Question

A certain metal sulfide, \(\mathrm{MS}_{n}\) (where \(n\) is a small integer), is widely used as a high-temperature lubricant. The substance is prepared by reaction of the metal pentachloride \(\left(\mathrm{MCl}_{5}\right)\) with sodium sulfide \(\left(\mathrm{Na}_{2} \mathrm{~S}\right)\). Heating the metal sulfide to \(700^{\circ} \mathrm{C}\) in air gives the metal trioxide \(\left(\mathrm{MO}_{3}\right)\) and sulfur dioxide \(\left(\mathrm{SO}_{2}\right)\), which reacts with \(\mathrm{Fe}^{3+}\) ion under aqueous acidic conditions to give sulfate ion \(\left(\mathrm{SO}_{4}^{2-}\right)\). Addition of aqueous \(\mathrm{BaCl}_{2}\) then forms a precipitate of \(\mathrm{BaSO}_{4}\). The unbalanced equations are: (1) \(\mathrm{MCl}_{5}(s)+\mathrm{Na}_{2} \mathrm{~S}(s) \longrightarrow \mathrm{MS}_{n}(s)+\mathrm{S}(l)+\mathrm{NaCl}(s)\) (2) \(\mathrm{MS}_{n}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{MO}_{3}(s)+\mathrm{SO}_{2}(g)\) (3) \(\mathrm{SO}_{2}(g)+\mathrm{Fe}^{3+}(a q) \longrightarrow \mathrm{Fe}^{2+}(a q)+\mathrm{SO}_{4}{ }^{2-}(a q)\) (in acid) (4) \(\mathrm{SO}_{4}{ }^{2-}(a q)+\mathrm{Ba}^{2+}(a q) \longrightarrow \mathrm{BaSO}_{4}(s)\) Assume that you begin with \(4.61 \mathrm{~g}\) of \(\mathrm{MCl}_{5}\) and that reaction (1) proceeds in \(91.3 \%\) yield. After oxidation of the \(\mathrm{MS}_{n}\) product, oxidation of \(\mathrm{SO}_{2}\), and precipitation of sulfate ion, \(7.19 \mathrm{~g}\) of \(\mathrm{BaSO}_{4}(s)\) is obtained. (a) How many moles of sulfur are present in the \(\mathrm{MS}_{n}\) sample? (b) Assuming several possible values for \(n(n=1,2,3 \ldots)\) what is the atomic weight of \(\mathbf{M}\) in each case? (c) What is the likely identity of the metal \(\mathbf{M}\), and what is the formula of the metal sulfide \(\mathrm{MS}_{n} ?\) (d) Balance all equations.

Short answer

(a) Use moles of BaSO4 to find moles of sulfur. (b) Calculate atomic weight of M for each n value. (c) Likely metal is found by comparing atomic weights. (d) Balance given equations.

Step-by-step solution

Determine moles of BaSO4

First, calculate the moles of \(\text{BaSO}_4\) using its mass and molar mass. The molar mass of \(\text{BaSO}_4\) (Barium Sulfate) is approximately 233.39 g/mol. Use the formula:\[\text{Moles of } \text{BaSO}_4 = \frac{7.19 \text{ g}}{233.39 \text{ g/mol}}\]

Equate moles of BaSO4 to moles of SO2

For the reaction \(\text{SO}_4^{2-} + \text{Ba}^{2+} \rightarrow \text{BaSO}_4\), the mole ratio is 1:1. Thus, moles of \(\text{BaSO}_4\) are equal to moles of \(\text{SO}_4^{2-}\), and this is also equal to the moles of \(\text{SO}_2\) as per the reaction (3). Calculate this using the answer from Step 1.

Determine moles of MCl5 used

Calculate the moles of \(\text{MCl}_5\). Given mass is 4.61 g, and the reaction yield is 91.3%. The molar mass of \(\text{MCl}_5\) is assumed to be unknown, but is required for further calculations. Denote this unknown molar mass as \(\mu\). Hence:\[\text{Moles of } \text{MCl}_5 = \frac{4.61 \text{ g}}{\mu}\] Multiply by 0.913 to consider the reaction yield: \[\text{Moles of } \text{MCl}_5 \text{ after yield} = 0.913 \times \frac{4.61}{\mu}\]

Relate moles of SO2 to MSn

Moles of \(\text{SO}_2\) determined in Step 2 represent moles of sulfur in \(\text{MS}_n\). So, moles of \(\text{SO}_2\) can be treated as moles of sulfur present in \(\text{MS}_n\).

Calculate atomic weight of M for each n

Assume several values for \(n\) such as 1, 2, or 3. Use moles of sulfur derived (Step 4) to establish a relationship between sulfur and metal. If \(n\) is 1, 2, or 3, determine the molar amounts accordingly. By using the law of conservation and stoichiometry, calculate the atomic mass of \(M\) using the following relationship:\[\frac{4.61}{\mu} = \text{Moles of Sulfur} = \text{Moles of } \text{MCl}_5\]Solve for the atomic mass of \(M\).

Identify likely metal M

Compare calculated atomic weights from Step 5 with typical atomic weights of metals to identify potential candidates for \(M\). Select the most appropriate metal that matches the atomic weight closely.

Balance chemical equations

Balance each chemical equation:(1) \(\text{MCl}_5 + 2\text{Na}_2 \text{S} \rightarrow \text{MS}_2 + \text{S} + 5\text{NaCl}\)(2) \(2\text{MS}_2 + 7\text{O}_2 \rightarrow 2\text{MO}_3 + 4\text{SO}_2\)(3) \(\text{SO}_2 + 2\text{Fe}^{3+} + 2\text{H}_2\text{O} \rightarrow 2\text{Fe}^{2+} + \text{SO}_4^{2-} + 4\text{H}^+\)(4) \(\text{SO}_4^{2-} + \text{Ba}^{2+} \rightarrow \text{BaSO}_4\)

Key concepts

Chemical Equation Balancing

To solve problems in stoichiometry and understand the changes occurring in a chemical reaction, balancing chemical equations is crucial. This ensures that the law of conservation of mass is followed, indicating that atoms are neither created nor destroyed in a reaction.
When balancing, we adjust the coefficients before the compounds to ensure that the number of each type of atom on the reactant side equals the number on the product side. Let's look at the reactions mentioned; (1) unbalanced forms include;

• \(\text{MCl}_5 + \text{Na}_2 \text{S} \rightarrow \text{MS}_n + \text{S} + \text{NaCl}\)
• Balanced: \(\text{MCl}_5 + 2\text{Na}_2 \text{S} \rightarrow \text{MS}_2 + \text{S} + 5\text{NaCl}\)

Each reaction must be balanced individually, using coefficients to ensure parity:

• For \(\text{MS}_n + \text{O}_2 \rightarrow \text{MO}_3 + \text{SO}_2\): Balanced as \(2\text{MS}_2 + 7\text{O}_2 \rightarrow 2\text{MO}_3 + 4\text{SO}_2\)
• \(\text{SO}_2 + \text{Fe}^{3+} + 2\text{H}_2\text{O} \rightarrow 2\text{Fe}^{2+} + \text{SO}_4^{2-} + 4\text{H}^+\)
• \(\text{SO}_4^{2-} + \text{Ba}^{2+} \rightarrow \text{BaSO}_4\)

Molar Mass Calculation

Understanding molar mass is essential for converting between mass and moles of a substance, enabling accurate stoichiometric calculations. To determine the molar mass of a compound, add the atomic masses of all the atoms in the given formula.
Consider \(\text{BaSO}_4\), whose individual atomic masses are approximately:

• Barium (Ba): 137.33 g/mol
• Sulfur (S): 32.07 g/mol
• Oxygen (O) \(\times 4\): 16.00 g/mol each

Adding these gives the molar mass of \(\text{BaSO}_4\) as approximately 233.39 g/mol.
In the problem, we calculate moles of \(\text{BaSO}_4\) using the formula: \( \frac{\text{mass of } \text{BaSO}_4}{\text{molar mass of } \text{BaSO}_4} \). By knowing the moles, we can interlink them to other species, such as \(\text{SO}_2\), through balanced chemical equations for accurate stoichiometry.

Chemical Reactions Analysis

Analyzing chemical reactions involves understanding reactants, products, and the processes leading to transformation. Here, the starting reactant \(\text{MCl}_5\) reacts with \(\text{Na}_2 \text{S}\) to form \(\text{MS}_n\), which upon heating reacts with oxygen to yield \(\text{MO}_3\) and \(\text{SO}_2\). Each transformation occurs in steps:
First, examine the overall reaction and determine the roles such as oxidizing and reducing agents. The yielded \(\text{SO}_2\) undergoes further oxidation to \(\text{SO}_4^{2-}\) when reacting with iron ions. This chain highlights how each reactant transforms and the importance of reaction conditions.
Observe that identifying the metal \(\text{M}\) involves interpreting data, calculated atomic masses, and characteristics such as volatile nature, reactivity, or commonly known uses.
Each step's success depends on accurate balancing, understanding molar relationships, and recognizing oxidation-reduction exchanges.