Lithium metal has a body centred cubic structure. Its density is `0.53 g cm^(-3)` and its molar mass is `6.94 g mol^(-1)`. Calculate the edge length of a unit cell of Lithium metal

Options

A. 153.6 pm
B. 351.6 pm
C. 527.4 pm
D. 263.7 pm

Answer

To calculate the edge length of a unit cell of lithium metal, which has a body-centered cubic (BCC) structure, we can follow these steps:

### Step 1: Identify the parameters
- **Density (D)** of lithium = \(0.53 \, \text{g/cm}^3\)
- **Molar mass (M)** of lithium = \(6.94 \, \text{g/mol}\)
- **Number of atoms per unit cell (Z)** for BCC = 2 (1 atom at the body center and 8 corner atoms contributing \( \frac{1}{8} \) each)

### Step 2: Use the formula for density
The formula for density in terms of the unit cell parameters is given by:
\[ D = \frac{Z \cdot M}{A^3 \cdot N_A} \]
Where:
- \(D\) = density
- \(Z\) = number of atoms per unit cell
- \(M\) = molar mass
- \(A\) = edge length of the unit cell
- \(N_A\) = Avogadro's number (\(6.022 \times 10^{23} \, \text{mol}^{-1}\))

### Step 3: Rearranging the formula to find \(A^3\)
Rearranging the density formula to solve for \(A^3\):
\[ A^3 = \frac{Z \cdot M}{D \cdot N_A} \]

### Step 4: Substitute the values
Now, substituting the known values into the equation:
\[ A^3 = \frac{2 \cdot 6.94 \, \text{g/mol}}{0.53 \, \text{g/cm}^3 \cdot 6.022 \times 10^{23} \, \text{mol}^{-1}} \]

### Step 5: Calculate \(A^3\)
Calculating the numerator:
\[ 2 \cdot 6.94 = 13.88 \, \text{g/mol} \]
Calculating the denominator:
\[ 0.53 \cdot 6.022 \times 10^{23} \approx 3.19466 \times 10^{23} \, \text{g/cm}^3 \cdot \text{mol}^{-1} \]
Now substituting back:
\[ A^3 = \frac{13.88}{3.19466 \times 10^{23}} \approx 4.344 \times 10^{-23} \, \text{cm}^3 \]

### Step 6: Calculate \(A\)
Taking the cube root to find \(A\):
\[ A = (4.344 \times 10^{-23})^{1/3} \approx 3.516 \times 10^{-8} \, \text{cm} \]

### Step 7: Convert to picometers
To convert centimeters to picometers:
\[ A = 3.516 \times 10^{-8} \, \text{cm} \times 10^{10} \, \text{pm/cm} = 351.6 \, \text{pm} \]

### Final Answer
The edge length of the unit cell of lithium metal is approximately **351.6 picometers (pm)**.

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