Problem 98 - Chemistry and Chemical Reactivity

Question

Stibnite, \(\mathrm{Sb}_{2} \mathrm{S}_{3},\) is a dark gray mineral from which antimony metal is obtained. What is the weight percent of antimony in the sulfide? If you have \(1.00 \mathrm{kg}\) of an ore that contains \(10.6 \%\) antimony, what mass of \(\mathrm{Sb}_{2} \mathrm{S}_{3}\) (in grams) is in the ore?

Step-by-Step Solution

Answer

The weight percent of antimony in Sb2S3 is approximately 71.69%. For a 1.00 kg ore sample containing 10.6% antimony, approximately 147.88 g of Sb2S3 is in the ore.

Step 1: Calculate the molar mass of Sb2S3

To find the molar mass of \ \( \mathrm{Sb}_2 \mathrm{S}_3 \ \), calculate the total mass by adding the masses of two antimony (Sb) atoms and three sulfur (S) atoms. The atomic mass of Sb is approximately 121.76 g/mol and S is 32.07 g/mol. So the molar mass of \ \( \mathrm{Sb}_2 \mathrm{S}_3 \ \) is:\[ \text{Molar mass of } \mathrm{Sb}_2 \mathrm{S}_3 = 2(121.76 \, \text{g/mol}) + 3(32.07 \, \text{g/mol}) \]\[ = 243.52 \, \text{g/mol} + 96.21 \, \text{g/mol} \= 339.73 \, \text{g/mol} \]

Step 2: Calculate the mass percent of antimony in Sb2S3

Use the molar mass of \ \( \mathrm{Sb}_2 \mathrm{S}_3 \ \) from Step 1 to determine the weight percent of antimony (Sb) in the compound. The mass of antimony in one mole of \ \( \mathrm{Sb}_2 \mathrm{S}_3 \ \) is the sum of the mass of two antimony atoms.\[ \text{Mass percent of Sb} = \left(\frac{2 \times 121.76 \, \text{g/mol}}{339.73 \, \text{g/mol}}\right) \times 100\% \= \left(\frac{243.52}{339.73}\right) \times 100\% \\approx 71.69\% \]

Step 3: Determine the mass of antimony in the ore

With 10.6% antimony in a 1.00 kg ore sample, calculate the total mass of antimony:\[ \text{Mass of Sb} = \left(\frac{10.6}{100}\right) \times 1000 \, \text{g} \= 106 \, \text{g} \]

Step 4: Calculate mass of Sb2S3 corresponding to the antimony

Find the mass of \ \( \mathrm{Sb}_2 \mathrm{S}_3 \ \) that contains 106 g of antimony, using the weight percent from Step 2:\[ \text{Mass of } \mathrm{Sb}_2 \mathrm{S}_3 = \left(\frac{106}{71.69}\right) \times 100 \\approx 147.88 \, \text{g} \]

Key Concepts

Molar Mass Calculation

When calculating the molar mass of a compound like \( \mathrm{Sb}_2 \mathrm{S}_3 \), we need to find the sum of the atomic masses of each atom in the formula. The molar mass represents the mass of one mole of a substance and is expressed in grams per mole (g/mol). To find the molar mass of \( \mathrm{Sb}_2 \mathrm{S}_3 \), add:

• the mass of two antimony (Sb) atoms
• the mass of three sulfur (S) atoms

The atomic mass of antimony (\( \mathrm{Sb} \)) is approximately 121.76 g/mol, while sulfur (\( \mathrm{S} \)) is 32.07 g/mol. Therefore, the calculation would look like this: \[2 \times 121.76 \, \text{g/mol} + 3 \times 32.07 \, \text{g/mol} = 243.52 \, \text{g/mol} + 96.21 \, \text{g/mol} = 339.73 \, \text{g/mol}\]Understanding molar mass is crucial for stoichiometry and helps us figure out how much of each element is in a compound.

Chemical Formulas

Chemical formulas are symbols and numbers that display the types and numbers of atoms in a molecule. For example, in \( \mathrm{Sb}_2 \mathrm{S}_3 \), it indicates there are two atoms of antimony and three atoms of sulfur involved in the molecule. These formulas help visualize and calculate how these elements are bound. Two essential concepts often used with chemical formulas include:

• Empirical formula which is the simplest integer ratio of the atoms in a compound.
• Molecular formula that shows the exact number of each type of atom in a molecule.

Chemical formulas enable us to determine the composition of a compound, which is necessary for quantifying substances in reactions. Knowing the chemical formula also guides us in calculating molar masses, as described above and in setting up stoichiometric problems.

Stoichiometry

Stoichiometry is the part of chemistry that deals with the quantitative relationships of the elements and compounds involved in chemical reactions. It involves calculations that relate to the masses of reactants and products in a chemical reaction.In problems like the one given, stoichiometry can help to determine the amount of a compound in a sample. For example, if you have a sample containing a known percentage of an element, stoichiometry can help determine how much of a compound contains that percentage. In this exercise, we used stoichiometry to connect the mass of antimony in the ore to the mass of \( \mathrm{Sb}_2 \mathrm{S}_3 \) needed to have that amount of antimony.By understanding stoichiometry, you can predict the quantities of materials required or produced in chemical reactions, enabling efficient and effective chemical processing. It's a key concept for chemists and is directly derived from having a firm grasp on molar masses and chemical formulas.