A metal has FCC crystal structure with ρ = 2.71 g/cm³ & 26.98 g/mol of atomic weight. A Avogadro’s No. is 6.023 × 10²³. The atomic radius of the metal is _______ nm (Round off to two decimal places).

The correct answer is 0.143.

Step-by-step Explanation:

1. Identify the given parameters:
- Crystal structure: Face-Centered Cubic (FCC)
- Density of the metal ($\rho$): $2.71{\text{g/cm}}^3$
- Atomic weight ($M$): $26.98\text{g/mol}$
- Avogadro’s Number ($N_A$): $6.023\times {10}^{23}{\text{atoms/mol}}^{}$

2. Determine the number of atoms per unit cell for FCC:
For a Face-Centered Cubic (FCC) crystal structure, the number of atoms per unit cell ($n$) is:

$n=4$

3. Relate density to unit cell dimensions:
The density ($\rho$) of a crystalline solid is calculated using the formula:

$\rho =\frac{n\cdot M}{V_c\cdot N_A}$

where $V_c$ is the volume of the cubic unit cell ($V_c=a^3$, with $a$ being the lattice parameter). Rearranging the formula to solve for $V_c$ gives:

$V_c=\frac{n\cdot M}{\rho \cdot N_A}$

4. Calculate the volume of the unit cell:
Substitute the given values into the equation:

$V_c=\frac{4\times 26.98}{2.71��6.023\times {10}^{23}}$

$V_c=\frac{107.92}{1.632233\times {10}^{24}}$

$V_c\approx 6.6118\times {10}^{-23}{\text{cm}}^3$

5. Find the lattice parameter ($a$):
Since $V_c=a^3$, we find $a$ by taking the cube root of the volume:

$a={(6.6118\times {10}^{-23})}^{1/3}$

To make calculation easier, write the number inside the parenthesis in terms of ${10}^{-24}$:

$a={(66.118\times {10}^{-24})}^{1/3}$

$a\approx 4.0436\times {10}^{-8}\text{cm}$

Convert the lattice parameter from centimeters to nanometers ($1\text{nm}={10}^{-7}\text{cm}$):

$a\approx 0.40436\text{nm}$

6. Calculate the atomic radius ($r$):
For an FCC crystal structure, the atoms touch along the face diagonal, establishing the relation:

$4r=a\sqrt{2}$

Solving for the atomic radius $r$:

$r=\frac{a\sqrt{2}}{4}=\frac{a}{2\sqrt{2}}$

Substitute the value of $a$ into this expression:

$r=\frac{0.40436}{2\times 1.4142}$

$r=\frac{0.40436}{2.8284}\approx 0.14296\text{nm}$

Rounding off to three decimal places yields:

$r\approx 0.143\text{nm}$