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)`.
If in "forsterite mineral" bivalent `Mg^(2+)` ions are to be replaced by unipositive `Na^(o+)` ions, and if `Na^(o+)` ions are occupying half of `TV_(s)` in`fcc` lattice, the arrangement of rest of the constituents is kept same, then the formula of the new solid is:

Options

A. `Na_(2)SiO_(4)`
B. `Na_(2)SiO_(3)`
C. `Na_(4)SiO_(4)`
D. `Na_(2)Si_(2)O_(6)`

Explanation

To solve the problem, we need to determine the formula of the new solid formed when bivalent \( \text{Mg}^{2+} \) ions in forsterite are replaced by unipositive \( \text{Na}^{+} \) ions, occupying half of the tetrahedral voids in the FCC lattice.

### Step-by-Step Solution:

1. **Identify the Composition of Forsterite:**
- Forsterite is represented as \( \text{Mg}_2\text{SiO}_4 \).
- In this structure:
- There are 2 \( \text{Mg}^{2+} \) ions.
- There is 1 \( \text{Si}^{4+} \) ion.
- There are 4 \( \text{O}^{2-} \) ions.

2. **Determine the Number of Tetrahedral and Octahedral Voids:**
- In an FCC lattice, there are 8 tetrahedral voids and 4 octahedral voids.
- Since \( \text{Na}^{+} \) ions occupy half of the tetrahedral voids:
- Number of \( \text{Na}^{+} \) ions = \( \frac{1}{2} \times 8 = 4 \).

3. **Determine the Number of Silicon and Oxygen Ions:**
- The number of \( \text{Si}^{4+} \) ions remains the same as in forsterite:
- Number of \( \text{Si}^{4+} \) ions = 1 (occupying one-fourth of the octahedral voids).
- The number of \( \text{O}^{2-} \) ions also remains the same:
- Number of \( \text{O}^{2-} \) ions = 4.

4. **Write the New Formula:**
- The new solid will have:
- 4 \( \text{Na}^{+} \) ions,
- 1 \( \text{Si}^{4+} \) ion,
- 4 \( \text{O}^{2-} \) ions.
- Therefore, the formula of the new solid is \( \text{Na}_4\text{SiO}_4 \).

5. **Check for Electrical Neutrality:**
- Cations: \( 4 \times (+1) = +4 \) from \( \text{Na}^{+} \).
- Anions: \( 4 \times (-2) = -8 \) from \( \text{O}^{2-} \).
- The total charge is balanced, confirming electrical neutrality.

### Final Answer:
The formula of the new solid is \( \text{Na}_4\text{SiO}_4 \).

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