In an ionic solid `r_(c)=1.6Å` and `r_(a)=1.86Å`. Find edge length of cubic unit cell `Å`

Explanation

To find the edge length of the cubic unit cell for the given ionic solid, we can follow these steps:

### Step 1: Understand the relationship between the ionic radii and the edge length
In a cubic unit cell, the cations and anions are arranged such that the distance along the body diagonal of the cube is equal to the sum of the radii of the cation and anion multiplied by 2 (since there are two cations and one anion along the body diagonal).

### Step 2: Write the formula for the body diagonal
The body diagonal \(d\) of a cubic unit cell can be expressed in terms of the edge length \(a\) as:
\[ d = a\sqrt{3} \]
This is derived from the Pythagorean theorem in three dimensions.

### Step 3: Set up the equation using ionic radii
The body diagonal can also be expressed in terms of the ionic radii:
\[ d = 2r_c + 2r_a \]
Where \(r_c\) is the radius of the cation and \(r_a\) is the radius of the anion.

### Step 4: Substitute the known values
Given:
- \(r_c = 1.6 \, \text{Å}\)
- \(r_a = 1.86 \, \text{Å}\)

Substituting these values into the equation gives:
\[ d = 2(1.6) + 2(1.86) = 3.2 + 3.72 = 6.92 \, \text{Å} \]

### Step 5: Equate the two expressions for the body diagonal
Now, we can set the two expressions for the body diagonal equal to each other:
\[ a\sqrt{3} = 6.92 \]

### Step 6: Solve for the edge length \(a\)
To find \(a\), rearrange the equation:
\[ a = \frac{6.92}{\sqrt{3}} \]

Calculating this gives:
\[ a \approx \frac{6.92}{1.732} \approx 4.00 \, \text{Å} \]

### Conclusion
The edge length of the cubic unit cell is approximately \(4.00 \, \text{Å}\).

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