Chromium crystallizes in a body centred cubic lattice, whose density is 7.20 `g//cm^3`. The length of the edge of unit cell is 288.4 pm. Calculate Avogadro's number.
(Atomic mass of chromium=52)

Explanation

To calculate Avogadro's number using the given data about chromium crystallizing in a body-centered cubic (BCC) lattice, we can follow these steps:

### Step 1: Understand the formula for density in a BCC lattice
The formula for density (d) in a crystalline solid is given by:
\[ d = \frac{z \cdot m}{a^3} \]
where:
- \( z \) = number of atoms per unit cell (for BCC, \( z = 2 \))
- \( m \) = molar mass of the substance (in grams per mole)
- \( a \) = edge length of the unit cell (in cm)

### Step 2: Convert the edge length from picometers to centimeters
The edge length of the unit cell is given as \( 288.4 \) pm. We need to convert this to centimeters:
\[ a = 288.4 \, \text{pm} = 288.4 \times 10^{-12} \, \text{m} = 288.4 \times 10^{-10} \, \text{cm} \]

### Step 3: Rearrange the density formula to find Avogadro's number
We can rearrange the density formula to express Avogadro's number (\( N_A \)):
\[ N_A = \frac{z \cdot m}{d \cdot a^3} \]

### Step 4: Substitute the known values into the formula
We know:
- \( z = 2 \) (for BCC)
- \( m = 52 \, \text{g/mol} \) (atomic mass of chromium)
- \( d = 7.20 \, \text{g/cm}^3 \) (density)
- \( a = 288.4 \times 10^{-10} \, \text{cm} \)

Now we can calculate \( a^3 \):
\[ a^3 = (288.4 \times 10^{-10})^3 = 2.396 \times 10^{-29} \, \text{cm}^3 \]

### Step 5: Calculate Avogadro's number
Now substituting all the values into the rearranged formula:
\[ N_A = \frac{2 \cdot 52}{7.20 \cdot 2.396 \times 10^{-29}} \]
Calculating the numerator:
\[ 2 \cdot 52 = 104 \]
Calculating the denominator:
\[ 7.20 \cdot 2.396 \times 10^{-29} = 1.725 \times 10^{-28} \]
Now, substituting these values:
\[ N_A = \frac{104}{1.725 \times 10^{-28}} \approx 6.03 \times 10^{23} \, \text{mol}^{-1} \]

### Final Answer
Thus, Avogadro's number is approximately:
\[ N_A \approx 6.03 \times 10^{23} \, \text{mol}^{-1} \]