key concepts

Hydration of Salts

Hydration of salts is an interesting chemical concept that involves water molecules associating with a salt compound. In chemistry, a hydrated salt has a specific number of water molecules attached to it in its solid form. For example, Epsom salt, commonly used in medicine, is magnesium sulfate with water molecules, denoted as \(\mathrm{MgSO}_4 \cdot 7\mathrm{H}_2\mathrm{O}\). Here, the salt is hydrated with seven water molecules.When hydrated salts are heated, these water molecules can be lost. This is called dehydration. Depending on the temperature, different numbers of water molecules will be lost. This property can be used to determine the level of hydration in a salt. The number of water molecules that remain after heating provides insight into a salt's hydrous state.

Moles Calculation

When working with chemical reactions and compounds, knowing the quantity of substance is crucial. This is where moles calculation comes in handy. A mole is a unit that measures the amount of a substance based on the number of particles, such as atoms or molecules, it contains. One mole has precisely \(6.022 \times 10^{23}\) particles, known as Avogadro's number.In practice, calculating moles involves using the mass of the substance and its molar mass. The molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol). To find the number of moles in a sample, you divide the sample's mass by its molar mass:\[\text{Moles} = \frac{\text{Mass of Sample}}{\text{Molar Mass}}.\]This calculation allows us to determine how many individual units, or molecules, are present in a given mass of a compound, which is essential for any stoichiometry problem.

Mass-Mole Conversion

In chemistry, the mass-mole conversion is a fundamental operation when analyzing substances. This process links the macroscopic world that we can measure with the microscopic world of atoms and molecules, utilizing the concept of the mole.To perform a mass-mole conversion, you need the mass of the substance and its molar mass. For instance, when you have the mass and need to find moles, you use the formula:\[\text{Moles} = \frac{\text{Mass of Substance}}{\text{Molar Mass}}.\]Conversely, if you know the moles and need the mass, you rearrange the formula:\[\text{Mass} = \text{Moles} \times \text{Molar Mass}.\]Understanding mass-mole conversions is crucial for calculating reactants/products in chemical reactions. Whether working in a laboratory or solving theoretical problems, mastering this conversion is essential for accurate measurements and predictions in stoichiometry.