Question

An element crystallizes in a bcc lattice of edge length 4 \AA. If the density of the element is 10 g cm\(^{-3}\), what is its atomic weight (g mol\(^{-1}\))?

• A. 96
• B. 124
• C. 192
• D. 235

Hint:

For cubic crystals use \(\rho = \frac{Z M}{N_A a^3}\) and always convert Å to cm.

Answer 1

The Correct Option is C

Step 1: Using density formula for crystal lattice.
Density is given by: \[ \rho = \frac{Z \times M}{N_A \times a^3} \] where \(Z = 2\) for bcc, \(M\) is molar mass, \(a\) is edge length.

Step 2: Converting edge length.
Given \(a = 4 \,\AA = 4 \times 10^{-8} \text{ cm}\).

Step 3: Substituting values.
\[ 10 = \frac{2M}{(6.022 \times 10^{23})(64 \times 10^{-24})} \]

Step 4: Simplifying expression.
\[ 64 \times 10^{-24} = 6.4 \times 10^{-23} \] So, \[ 10 = \frac{2M}{(6.022 \times 6.4)\times 10^{0}} \]

Step 5: Solving for M.
\[ 10 = \frac{2M}{38.53} \Rightarrow M = \frac{10 \times 38.53}{2} \approx 192 \]

Step 6: Final conclusion.
Thus, molar mass of the element is \(192 \,\text{g mol}^{-1}\).
Final Answer: \[ \boxed{192} \]