Question

A sample of the compound MSO \(_{4}\) weighing \(0.1131 \mathrm{g}\) reacts with barium chloride and yields \(0.2193 \mathrm{g} \mathrm{BaSO}_{4}\) What must be the atomic mass of the metal M? [Hint: All the \(\mathrm{SO}_{4}^{2-}\) from the MSO \(_{4}\) appears in the \(\mathrm{BaSO}_{4}\) ]

Short answer

The atomic mass of M is \(24.4 g/mol\).

Step-by-step solution

Calculate Mass of \(\mathrm{SO}_{4}^{2-}\) in \(\mathrm{BaSO}_{4}\)

To find the mass of \(\mathrm{SO}_{4}^{2-}\), the molecular mass of \(\mathrm{BaSO}_{4}\), which is 233.39 g/mol, is used. Of that, 96.06 g/mol is attributable to \(\mathrm{SO}_{4}^{2-}\). By calculating the percentage, it is found that \(\mathrm{SO}_{4}^{2-}\) accounts for \(96.06/233.39 = 0.4116 = 41.16\%\) of the mass of \(\mathrm{BaSO}_{4}\). Therefore, the mass of \(\mathrm{SO}_{4}^{2-}\) in 0.2193g of \(\mathrm{BaSO}_{4}\) is \(0.2193g * 0.4116 = 0.0902g\).

Determine the Mass of M

Since the mass of \(\mathrm{SO}_{4}^{2-}\) in MSO4 equals the mass of \(\mathrm{SO}_{4}^{2-}\) in \(\mathrm{BaSO}_{4}\), the mass of the metal M in MSO4 is simply the total mass of MSO4 minus the mass of \(\mathrm{SO}_{4}^{2-}\). Therefore, the mass of M is \(0.1131g - 0.0902g = 0.0229g\).

Calculate the Atomic Mass of M

The atomic mass of M will be the mass of M divided by the number of moles of M. The number of moles of \(\mathrm{SO}_{4}^{2-}\) can be calculated from the mass of \(\mathrm{SO}_{4}^{2-}\) using its molar mass \(96.06g/mol\), thereby giving \(0.0902g/96.06g/mol = 0.00094mol\). Hence, the atomic mass of M equals \(0.0229g / 0.00094mol = 24.4g/mol\).

Key concepts

Barium Sulfate Precipitation

When a compound containing sulfate ions, such as MSO_4, reacts with barium chloride, a chemical reaction occurs. The sulfate ions pair with barium ions to form barium sulfate, written as BaSO_4. This compound is insoluble in water, meaning it forms a solid precipitate. The formation of a visible precipitate helps verify the presence of sulfate ions in a solution.
The overall equation for the reaction is:\[ \text{MSO}_4(aq) + \text{BaCl}_2(aq) \rightarrow \text{BaSO}_4(s) + \text{MCl}_2(aq) \]This equation shows that barium sulfate precipitates out of the solution while the remaining ions stay dissolved. This allows scientists and chemists to isolate and measure the sulfate ions through the precipitated barium sulfate.

Molecular Mass

Molecular mass, or molecular weight, is the sum of the atomic masses of all atoms in a molecule. It's typically expressed in grams per mole (g/mol), allowing chemists to convert between mass and amount.
In this context, the molecular mass of barium sulfate (BaSO_4) is crucial. It is calculated by adding the atomic masses of barium (137.33 g/mol), sulfur (32.07 g/mol), and four oxygen atoms (4 x 16.00 g/mol). The total is 233.39 g/mol.

• The barium accounts for the bulk of this mass, contributing 137.33 g/mol.
• Sulfur adds 32.07 g/mol.
• Oxygen's contribution is 64.00 g/mol.

Understanding molecular mass is essential for quantitative chemical analysis, enabling precise calculation of the amounts of substances involved in chemical reactions.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It allows chemists to predict the amounts needed or produced based on balanced chemical equations.
In the barium sulfate precipitation reaction, stoichiometry tells us that one mole of MSO_4 reacts with one mole of BaCl_2 to yield one mole of BaSO_4. This 1:1:1 mole ratio ensures that you can calculate the mass of any component involved in the reaction using molar masses.
For example, knowing the stoichiometry and mass of BaSO_4 allows determination of the mass of sulfate ions in the original MSO_4 compound. Thus, stoichiometry is a fundamental concept that bridges chemical equations with real-world measurements and predictions.

Moles Calculation

The concept of moles is a core idea in chemistry, acting as a bridge between the atomic world and macroscopic measurements. A mole is equivalent to Avogadro's number (approximately \(6.022 \times 10^{23}\)) of particles, whether atoms, molecules, or ions.
Calculating moles involves dividing a substance's mass by its molar mass to determine the number of moles present. For instance, the mass of sulfate ions (\(\text{SO}_4^{2-}\)) in the problem is 0.0902 g. Using the molar mass of sulfate (96.06 g/mol), the moles are calculated as:\[ \text{Moles of } \text{SO}_4^{2-} = \frac{0.0902 \text{ g}}{96.06 \text{ g/mol}} = 0.00094 \text{ mol} \]This calculation is instrumental in determining other unknowns in a chemical reaction, such as the atomic mass of an unknown metal in a compound. It's an essential step in applying theoretical chemistry to practical problems.