A uni-univalent ionic crystal AX is composed of the following radii (arbitrary units) :
`{:(A^+,A^-),(1.0,2.0):}`
Assuming that ions are hard spheres , predict giving reasons whether the crystal will have sodium chloride cesium chloride structure. Calculate the volume of the unit cell.

Explanation

To determine whether the ionic crystal AX with given ionic radii will have a sodium chloride (NaCl) or cesium chloride (CsCl) structure, we will follow these steps:

### Step 1: Identify the Ionic Radii
We are given the ionic radii:
- Radius of cation \( A^+ = 1.0 \) (arbitrary units)
- Radius of anion \( A^- = 2.0 \) (arbitrary units)

### Step 2: Calculate the Radius Ratio
The radius ratio \( r \) is calculated using the formula:
\[ r = \frac{r_{cation}}{r_{anion}} = \frac{r^+}{r^-} \]
Substituting the values:
\[ r = \frac{1.0}{2.0} = 0.5 \]

### Step 3: Determine the Structure Type
To determine whether the crystal structure is similar to NaCl or CsCl, we compare the radius ratio to known ranges:
- For NaCl structure, the radius ratio should be between \( 0.414 \) and \( 0.732 \).
- For CsCl structure, the radius ratio is typically greater than \( 0.732 \).

Since our calculated radius ratio \( 0.5 \) falls within the range for NaCl, we conclude that the crystal structure is similar to NaCl.

### Step 4: Calculate the Edge Length of the Unit Cell
The edge length \( a \) of the unit cell for NaCl structure can be calculated using the formula:
\[ a = 2(r^+ + r^-) \]
Substituting the values:
\[ a = 2(1.0 + 2.0) = 2 \times 3.0 = 6.0 \text{ (arbitrary units)} \]

### Step 5: Calculate the Volume of the Unit Cell
The volume \( V \) of the cubic unit cell is given by:
\[ V = a^3 \]
Substituting the edge length:
\[ V = (6.0)^3 = 216.0 \text{ (arbitrary units)}^3 \]

### Final Answer
The ionic crystal AX behaves like sodium chloride (NaCl) and the volume of the unit cell is \( 216.0 \) (arbitrary units)³.

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