Question

What is the molar mass of a metal having density $8.57\ \mathrm{g\ cm}^{-3}$ and edge length $3.3\ \mathrm{\AA}$? (packing efficiency = 68%)

• A. 63 g mol$^{-1}$
• B. 93 g mol$^{-1}$
• C. 29 g mol$^{-1}$
• D. 39 g mol$^{-1}$

Hint:

Connecting packing efficiencies directly to lattice type saves precious time! Always remember: $52.4\% \rightarrow \text{simple cubic } (n=1)$, $68\% \rightarrow \text{bcc } (n=2)$, and $74\% \rightarrow \text{fcc/hcp } (n=4)$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks us to calculate the molar mass ($M$) of a metal, given its density ($\rho = 8.57\text{ g cm}^{-3}$), the unit cell edge length ($a = 3.3\text{ \r{A}}$), and its packing efficiency ($68\%$).

Step 2: Key Formula or Approach:
1. A packing efficiency of $68\%$ indicates that the metal crystallizes in a body-centered cubic (bcc) lattice structure. For a bcc unit cell, the number of atoms per unit cell ($n$) is exactly 2.
2. The standard formula relating density to the crystal parameters is: $$\rho = \frac{n \times M}{a^3 \times N_A}$$ Rearranging this equation to solve for the molar mass ($M$) yields: $$M = \frac{\rho \times a^3 \times N_A}{n}$$ Where Avogadro's number $N_A = 6.022 \times 10^{23}\text{ atoms mol}^{-1}$.

Step 3: Detailed Explanation:
First, convert the edge length from angstroms (\r{A}) to centimeters ($\text{cm}$): $$a = 3.3\text{ \r{A}} = 3.3 \times 10^{-8}\text{ cm}$$ Calculate the volume of the unit cell ($a^3$): $$a^3 = (3.3 \times 10^{-8}\text{ cm})^3 = 35.937 \times 10^{-24}\text{ cm}^3$$ Now, substitute all known values into the rearranged molar mass equation: $$M = \frac{8.57 \times (35.937 \times 10^{-24}) \times (6.022 \times 10^{23})}{2}$$ Simplify the powers of 10: $$10^{-24} \times 10^{23} = 10^{-1} = 0.1$$ Multiply the remaining terms together: $$M = \frac{8.57 \times 35.937 \times 6.022 \times 0.1}{2}$$ $$M = \frac{185.47}{2} \approx 92.73\text{ g mol}^{-1}$$ Rounding this to the nearest integer gives $93\text{ g mol}^{-1}$.

Step 4: Final Answer:
The molar mass of the metal is approximately $93\text{ g mol}^{-1}$, which matches option (B).