Potassium has a bcc structure with nearest neighour distance `4.52 Å`its atomic weight is `39` its density (in kg `m^(-3)`) will be

Options

A. 454
B. 804
C. 852
D. 908

Answer

To find the density of potassium with a body-centered cubic (BCC) structure, we can follow these steps:

### Step 1: Understand the BCC Structure
In a BCC structure, there are 2 atoms per unit cell. This is denoted by \( Z = 2 \).

### Step 2: Calculate the Edge Length (a)
The nearest neighbor distance \( d \) in a BCC structure is related to the edge length \( a \) by the formula:
\[ d = \frac{\sqrt{3}}{2} a \]
Given that the nearest neighbor distance \( d = 4.52 \, \text{Å} \), we can rearrange the formula to find \( a \):
\[ a = \frac{2d}{\sqrt{3}} = \frac{2 \times 4.52}{\sqrt{3}} \]
Calculating this gives:
\[ a = \frac{9.04}{1.732} \approx 5.22 \, \text{Å} \]

### Step 3: Convert Edge Length to Meters
Since we need to use SI units for density, convert \( a \) from Ångströms to meters:
\[ a = 5.22 \, \text{Å} = 5.22 \times 10^{-10} \, \text{m} \]

### Step 4: Use the Density Formula
The density \( \rho \) of a crystal can be calculated using the formula:
\[ \rho = \frac{Z \cdot M}{N_A \cdot a^3} \]
Where:
- \( Z = 2 \) (number of atoms per unit cell)
- \( M = 39 \, \text{g/mol} = 39 \times 10^{-3} \, \text{kg/mol} \) (molar mass of potassium)
- \( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \) (Avogadro's number)
- \( a = 5.22 \times 10^{-10} \, \text{m} \)

### Step 5: Calculate \( a^3 \)
First, calculate \( a^3 \):
\[ a^3 = (5.22 \times 10^{-10})^3 \approx 1.42 \times 10^{-28} \, \text{m}^3 \]

### Step 6: Substitute Values into the Density Formula
Now substitute the values into the density formula:
\[ \rho = \frac{2 \cdot (39 \times 10^{-3})}{(6.022 \times 10^{23}) \cdot (1.42 \times 10^{-28})} \]

### Step 7: Calculate the Density
Calculating this gives:
\[ \rho \approx \frac{0.078}{8.54 \times 10^{-6}} \approx 9140 \, \text{kg/m}^3 \]

### Step 8: Convert to kg/m³
Since we need the density in kg/m³, we already have it in the correct unit.

### Final Answer
The density of potassium is approximately \( 9140 \, \text{kg/m}^3 \).