Search any question & find its solution
Question:
Answered & Verified by Expert
Calculate the efficiency of packing in case of a metal crystal for (i) simple cubic, (ii) body centred cubic, and (iii) face centred cubic (with the assumptions that atoms are touching each other).
Solution:
1717 Upvotes
Verified Answer
Packing efficiency: It is the percentage of total space filled by the particles.
(i) In simple cubic lattice :
Let the edge length of the cube be ' \(a\) ' and the radius of each particle be ' \(r\) '.
\(\therefore a=2 r\)
No. of spheres per unit cell \(=1\)
Volume of the occupied unit cell \(=\frac{4}{3} \pi r^3\)
Volume of cube \(=a^3=(2 r)^3=8 r^3\)
\(\therefore\) Packing efficiency \(=\frac{\begin{array}{c}\text { volume of } \\ \text { one particle }\end{array}}{\begin{array}{c}\text { volume of cubic } \\ \text { unit cell }\end{array}}\)
\(=\frac{4 / 3 \pi r^3}{8 r^3}=0.524\), i.e., \(52.4 \%\).
(ii) In \(b c c\) : Length of body diagonal \(\mathrm{AD}=4 r\)
From right angled triangle \(\mathrm{ABC}\),
\(\mathrm{AC}=\sqrt{\mathrm{AB}^2+\mathrm{BC}^2}=\sqrt{a^2+a^2}=\sqrt{2} a\)
Body diagonal, AD
\(\begin{aligned}
&=\sqrt{\mathrm{AC}^2+\mathrm{CD}^2}=\sqrt{2 a^2+a^2}=\sqrt{3 a^2}=\sqrt{3} a \\
\therefore \quad & \sqrt{3} a=4 r \quad a=4 r / \sqrt{3}
\end{aligned}\)
\(\therefore \quad\) Volume of unit cell \(=a^3=\left(\frac{4 r}{\sqrt{3}}\right)^3=\frac{64 r^3}{3 \sqrt{3}}\) No. of spheres per unit cell \(=2\)
\(\therefore \quad\) Volume of two spheres \(=2 \times \frac{4}{3} \pi r^3=\frac{8}{3} \pi r^3\)
\(\therefore\) Packing efficiency
\(=\frac{8 / 3 \pi r^3}{64 r^3 / 3 \sqrt{3}}=0.68 \text {, i.e., } 68 \%\)
(iii) In fcc: Let the edge length of unit cell \(=a\)
Let the radius of each sphere \(=r\)
\(\therefore\) length of face diagonal \(\mathrm{AC}=4 r\) From right angled triangle \(\mathrm{ABC}\),
\(\mathrm{AC}=\sqrt{\mathrm{AB}^2+\mathrm{BC}^2}=\sqrt{a^2+a^2}=\sqrt{2 a^2}=\sqrt{2} a\)
\(\therefore \quad \sqrt{2} a=4 r\)
\(\therefore \quad a=\frac{4 r}{\sqrt{2}}\)
\(\therefore \quad\) Volume of the unit cell \(=a^3=\left(\frac{4}{\sqrt{2}} r\right)^3=\frac{64 r^3}{2 \sqrt{2}}=16 \sqrt{2} r^3\)
No. of unit cell in \(f c c=4\)
\(\therefore \quad\) Volume of four spheres \(=4 \times \frac{4}{3} \pi \mathrm{r}^3=\frac{16}{3} \pi \mathrm{r}^3\)
\(\therefore\) Packing efficiency
\(=\frac{16 \pi r^3 / 3}{16 \sqrt{2} r^3}=0.74 \text {, i.e., } 74 \%\)
(i) In simple cubic lattice :
Let the edge length of the cube be ' \(a\) ' and the radius of each particle be ' \(r\) '.
\(\therefore a=2 r\)
No. of spheres per unit cell \(=1\)
Volume of the occupied unit cell \(=\frac{4}{3} \pi r^3\)
Volume of cube \(=a^3=(2 r)^3=8 r^3\)
\(\therefore\) Packing efficiency \(=\frac{\begin{array}{c}\text { volume of } \\ \text { one particle }\end{array}}{\begin{array}{c}\text { volume of cubic } \\ \text { unit cell }\end{array}}\)
\(=\frac{4 / 3 \pi r^3}{8 r^3}=0.524\), i.e., \(52.4 \%\).
(ii) In \(b c c\) : Length of body diagonal \(\mathrm{AD}=4 r\)
From right angled triangle \(\mathrm{ABC}\),
\(\mathrm{AC}=\sqrt{\mathrm{AB}^2+\mathrm{BC}^2}=\sqrt{a^2+a^2}=\sqrt{2} a\)
Body diagonal, AD
\(\begin{aligned}
&=\sqrt{\mathrm{AC}^2+\mathrm{CD}^2}=\sqrt{2 a^2+a^2}=\sqrt{3 a^2}=\sqrt{3} a \\
\therefore \quad & \sqrt{3} a=4 r \quad a=4 r / \sqrt{3}
\end{aligned}\)
\(\therefore \quad\) Volume of unit cell \(=a^3=\left(\frac{4 r}{\sqrt{3}}\right)^3=\frac{64 r^3}{3 \sqrt{3}}\) No. of spheres per unit cell \(=2\)
\(\therefore \quad\) Volume of two spheres \(=2 \times \frac{4}{3} \pi r^3=\frac{8}{3} \pi r^3\)
\(\therefore\) Packing efficiency
\(=\frac{8 / 3 \pi r^3}{64 r^3 / 3 \sqrt{3}}=0.68 \text {, i.e., } 68 \%\)
(iii) In fcc: Let the edge length of unit cell \(=a\)
Let the radius of each sphere \(=r\)
\(\therefore\) length of face diagonal \(\mathrm{AC}=4 r\) From right angled triangle \(\mathrm{ABC}\),
\(\mathrm{AC}=\sqrt{\mathrm{AB}^2+\mathrm{BC}^2}=\sqrt{a^2+a^2}=\sqrt{2 a^2}=\sqrt{2} a\)
\(\therefore \quad \sqrt{2} a=4 r\)
\(\therefore \quad a=\frac{4 r}{\sqrt{2}}\)
\(\therefore \quad\) Volume of the unit cell \(=a^3=\left(\frac{4}{\sqrt{2}} r\right)^3=\frac{64 r^3}{2 \sqrt{2}}=16 \sqrt{2} r^3\)
No. of unit cell in \(f c c=4\)
\(\therefore \quad\) Volume of four spheres \(=4 \times \frac{4}{3} \pi \mathrm{r}^3=\frac{16}{3} \pi \mathrm{r}^3\)
\(\therefore\) Packing efficiency
\(=\frac{16 \pi r^3 / 3}{16 \sqrt{2} r^3}=0.74 \text {, i.e., } 74 \%\)
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.