Problem 103 - Chemistry and Chemical Reactivity

Question

Copper metal can be prepared by roasting copper ore, which can contain cuprite \(\left(\mathrm{Cu}_{2} \mathrm{S}\right)\) and copper(II) sulfide. $$\begin{aligned}\mathrm{Cu}_{2} \mathrm{S}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{g}) & \rightarrow 2 \mathrm{Cu}(\mathrm{s})+\mathrm{SO}_{2}(\mathrm{g}) \\\\\mathrm{CuS}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{g}) & \rightarrow \mathrm{Cu}(\mathrm{s})+\mathrm{SO}_{2}(\mathrm{g})\end{aligned}$$ Suppose an ore sample contains \(11.0 \%\) impurity in addition to a mixture of CuS and \(\mathrm{Cu}_{2} \mathrm{S}\). Heating \(100.0 \mathrm{g}\) of the mixture produces \(75.4 \mathrm{g}\) of copper metal with a purity of \(89.5 \% .\) What is the weight percent of CuS in the ore? The weight percent of \(\mathrm{Cu}_{2} \mathrm{S} ?\)

Step-by-Step Solution

Answer

Weight percent of CuS: 26.1%, Cu_2S: 73.9%.

Step 1: Determine Pure Copper Produced

Given that the purity of copper metal is 89.5%, calculate the actual mass of pure copper produced from heating 100.0 g of the ore:\[\text{Pure \( Cu \) mass} = 75.4 \times \frac{89.5}{100} = 67.483 \text{ g}\]

Step 2: Establish Total Impure Ore Mass

The ore sample has an 11.0% impurity, so the effective mass of pure ore is:\[\text{Pure ore mass} = 100.0 \times \left(1 - \frac{11.0}{100}\right) = 89.0 \text{ g}\]

Step 3: Set Up System of Equations

Let \( x \) be the mass of \( Cu_2S \) and \( y \) be the mass of \( CuS \) in the ore. We have two equations:1. Total mass of pure ore: \[ x + y = 89.0 \]2. Mass of copper produced: \[ 0.8x + 0.64y = 67.483 \] These expressions come from the proportions of copper in \( Cu_2S \) and \( CuS \).

Step 4: Solve the System of Equations

First, solve one of the equations for one of the variables. From equation (1), solve for \( y \):\[ y = 89.0 - x \]Substitute into equation (2):\[0.8x + 0.64(89.0 - x) = 67.483\]Simplify and solve for \( x \):\[0.8x + 56.96 - 0.64x = 67.483\]\[0.16x = 10.523\]\[x = 65.77 \text{ g}\]Now, substitute \( x \) back into equation (1) to find \( y \):\[y = 89.0 - 65.77 = 23.23 \text{ g}\]

Step 5: Calculate Weight Percentages

Calculate the weight percent of \( CuS \) and \( Cu_2S \):- Weight percent of \( CuS \): \[ \text{Weight percent of \( CuS \)} = \frac{23.23}{89.0} \times 100 \approx 26.1\% \]- Weight percent of \( Cu_2S \): \[ \text{Weight percent of \( Cu_2S \)} = \frac{65.77}{89.0} \times 100 \approx 73.9\% \]

Key Concepts

Understanding Stoichiometry

Stoichiometry is like a recipe for chemical reactions. It helps us understand how much of each ingredient (chemical) is needed and what is produced. In the context of copper extraction, we use stoichiometry to calculate how much copper can be produced from copper ores like Cuprite (\( \text{Cu}_2\text{S} \)) and Copper(II) sulfide (\( \text{CuS} \)). By examining balanced chemical equations, we can determine the relationships between reactants and products. This ensures we know precisely how many grams of copper are produced when reacting a specific amount of ore. This is crucial for efficient extraction in real-world applications.

Key Chemical Reactions

Chemical reactions are the backbone of processes like copper extraction. In this example, we consider two reactions:

• Cuprite reaction: \[\text{Cu}_2\text{S}(\text{s}) + \text{O}_2(\text{g}) \rightarrow 2\text{Cu}(\text{s}) + \text{SO}_2(\text{g})\]
• Copper(II) sulfide reaction: \[\text{CuS}(\text{s}) + \text{O}_2(\text{g}) \rightarrow \text{Cu}(\text{s}) + \text{SO}_2(\text{g})\]

In both reactions, copper minerals react with oxygen to produce elemental copper and sulfur dioxide (\( \text{SO}_2 \)). These reactions demonstrate oxidation, where oxygen helps break down ore compounds, releasing copper metal from its sulfide compounds. Understanding these reactions is key to predicting the amount of copper obtained from the ore.

Ore Composition Analysis

Ore composition analysis involves determining the proportion of different minerals in a sample. The ore in this problem contains both \( \text{Cu}_2\text{S} \) and \( \text{CuS} \), alongside impurities making up 11.0% of its mass. By analyzing these components, engineers can predict the efficiency of the extraction process. Knowing the exact quantity of each ore component allows us to use stoichiometry to find out how much pure copper can be retrieved from specific amounts of ore.

Calculating Purity

Purity calculation tells us how much of a useful product (like copper) is in a mixture. In the context of copper extraction, purity shows the proportion of actual copper metal in the extracted material. For example, if the purity is 89.5%, this means that 89.5% of the metal extracted is pure copper. To calculate the purity in this exercise, the mass of the pure copper produced was compared to the total metal mass recovered. Such calculations ensure the quality of the extracted metal and help in evaluating the extraction process's efficiency.