Problem 101 - Chemistry and Chemical Reactivity

Question

A Copper metal can be prepared by roasting copper ore, which can contain cuprite \(\left(\mathrm{Cu}_{2} \mathrm{S}\right)\) and copper (11) sulfide. $$ \begin{aligned} \mathrm{Cu}_{2} \mathrm{S}(\mathrm{s})+\mathrm{O}_{2}(g) & \rightarrow 2 \mathrm{Cu}(\mathrm{s})+\mathrm{SO}_{2}(\mathrm{g}) \\ \mathrm{CuS}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{g}) & \rightarrow \mathrm{Cu}(\mathrm{s})+\mathrm{SO}_{2}(\mathrm{g}) \end{aligned} $$ Suppose an ore sample contains \(11.0 \%\) impurity in addition to a mixture of CuS and 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

CuS: 42.01%; Cu2S: 57.99%.

Step 1: Determine Pure Copper Produced

Given that heating the mixture produces 75.4 g of copper metal that is 89.5% pure, we first calculate the weight of pure copper:\[\text{Weight of pure Cu} = 75.4\, \text{g} \times 0.895 = 67.483\, \text{g}\]

Step 2: Establish Equations for Reactions

The reactions are:1. \( \text{Cu}_2\text{S} + \text{O}_2 \rightarrow 2 \text{Cu} + \text{SO}_2 \) (produces 2 moles of Cu)2. \( \text{CuS} + \text{O}_2 \rightarrow \text{Cu} + \text{SO}_2 \) (produces 1 mole of Cu)

Step 3: Calculate Molar Masses

We calculate the molar masses needed:- Molar mass of \( \text{Cu}_2\text{S} = (2 \times 63.55) + 32.07 = 159.17 \text{ g/mol} \)- Molar mass of \( \text{CuS} = 63.55 + 32.07 = 95.62 \text{ g/mol} \)- Molar mass of Cu = 63.55 g/mol.

Step 4: Write System of Equations for the Masses

Let \( x \) be the mass of \( \text{Cu}_2\text{S} \) and \( y \) be the mass of \( \text{CuS} \).1. Total mass constraint: \[ x + y + 11.0\% \times 100.0\, \text{g} = 100.0\, \text{g} \] \[ x + y = 89.0\, \text{g} \] 2. Copper purity constraint (using moles to grams conversion): \[ 2\left(\frac{x}{159.17}\right) \times 63.55 + \left(\frac{y}{95.62}\right) \times 63.55 = 67.483\, \text{g} \]

Step 5: Solve the System of Equations

Using the equations from Step 4:1. \( x + y = 89.0 \)2. \( \frac{127.10x}{159.17} + \frac{63.55y}{95.62} = 67.483 \)Solving this system gives:- \( x = 51.62 \text{ g} \) (mass of \( \text{Cu}_2\text{S} \))- \( y = 37.38 \text{ g} \) (mass of \( \text{CuS} \))

Step 6: Calculate Percentage Composition

Finally, we calculate the weight percent of each component:- Weight percent of \( \text{Cu}_2\text{S} \): \[ \frac{51.62}{89.0} \times 100\% = 57.99\% \]- Weight percent of \( \text{CuS} \): \[ \frac{37.38}{89.0} \times 100\% = 42.01\% \]

Step 7: Conclusion

The weight percent of \( \text{Cu}_2\text{S} \) in the ore is approximately 57.99%, and the weight percent of \( \text{CuS} \) is approximately 42.01%.

Key Concepts

Chemical Reactions

Chemical reactions are processes where substances, known as reactants, are transformed into different substances, called products. In the extraction of copper, the chemical reactions involve the oxidation of copper ores. These reactions specifically are:

1. \(\text{Cu}_2\text{S} + \text{O}_2 \rightarrow 2\text{Cu} + \text{SO}_2\)
2. \(\text{CuS} + \text{O}_2 \rightarrow \text{Cu} + \text{SO}_2\)

Both reactions convert copper sulfide compounds into pure copper metal and sulfur dioxide gas. This process is part of roasting, an important step in metallurgical applications.

Observing these reactions is key to understanding how metals are extracted from their ores and how energy and matter interact during chemical changes.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. This mathematical method helps determine the proportions of elements involved. In our case, stoichiometry shows us the amount of copper produced from given amounts of copper ores:

- For \(\text{Cu}_2\text{S}\), each mole produces 2 moles of copper.
- For \(\text{CuS}\), each mole produces 1 mole of copper.

This relationship is crucial for calculating how much copper can be obtained from a specific mass of ore. It involves the use of balanced equations to ensure mass conservation, highlighting the importance of precise measurements in chemical processes.

Molar Mass

The molar mass of a substance is the weight of one mole of that substance, usually expressed in grams per mole (g/mol). Calculating molar masses is essential for stoichiometry and understanding chemical reactions. For this extraction, we have:

• Molar mass of \(\text{Cu}_2\text{S}\): \((2 \times 63.55) + 32.07 = 159.17 \, \text{g/mol}\)
• Molar mass of \(\text{CuS}\): \(63.55 + 32.07 = 95.62 \, \text{g/mol}\)
• Molar mass of Cu: \(63.55 \, \text{g/mol}\)

These values are used to convert between mass and moles, allowing chemists to calculate how much of each reactant is needed and what mass of products will be formed. Understanding molar mass is foundational in chemistry, giving insights about the substance and its role in reactions.

Purity Calculation

Purity calculation determines the proportion of a desired substance within a sample. It is expressed as a percentage, indicating the efficiency of the extraction process. In the exercise, we find the purity of copper produced from roasting:

The calculation is:

Given 75.4 g of copper, with 89.5% purity, the pure copper weight is:
\[ \text{Weight of pure Cu} = 75.4 \, \text{g} \times 0.895 = 67.483 \, \text{g} \]

Purity affects the quality and utility of the final metal product. For industrial applications, understanding and calculating purity is crucial for assessing the economic viability and efficiency of metallurgical processes.