Calculate the empirical formula of a mineral which has the following percentage composition :
CuO = 44.82%, `SiO_(2)` = 34.83% and water = 20.35% (at. wt. of Cu = 63.5, Si = 28).

Explanation

To calculate the empirical formula of the mineral with the given percentage composition of CuO, SiO₂, and water, we will follow these steps:

### Step 1: Assume a Sample Size
Assume the total weight of the sample is 100 grams. This makes it easier to convert the percentage composition directly into grams.

### Step 2: Calculate the Mass of Each Component
- Mass of CuO = 44.82% of 100 g = 44.82 g
- Mass of SiO₂ = 34.83% of 100 g = 34.83 g
- Mass of water (H₂O) = 20.35% of 100 g = 20.35 g

### Step 3: Calculate Moles of Each Component
To find the number of moles, use the formula:
\[ \text{Moles} = \frac{\text{Mass (g)}}{\text{Molar Mass (g/mol)}} \]

#### Moles of CuO
- Molar mass of CuO = Atomic mass of Cu (63.5 g/mol) + Atomic mass of O (16 g/mol) = 79.5 g/mol
- Moles of CuO = \( \frac{44.82 \text{ g}}{79.5 \text{ g/mol}} \approx 0.563 \text{ moles} \)

#### Moles of SiO₂
- Molar mass of SiO₂ = Atomic mass of Si (28 g/mol) + 2 × Atomic mass of O (16 g/mol) = 60 g/mol
- Moles of SiO₂ = \( \frac{34.83 \text{ g}}{60 \text{ g/mol}} \approx 0.580 \text{ moles} \)

#### Moles of H₂O
- Molar mass of H₂O = 2 × Atomic mass of H (1 g/mol) + Atomic mass of O (16 g/mol) = 18 g/mol
- Moles of H₂O = \( \frac{20.35 \text{ g}}{18 \text{ g/mol}} \approx 1.130 \text{ moles} \)

### Step 4: Find the Simplest Mole Ratio
Now, we will divide the number of moles of each component by the smallest number of moles calculated.

- Smallest number of moles = 0.563 (for CuO)

#### Ratios
- Ratio of CuO = \( \frac{0.563}{0.563} = 1 \)
- Ratio of SiO₂ = \( \frac{0.580}{0.563} \approx 1.03 \) (approximately 1)
- Ratio of H₂O = \( \frac{1.130}{0.563} \approx 2.01 \) (approximately 2)

### Step 5: Write the Empirical Formula
From the ratios calculated:
- CuO: 1
- SiO₂: 1
- H₂O: 2

Thus, the empirical formula can be written as:
\[ \text{Empirical Formula} = \text{CuO} \cdot \text{SiO}_2 \cdot \text{H}_2\text{O}_2 \]

### Final Answer
The empirical formula of the mineral is:
\[ \text{CuSiO}_2 \cdot 2\text{H}_2\text{O} \]

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