A certain sample of cuprous sulphide is found to have composition `Cu_(1.8)S,`, because of incroporation of `Cu^(2+)` ion in the lattice, What is the mole % of `Cu^(2+)` in total content in this crystal?

Options

A. 0.998
B. 0.1111
C. 0.8888
D. none of these

Answer

To find the mole percentage of \( \text{Cu}^{2+} \) in the given sample of cuprous sulphide (\( \text{Cu}_{1.8}\text{S} \)), we can follow these steps:

### Step 1: Understand the Composition
The original cuprous sulphide has the formula \( \text{Cu}_2\text{S} \), where copper exists in the +1 oxidation state (\( \text{Cu}^+ \)). In the given sample \( \text{Cu}_{1.8}\text{S} \), some of the \( \text{Cu}^+ \) ions have been replaced by \( \text{Cu}^{2+} \) ions.

### Step 2: Define Variables
Let \( x \) be the number of \( \text{Cu}^{2+} \) ions in the crystal. Therefore, the number of \( \text{Cu}^+ \) ions will be \( 1.8 - x \).

### Step 3: Charge Neutrality Condition
The crystal must be electrically neutral. The total positive charge from the copper ions must equal the total negative charge from the sulphide ions.

- The charge from \( x \) \( \text{Cu}^{2+} \) ions is \( 2x \).
- The charge from \( (1.8 - x) \) \( \text{Cu}^+ \) ions is \( (1.8 - x) \).
- The charge from the sulphide ion (\( \text{S}^{2-} \)) is \( -2 \).

Setting up the equation for charge neutrality:
\[ 2x + (1.8 - x) - 2 = 0 \]

### Step 4: Solve for \( x \)
Simplifying the equation:
\[ 2x + 1.8 - x - 2 = 0 \]
\[ x - 0.2 = 0 \]
\[ x = 0.2 \]

This means there are 0.2 moles of \( \text{Cu}^{2+} \) ions in the crystal.

### Step 5: Calculate Total Copper Content
The total copper content in the crystal is given as \( 1.8 \) moles (from \( \text{Cu}_{1.8}\text{S} \)).

### Step 6: Calculate Mole Fraction of \( \text{Cu}^{2+} \)
The mole fraction of \( \text{Cu}^{2+} \) is given by:
\[ \text{Mole fraction of } \text{Cu}^{2+} = \frac{x}{\text{Total Cu}} = \frac{0.2}{1.8} \]

### Step 7: Calculate the Value
Calculating the fraction:
\[ \frac{0.2}{1.8} = \frac{2}{18} = \frac{1}{9} \approx 0.1111 \]

### Step 8: Convert to Mole Percentage
To convert the mole fraction to mole percentage:
\[ \text{Mole \% of } \text{Cu}^{2+} = \left( \frac{0.2}{1.8} \right) \times 100 \approx 11.11\% \]

### Final Answer
The mole percentage of \( \text{Cu}^{2+} \) in the total content of the crystal is approximately \( 11.11\% \).

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