CsCl crystallizes in a cubic lattice and has a `Cl^(-)` at each corner and `Cs^(+)` at the centre of unit cell. If `r_(Cs^(+))=1.69 Å` and `r_(Cl^(-))=1.81 Å`, what is the value of edge - length 'a' of the cube ? Compare this values of 'a' (calculated) from the observed density of CsCl. `3.97 gm//cm^(3). [M_((CsCl))=168.5]`

Answer

To find the edge length 'a' of the cubic unit cell of CsCl, we will follow these steps:

### Step 1: Understanding the Structure
In a CsCl crystal structure:
- The chloride ions (Cl^-) are located at each corner of the cubic unit cell.
- The cesium ions (Cs^+) are located at the center of the unit cell.

### Step 2: Relationship Between Ionic Radii and Edge Length
The edge length 'a' of the cubic unit cell can be related to the ionic radii of Cs^+ and Cl^- using the geometry of the unit cell. In a simple cubic lattice, the distance between the center of the Cs^+ ion and the Cl^- ion at the corner can be expressed as:
\[ a = 2(r_{Cs^+} + r_{Cl^-}) \]

### Step 3: Substitute the Given Values
We are given:
- \( r_{Cs^+} = 1.69 \, \text{Å} \)
- \( r_{Cl^-} = 1.81 \, \text{Å} \)

Now substituting these values into the equation:
\[ a = 2(1.69 \, \text{Å} + 1.81 \, \text{Å}) \]
\[ a = 2(3.50 \, \text{Å}) \]
\[ a = 7.00 \, \text{Å} \]

### Step 4: Convert to cm for Density Calculation
To compare with the density, we convert the edge length from Ångströms to centimeters:
\[ a = 7.00 \, \text{Å} = 7.00 \times 10^{-8} \, \text{cm} \]

### Step 5: Calculate the Volume of the Unit Cell
The volume \( V \) of the cubic unit cell is given by:
\[ V = a^3 = (7.00 \times 10^{-8} \, \text{cm})^3 = 3.43 \times 10^{-22} \, \text{cm}^3 \]

### Step 6: Calculate the Mass of the Unit Cell
The molar mass of CsCl is given as 168.5 g/mol. The mass of one unit cell can be calculated using the formula:
\[ \text{Mass of unit cell} = \frac{\text{Molar mass}}{N_A} \]
Where \( N_A \) (Avogadro's number) is approximately \( 6.022 \times 10^{23} \, \text{mol}^{-1} \).
\[ \text{Mass of unit cell} = \frac{168.5 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{mol}^{-1}} = 2.80 \times 10^{-22} \, \text{g} \]

### Step 7: Calculate the Density
Density \( \rho \) can be calculated using the formula:
\[ \rho = \frac{\text{Mass}}{\text{Volume}} = \frac{2.80 \times 10^{-22} \, \text{g}}{3.43 \times 10^{-22} \, \text{cm}^3} \approx 0.816 \, \text{g/cm}^3 \]

### Step 8: Compare with the Observed Density
The observed density of CsCl is given as \( 3.97 \, \text{g/cm}^3 \). The calculated density is significantly lower than the observed density, indicating that the calculated edge length might not accurately reflect the actual structure or that there are other factors affecting the density.

### Summary
- The calculated edge length \( a \) of the CsCl unit cell is \( 7.00 \, \text{Å} \).
- The calculated density is \( 0.816 \, \text{g/cm}^3 \), which is lower than the observed density of \( 3.97 \, \text{g/cm}^3 \).