Tungsten has bcc lattice. Each edge of the unit cell is 316 pm and the density of the metal is 19.35g `cm^(-3)`. How many atoms are present in 50 g of this element?

Explanation

To find the number of atoms present in 50 g of tungsten, we will follow these steps:

### Step 1: Convert the edge length of the unit cell from picometers to centimeters.
Given:
- Edge length (a) = 316 pm

To convert picometers to centimeters:
\[ 1 \text{ pm} = 10^{-10} \text{ cm} \]
Thus,
\[ a = 316 \times 10^{-10} \text{ cm} \]

### Step 2: Calculate the volume of the unit cell.
The volume (V) of the cubic unit cell is given by:
\[ V = a^3 \]
Substituting the value of a:
\[ V = (316 \times 10^{-10})^3 \]
Calculating this:
\[ V = 3.2 \times 10^{-20} \text{ cm}^3 \]

### Step 3: Use the density formula to find the atomic mass (m).
The density (d) of tungsten is given as 19.35 g/cm³. The formula for density is:
\[ d = \frac{z \cdot m}{V \cdot N_a} \]
Where:
- \( z \) = number of atoms per unit cell (for BCC, \( z = 2 \))
- \( N_a \) = Avogadro's number \( (6.02 \times 10^{23} \text{ atoms/mol}) \)

Rearranging the formula to find m:
\[ m = \frac{d \cdot V \cdot N_a}{z} \]
Substituting the known values:
\[ m = \frac{19.35 \cdot 3.2 \times 10^{-20} \cdot 6.02 \times 10^{23}}{2} \]

### Step 4: Calculate the atomic mass (m).
Calculating this expression:
\[ m = \frac{19.35 \cdot 3.2 \cdot 6.02}{2} \]
\[ m \approx 186.5 \text{ g/mol} \]

### Step 5: Calculate the number of moles in 50 g of tungsten.
Using the formula:
\[ \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} \]
Substituting the values:
\[ \text{Number of moles} = \frac{50 \text{ g}}{186.5 \text{ g/mol}} \]

### Step 6: Calculate the number of atoms in 50 g of tungsten.
To find the number of atoms:
\[ \text{Number of atoms} = \text{Number of moles} \times N_a \]
Substituting the values:
\[ \text{Number of atoms} = \left(\frac{50}{186.5}\right) \times 6.02 \times 10^{23} \]
Calculating this gives:
\[ \text{Number of atoms} \approx 1.614 \times 10^{23} \]

### Final Answer:
Thus, the number of atoms present in 50 g of tungsten is approximately:
\[ 1.614 \times 10^{23} \text{ atoms} \]

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