A sample of clay is found to have the formula `Al_(2)O_(3).K_(2)O.6SiO_(2)` . Calculate the percentage of alumina `(Al_(2)O_(3))`, potassium oxide `(K_(2)O)` and silica `(SiO_(2))` in the sample.

Explanation

To calculate the percentage of alumina \((Al_2O_3)\), potassium oxide \((K_2O)\), and silica \((SiO_2)\) in the sample with the formula \(Al_2O_3.K_2O.6SiO_2\), we will follow these steps:

### Step 1: Calculate the Molar Mass of the Compound
The molar mass of the compound \(Al_2O_3.K_2O.6SiO_2\) can be calculated by adding the molar masses of each component.

1. **Alumina \((Al_2O_3)\)**:
- Aluminum (Al): \(27 \, \text{g/mol} \times 2 = 54 \, \text{g/mol}\)
- Oxygen (O): \(16 \, \text{g/mol} \times 3 = 48 \, \text{g/mol}\)
- Total for \(Al_2O_3 = 54 + 48 = 102 \, \text{g/mol}\)

2. **Potassium Oxide \((K_2O)\)**:
- Potassium (K): \(39 \, \text{g/mol} \times 2 = 78 \, \text{g/mol}\)
- Oxygen (O): \(16 \, \text{g/mol} \times 1 = 16 \, \text{g/mol}\)
- Total for \(K_2O = 78 + 16 = 94 \, \text{g/mol}\)

3. **Silica \((SiO_2)\)**:
- Silicon (Si): \(28 \, \text{g/mol} \times 1 = 28 \, \text{g/mol}\)
- Oxygen (O): \(16 \, \text{g/mol} \times 2 = 32 \, \text{g/mol}\)
- Total for \(SiO_2 = 28 + 32 = 60 \, \text{g/mol}\)
- Since there are 6 SiO2 in the formula: \(60 \, \text{g/mol} \times 6 = 360 \, \text{g/mol}\)

4. **Total Molar Mass of the Compound**:
\[ Molar \, Mass = Molar \, Mass \, of \, Al_2O_3 + Molar \, Mass \, of \, K_2O + Molar \, Mass \, of \, 6SiO_2 \]
\[ Molar \, Mass = 102 + 94 + 360 = 556 \, \text{g/mol} \]

### Step 2: Calculate the Percentage Composition
Now, we will calculate the percentage of each component in the sample.

1. **Percentage of Alumina \((Al_2O_3)\)**:
\[ \text{Percentage of } Al_2O_3 = \left( \frac{Molar \, Mass \, of \, Al_2O_3}{Total \, Molar \, Mass} \right) \times 100 \]
\[ \text{Percentage of } Al_2O_3 = \left( \frac{102}{556} \right) \times 100 \approx 18.35\% \]

2. **Percentage of Potassium Oxide \((K_2O)\)**:
\[ \text{Percentage of } K_2O = \left( \frac{Molar \, Mass \, of \, K_2O}{Total \, Molar \, Mass} \right) \times 100 \]
\[ \text{Percentage of } K_2O = \left( \frac{94}{556} \right) \times 100 \approx 16.90\% \]

3. **Percentage of Silica \((SiO_2)\)**:
\[ \text{Percentage of } SiO_2 = \left( \frac{Molar \, Mass \, of \, 6SiO_2}{Total \, Molar \, Mass} \right) \times 100 \]
\[ \text{Percentage of } SiO_2 = \left( \frac{360}{556} \right) \times 100 \approx 64.00\% \]

### Final Results:
- Percentage of Alumina \((Al_2O_3)\) = **18.35%**
- Percentage of Potassium Oxide \((K_2O)\) = **16.90%**
- Percentage of Silica \((SiO_2)\) = **64.00%**

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