To find the density of CsBr, which crystallizes in a body-centered cubic (BCC) lattice, we can follow these steps:
### Step 1: Convert the unit cell length to centimeters
The unit cell length is given as 420 picometers (pm). We need to convert this to centimeters (cm).
\[
\text{Unit cell length} = 420 \, \text{pm} = 420 \times 10^{-10} \, \text{cm} = 4.20 \times 10^{-8} \, \text{cm}
\]
### Step 2: Calculate the molar mass of CsBr
The atomic mass of cesium (Cs) is 133 amu and that of bromine (Br) is 80 amu. The molar mass of CsBr can be calculated as follows:
\[
\text{Molar mass of CsBr} = \text{Mass of Cs} + \text{Mass of Br} = 133 \, \text{g/mol} + 80 \, \text{g/mol} = 213 \, \text{g/mol}
\]
### Step 3: Determine the number of formula units per unit cell (Z)
For a body-centered cubic lattice, the number of formula units per unit cell (Z) is 2.
### Step 4: Use the density formula
The density (\( \rho \)) can be calculated using the formula:
\[
\rho = \frac{Z \cdot M}{a^3 \cdot N_A}
\]
Where:
- \( Z = 2 \) (number of formula units per unit cell)
- \( M = 213 \, \text{g/mol} \) (molar mass of CsBr)
- \( a = 4.20 \times 10^{-8} \, \text{cm} \) (edge length of the unit cell)
- \( N_A = 6.02 \times 10^{23} \, \text{mol}^{-1} \) (Avogadro's number)
### Step 5: Calculate \( a^3 \)
First, we need to calculate \( a^3 \):
\[
a^3 = (4.20 \times 10^{-8} \, \text{cm})^3 = 7.388 \times 10^{-24} \, \text{cm}^3
\]
### Step 6: Substitute values into the density formula
Now, substituting the values into the density formula:
\[
\rho = \frac{2 \cdot 213 \, \text{g/mol}}{7.388 \times 10^{-24} \, \text{cm}^3 \cdot 6.02 \times 10^{23} \, \text{mol}^{-1}}
\]
Calculating the denominator:
\[
7.388 \times 10^{-24} \, \text{cm}^3 \cdot 6.02 \times 10^{23} \, \text{mol}^{-1} = 4.444 \times 10^{-1} \, \text{cm}^3/\text{mol}
\]
Now substituting back into the density equation:
\[
\rho = \frac{426 \, \text{g/mol}}{4.444 \times 10^{-1} \, \text{cm}^3/\text{mol}} = 9.57 \, \text{g/cm}^3
\]
### Step 7: Final result
The density of CsBr is approximately:
\[
\rho \approx 9.57 \, \text{g/cm}^3
\]
