Question

When $1.164\mathrm{g}$ of a certain metal sulfide was roasted in air, $0.972\mathrm{g}$ of the metal oxide was formed. If the oxidation number of the metal is $+2,$ calculate the molar mass of the metal.

Short answer

The molar mass of the metal is approximately $146.44\text{g/mol}.$

Step-by-step solution

Write the Chemical Reaction

When a metal sulfide is roasted in air, it is converted to a metal oxide. The general reaction is: $$MS+O_2\to MO+S{O}_2$$ where $M$ is the metal.

Determine the Masses in Reaction

We have $1.164\text{ g}$ of metal sulfide $MS$ and $0.972\text{ g}$ of metal oxide $MO$.

Identify the Stoichiometry

The reaction converts $MS$ to $MO$ and $S{O}_2$. The metal $M$ appears in both $MS$ and $MO$, indicating that $M$ is conserved.

Calculate the Mass of Sulfur

The difference in mass between $MS$ and $MO$ is due to the loss of sulfur: $$1.164\text{ g}-0.972\text{ g}=0.192\text{ g}$$ This is the mass of sulfur present initially.

Use Molar Mass of Sulfur

The molar mass of sulfur $(S)$ is approximately $32.07\text{ g/mol}$. Calculate the moles of sulfur: $$\text{Moles of }S=\frac{0.192\text{ g}}{32.07\text{ g/mol}}\approx 0.00598\text{ moles}$$

Calculate the Moles of Metal Sulfide

In $MS$, the moles of sulfur also equal the moles of metal sulfide $MS$ because it is a 1:1 ratio. So, moles of $MS=0.00598$.

Find the Moles of Metal Oxide

In the product $MO$, the moles of $MO$ are equal to the initial moles of $MS$, so $0.00598$.

Calculate the Molar Mass of Metal Oxide

Using the mass of $MO$: $$\text{Molar Mass of }MO=\frac{0.972\text{ g}}{0.00598\text{ moles}}\approx 162.44\text{ g/mol}$$

Deduce the Molar Mass of the Metal

The molar mass of $MO$ is the sum of molar masses of $M$ and $O$ (oxygen $\approx 16.00\text{ g/mol}$): $$M+16.00=162.44$$ Solve for $M$: $$M=162.44-16.00=146.44\text{ g/mol}$$

Verify Hypothesis

Since we assumed the oxidation number of $M$ is $+2$, this solution aligns with the balanced reaction equation and stoichiometry.

Key concepts

Metal Sulfide Oxidation

When a metal sulfide undergoes oxidation, it reacts with oxygen to transform into a metal oxide. This process is often performed by roasting the metal sulfide in air, which is rich in oxygen. During this reaction, the sulfur part of the sulfide compound converts into sulfur dioxide $S{O}_2$, which gets released into the atmosphere. The overall chemical reaction typically looks like this: $$MS+O_2\to MO+S{O}_2$$ where $M$ is the metal. This process is significant in industrial applications and helps extract metals from their sulfide ores. Understanding this conversion is crucial for studying how raw materials are refined into usable metal forms.

Stoichiometry

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It involves using the laws of conservation of mass, where the number of moles and mass of reactants equals the mass of products. For the metal sulfide oxidation reaction $(MS+O_2\to MO+S{O}_2)$, stoichiometry helps us determine the amounts of each substance involved. In this reaction, the metal and sulfur combine in fixed proportions, forming products that follow the same ratio.
The reaction ratio can be used to calculate unknown quantities as long as one mass or mole amount is known, allowing us to find the balanced form of any chemical equation. It also helps in predicting amounts of products formed in real-world chemical processing industries.

Sulfur Content Determination

Determining the sulfur content in a metal sulfide compound is essential for several analytical methods. It can be done by measuring the mass difference before and after the oxidation process. In our example, $1.164extg$ of the metal sulfide was oxidized to $0.972extg$ of metal oxide. The mass difference is solely due to sulfur loss, amounting to $0.192extg$ of sulfur that escaped as $S{O}_2$.
Knowing the molar mass of sulfur $(32.07extg/mol)$, we can calculate the moles of sulfur $(\approx 0.00598extmoles)$. This sulfur content determination is essential in fields like environmental monitoring, metallurgy, and material sciences, where precise composition analysis is required for quality control and environmental impact assessments.

Chemical Reaction Balancing

Balancing chemical reactions ensures that atoms are conserved throughout a chemical process, adhering to the law of conservation of mass. A balanced chemical equation has equal numbers of each type of atom on both the reactant and product sides. For the metal sulfide oxidation reaction $(MS+O_2\to MO+S{O}_2)$, it is already balanced as it obeys stoichiometry rules and stoichiometric coefficients of 1 for each compound.
Balanced equations are foundational for stoichiometry calculations and accurate predictions of reaction yields. Understanding how to balance equations is crucial for chemists in designing experiments, scaling reactions for production, and simulating processes across various industries. It is a fundamental skill for anyone working with chemical transformations and product development.