Question

Tungsten Has An Atomic Radius Of $0.136nm$. The Density Of Tungsten Is $194g/c{m}^3$. What Is Crystal Structure Of Tungsten? (Atomic Weight Of Tungsten $=184$).

• A. Simple Cubic
• B. Body Centered Cubic
• C. Face Centered Cubic
• D. None Of These

Answer 1

The correct option is B

Body Centered Cubic

Explanation for correct option:

(B) Body Centered Cubic

Step I - Given Data

• Atomic Radius of Tungsten $\left(r\right)$ $$\begin{gathered} =0.136nm \\ =0.136\times {10}^{-7}cm \end{gathered}$$
• Density of Tungsten $\left(\rho \right)$$=19.4g/c{m}^3$
• Atomic Weight Of Tungsten $=184g/mol$

Step II - Formula for Density

• $\rho =\frac{Z\times MolecularWeight}{a^3\times N_A}$
• $Z=$Number of atoms per unit cell
• $a=$Lattice Constant
• $N_A=$Avogadro Number $=6.023\times {10}^{23}$

Step III - For $\mathrm{BCC}$ Structure

Figure: Body Centered Cubic Structure

• $Z=8\times \frac{1}{8}+1=2$
• $4r=\sqrt{3}a$
• $a=\frac{4r}{\sqrt{3}}$
• Denstiy of $BCC$ Structure, $\rho =\frac{2\times 184}{{\left({\displaystyle \frac{4\times 0.136\times {10}^{-7}}{\sqrt{3}}}\right)}^3\times 6.023\times {10}^{23}}$

$=19.72g/c{m}^3$

• This value is nearest to the actual density of Tungsten $\left(\mathrm{W}\right)$ i.e. $19.4g/c{m}^3$.

Explanation for incorrect option:

(A) Simple Cubic

• $Z=8\times \frac{1}{8}=1$
• $a=2r$
• $\rho =\frac{1\times 184}{{\left({\displaystyle 2\times 0.136\times {10}^{-7}}\right)}^3\times 6.023\times {10}^{23}}$

$=15.189g/c{m}^3$

• It is not close to the actual density of Tungsten.

(C) Face Centered Cubic

Figure: Face View of FCC structure

• $Z=8\times \frac{1}{8}+6\times \frac{1}{2}=4$
• $$\begin{gathered} 4r=\sqrt{2}a \\ a=\frac{4r}{\sqrt{2}} \end{gathered}$$
• $\rho =\frac{4\times 184}{{\left({\displaystyle \frac{4\times 0.136\times {10}^{-7}}{\sqrt{2}}}\right)}^3\times 6.023\times {10}^{23}}$

$=21.47g/c{m}^3$

• It is not close to the actual density of Tungsten.

Hence, option (B) is correct.