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The problem requires proving that the cation-to-anion radius ratio is 0.414 for a coordination number of 6, using the NaCl (halite) structure as an example.

To begin solving this:

### Step-by-Step Solution:
1. **NaCl Crystal Structure**: 
   NaCl has a face-centered cubic (FCC) structure where each sodium ion (Na⁺) is surrounded by six chloride ions (Cl⁻), leading to a coordination number of 6.

2. **Geometrical Analysis**:
   In an FCC lattice, the cations and anions touch each other along the diagonal of the unit cell's face. Therefore, the distance between the centers of the adjacent ions is critical.

3. **Relation between Radii and Distance**:
   For a coordination number of 6:
   - The **Na⁺ ions** and **Cl⁻ ions** alternate in the crystal.
   - The edge length of the unit cell is related to the ionic radii by considering the diagonal of the face of the cubic unit cell. The distance between neighboring ions is given by:
   \[
   \text{Face diagonal} = 2r_{\text{anion}} + 2r_{\text{cation}}
   \]
   
4. **Deriving the Ratio**:
   By simple geometry (Pythagoras theorem for the face diagonal):
   \[
   \text{Face diagonal} = \sqrt{2} \cdot \text{edge length}
   \]
   The edge length is twice the radius of the anion (since anions are on the corners). Therefore, we have:
   \[
   \sqrt{2} \cdot 2r_{\text{anion}} = 2r_{\text{anion}} + 2r_{\text{cation}}
   \]
   Simplifying:
   \[
   \sqrt{2} r_{\text{anion}} = r_{\text{anion}} + r_{\text{cation}}
   \]
   Dividing through by \( r_{\text{anion}} \):
   \[
   \sqrt{2} = 1 + \frac{r_{\text{cation}}}{r_{\text{anion}}}
   \]
   Therefore, the cation-to-anion radius ratio is:
   \[
   \frac{r_{\text{cation}}}{r_{\text{anion}}} = \sqrt{2} - 1 \approx 0.414
   \]

Thus, we've shown that the ratio of the cation radius to the anion radius in a structure with a coordination number of 6, like NaCl, is indeed 0.414.

Let me know if you'd like any further details or explanations. Here are some related questions:

1. How does the coordination number affect the stability of ionic compounds?
2. Can the same radius ratio be used for different coordination numbers?
3. What is the relationship between lattice energy and ionic radii?
4. How would the radius ratio change if the cation were larger than the anion?
5. How does the crystal structure differ for compounds with coordination numbers other than 6?

**Tip:** Always check for consistency between geometric models and empirical values when working with crystal structures.

Usando la estructura de la halita (NaCl), demuestre que la relación (radio catión)/(radio anión) para un número de coordinación de 6 es 0.414

Learn how to demonstrate the cation-to-anion radius ratio (0.414) in NaCl for a coordination number of 6. This detailed solution uses geometry and the Pythagorean theorem in the context of crystal lattice structures, suitable for undergraduate chemistry and physics students.

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