Question

An element is found to crystallize with BCC structure having density $8.55\ \text{g\ cm}^{-3}$. What is the edge length of unit cell? (At. mass of element = 93)

• A. $(3.61 \times 10^{-23})^{1/3}\ \text{cm}$
• B. $(3.91 \times 10^{-20})^{1/3}\ \text{cm}$
• C. $(3.01 \times 10^{-24})^{1/3}\ \text{cm}$
• D. $(3.30 \times 10^{-20})^{1/3}\ \text{cm}$

Hint:

When an answer features a complicated fractional exponent like $(\dots)^{1/3}$, do not waste valuable time executing complex cube root estimations! Simply calculate the interior base number ($a^3$), match it to the options, and append the exponent.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question gives the crystal structure (BCC), the mass density ($\rho$), and the atomic mass ($M$) of an element. We need to solve for the unit cell edge length ($a$) expressed as a cube root.

Step 2: Key Formula or Approach:
The density of a crystalline solid unit cell is governed by the standard formula:
$$\rho = \frac{z \times M}{a^3 \times N_A} \implies a^3 = \frac{z \times M}{\rho \times N_A}$$ Where:
$z =$ number of constituent atoms per unit cell. For a Body-Centered Cubic (BCC) lattice, $z = 2$.
$M =$ atomic mass of the element ($93\ \text{g\ mol}^{-1}$).
$\rho =$ mass density of the crystal ($8.55\ \text{g\ cm}^{-3}$).
$N_A =$ Avogadro's constant ($6.022 \times 10^{23}\ \text{mol}^{-1}$).

Step 3: Detailed Explanation:
Let's substitute our known parameters directly into the rearranged formula to evaluate the volume of the unit cell ($a^3$):
$$a^3 = \frac{2 \times 93}{8.55 \times 6.022 \times 10^{23}}$$ Simplify the values in the numerator and denominator:
$$\text{Numerator} = 2 \times 93 = 186$$ $$\text{Denominator} = 8.55 \times 6.022 \times 10^{23} = 51.488 \times 10^{23}$$ Divide the core terms:
$$a^3 = \frac{186}{51.488} \times 10^{-23}$$ $$a^3 \approx 3.6124 \times 10^{-23}\ \text{cm}^3$$ To isolate the edge length $a$, take the cube root of both sides of the expression:
$$a = (3.61 \times 10^{-23})^{1/3}\ \text{cm}$$ This matches option (A).

Step 4: Final Answer:
The edge length of the unit cell is $(3.61 \times 10^{-23})^{1/3}\ \text{cm}$, which corresponds to option (A).