The `"OLIVINE"` series of minerals consists of crystal in which `Fe^(2+)` and `Mg^(2+)` ions may substitute for each other causing susbstitutional impurity defects without changing the volume of unit cell. In `"OLIVINE"` series of minerals, `O^(2-)` ions exist as fcc with `Si^(4+)` occupying one-fourth of `OVs` and divalent metal ions occupying one-fourth of `OVs` and divalent metal ions occupying one-fourth of `TVs`. The density of "forsterite" (magnesium silicate) is `3.21 g cm^(-3)` land that of "fayalite" (ferrous silicate) is `4.34 g cm^(-3)`.
The formula of "fayalite mineral" is

Options

A. `Fe_(2)SiO_(4)`
B. `FeSiO_(4)`
C. `Fe_(2)SiO_(6)`
D. `Fe_SiO_(3)`

Explanation

To find the formula of the fayalite mineral in the olivine series, we will follow these steps:

### Step 1: Determine the number of ions in the unit cell
In the olivine structure:
- The oxide ions (O²⁻) form a face-centered cubic (FCC) lattice.
- The number of O²⁻ ions in an FCC unit cell is calculated as follows:
- There are 8 corner atoms contributing 1/8 each and 6 face-centered atoms contributing 1/2 each.
- Total O²⁻ ions = \(8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4\).

### Step 2: Determine the number of Si⁴⁺ ions
- It is given that Si⁴⁺ ions occupy one-fourth of the octahedral voids.
- In an FCC lattice, the number of octahedral voids is equal to the number of atoms in the lattice, which is 4.
- Therefore, the number of Si⁴⁺ ions = \( \frac{1}{4} \times 4 = 1\).

### Step 3: Determine the number of Fe²⁺ ions
- The divalent metal ions (Fe²⁺ or Mg²⁺) occupy one-fourth of the tetrahedral voids.
- The number of tetrahedral voids in an FCC lattice is twice the number of atoms, which is \(2 \times 4 = 8\).
- Therefore, the number of Fe²⁺ ions = \( \frac{1}{4} \times 8 = 2\).

### Step 4: Write the empirical formula
From the above calculations:
- Number of Fe²⁺ ions = 2
- Number of Si⁴⁺ ions = 1
- Number of O²⁻ ions = 4

Thus, the empirical formula of fayalite can be written as:
\[ \text{Fe}_2\text{Si}_1\text{O}_4 \]

### Step 5: Check for electrical neutrality
- Cation charges: \(2 \times (+2) + 1 \times (+4) = 4 + 4 = +8\)
- Anion charges: \(4 \times (-2) = -8\)

Since the total positive charge (+8) equals the total negative charge (-8), the formula maintains electrical neutrality.

### Final Answer
The formula of the fayalite mineral is:
\[ \text{Fe}_2\text{SiO}_4 \]

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