Tungsten has a density of 19.35 g `cm^(-3)` and the length of the side of the unit cell is 316 pm. The unit cell is a body centred unit cell. How many atoms does 50 grams of the element contain?

Explanation

To solve the problem of how many atoms are present in 50 grams of tungsten, we will follow these steps:

### Step 1: Convert the unit cell length from picometers to centimeters
The length of the side of the unit cell is given as 316 pm (picometers). We need to convert this to centimeters for consistency with the density units.

\[ 1 \text{ pm} = 10^{-12} \text{ m} = 10^{-10} \text{ cm} \]
Therefore,
\[ 316 \text{ pm} = 316 \times 10^{-10} \text{ cm} = 3.16 \times 10^{-8} \text{ cm} \]

### Step 2: Calculate the volume of the unit cell
The volume \( V \) of the unit cell can be calculated using the formula for the volume of a cube:

\[ V = a^3 \]
where \( a \) is the length of the side of the unit cell.

\[ V = (3.16 \times 10^{-8} \text{ cm})^3 = 3.16^3 \times 10^{-24} \text{ cm}^3 \approx 3.16 \times 10^{-24} \text{ cm}^3 \]

### Step 3: Use the density to find the molar mass
The density \( D \) of tungsten is given as 19.35 g/cm³. The formula relating density, molar mass \( M \), and the number of atoms per unit cell \( Z \) is:

\[ D = \frac{Z \cdot M}{V \cdot N_A} \]
where \( N_A \) is Avogadro's number (\( 6.022 \times 10^{23} \text{ mol}^{-1} \)) and \( Z \) for a body-centered cubic (BCC) unit cell is 2.

Rearranging the formula to find the molar mass \( M \):

\[ M = \frac{D \cdot V \cdot N_A}{Z} \]

Substituting the known values:

\[ M = \frac{19.35 \text{ g/cm}^3 \cdot 3.16 \times 10^{-24} \text{ cm}^3 \cdot 6.022 \times 10^{23} \text{ mol}^{-1}}{2} \]

Calculating this gives:

\[ M \approx 183.84 \text{ g/mol} \]

### Step 4: Calculate the number of moles in 50 grams of tungsten
To find the number of moles \( n \):

\[ n = \frac{\text{mass}}{\text{molar mass}} = \frac{50 \text{ g}}{183.84 \text{ g/mol}} \approx 0.272 \text{ mol} \]

### Step 5: Calculate the total number of atoms
To find the total number of atoms \( N \):

\[ N = n \cdot N_A = 0.272 \text{ mol} \cdot 6.022 \times 10^{23} \text{ atoms/mol} \approx 1.64 \times 10^{23} \text{ atoms} \]

### Final Answer
The total number of atoms in 50 grams of tungsten is approximately \( 1.64 \times 10^{23} \) atoms.

---