❖ <u>Packing Efficiency</u> ❖
➪ The percentage of total space occupied by particles is called <u>packing efficiency</u>.
![\\ \qquad{\rule{200pt}{3pt}}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Cqquad%7B%5Crule%7B200pt%7D%7B3pt%7D%7D%20)
![\bigstar \boxed{ \sf \: \bigg(Packing \; effeciency = \dfrac{Total \; value \; of \; sphere}{Volume \; of \; unit \; cell} \times 100 \bigg)}](https://tex.z-dn.net/?f=%5Cbigstar%20%5Cboxed%7B%20%5Csf%20%5C%3A%20%20%5Cbigg%28Packing%20%5C%3B%20effeciency%20%3D%20%5Cdfrac%7BTotal%20%5C%3B%20value%20%5C%3B%20of%20%5C%3B%20sphere%7D%7BVolume%20%5C%3B%20of%20%5C%3B%20unit%20%5C%3B%20cell%7D%20%5Ctimes%20100%20%5Cbigg%29%7D)
❖ Packing efficiency of simple cubic structure (SCC).❖
![\rm \: {Let \: } \bf{r } \: \rm{ be \: the \: radius \: of \: a \: sphere }\: \\ \rm and \: \bf{a} \: \rm {be \: the \: edge \: of \: unit \: cell.} \\ \\ \; \bigstar \boxed{ \sf{Volume_{(sphere)} = \dfrac{4}{3} \pi r^3}} \bigstar](https://tex.z-dn.net/?f=%20%5Crm%20%5C%3A%20%7BLet%20%5C%3A%20%7D%20%5Cbf%7Br%20%7D%20%5C%3A%20%5Crm%7B%20be%20%20%5C%3A%20the%20%5C%3A%20%20radius%20%20%5C%3A%20of%20%20%5C%3A%20a%20%5C%3A%20%20sphere%20%20%7D%5C%3A%20%20%5C%5C%20%5Crm%20and%20%5C%3A%20%5Cbf%7Ba%7D%20%5C%3A%20%20%5Crm%20%7Bbe%20%5C%3A%20%20the%20%5C%3A%20%20edge%20%5C%3A%20%20of%20%20%5C%3A%20unit%20%5C%3A%20%20cell.%7D%20%5C%5C%20%20%5C%5C%20%20%5C%3B%20%20%5Cbigstar%20%5Cboxed%7B%20%5Csf%7BVolume_%7B%28sphere%29%7D%20%3D%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20r%5E3%7D%7D%20%5Cbigstar)
❒ Since, simple cubic unit cell contain 1 atom (sphere). So, the total volume of sphere will be :
![: : \implies \sf \: 1 \times \dfrac{4}{3}\pi r^3 = \dfrac{4}{3}\pi r^3](https://tex.z-dn.net/?f=%20%20%20%3A%20%20%3A%20%5Cimplies%20%5Csf%20%5C%3A%201%20%5Ctimes%20%5Cdfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3%20%3D%20%5Cdfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3)
- Volume of unit cell = (2r)³
- Volume of unit cell = 8r³
❒ <u>Now, we know that</u>,❒
![\bigstar \boxed{ \sf \: \bigg(Packing \; effeciency = \dfrac{Total \; value \; of \; sphere}{Volume \; of \; unit \; cell} \times 100 \bigg)}](https://tex.z-dn.net/?f=%5Cbigstar%20%5Cboxed%7B%20%5Csf%20%5C%3A%20%20%5Cbigg%28Packing%20%5C%3B%20effeciency%20%3D%20%5Cdfrac%7BTotal%20%5C%3B%20value%20%5C%3B%20of%20%5C%3B%20sphere%7D%7BVolume%20%5C%3B%20of%20%5C%3B%20unit%20%5C%3B%20cell%7D%20%5Ctimes%20100%20%5Cbigg%29%7D)
➪ Substituting the known values in the formula, we get the following results:
![: : \implies \rm \: {\dfrac{\dfrac{4}{3}\pi r^3}{8r^3} \times 100} \\ \\ : : \implies \dfrac{\pi}{6} \times 100 \\\\\ : : \implies \bf {52.4 \%}](https://tex.z-dn.net/?f=%20%20%3A%20%20%3A%20%5Cimplies%20%5Crm%20%5C%3A%20%7B%5Cdfrac%7B%5Cdfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3%7D%7B8r%5E3%7D%20%5Ctimes%20100%7D%20%5C%5C%20%5C%5C%20%3A%20%20%3A%20%5Cimplies%20%20%20%5Cdfrac%7B%5Cpi%7D%7B6%7D%20%5Ctimes%20100%20%5C%5C%5C%5C%5C%20%20%20%3A%20%20%3A%20%5Cimplies%20%5Cbf%20%7B52.4%20%5C%25%7D)
❒ Hence, the packing efficiency of simple cubic structure is 52.4%.
![\\ \qquad{\rule{200pt}{3pt}}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Cqquad%7B%5Crule%7B200pt%7D%7B3pt%7D%7D%20)
➪ Packing efficiency of cubic close packing (SCC)/face centred cubic structure (FCC).
![\rm \: {Let \: } \bf{r } \: \rm{ be \: the \: radius \: of \: a \: sphere }\: \\ \rm and \: \bf{a} \: \rm {be \: the \: edge \: of \: unit \: cell.} \\ \\ \; \bigstar \boxed{ \sf{Volume_{(sphere)} = \dfrac{4}{3} \pi r^3}} \bigstar](https://tex.z-dn.net/?f=%20%5Crm%20%5C%3A%20%7BLet%20%5C%3A%20%7D%20%5Cbf%7Br%20%7D%20%5C%3A%20%5Crm%7B%20be%20%20%5C%3A%20the%20%5C%3A%20%20radius%20%20%5C%3A%20of%20%20%5C%3A%20a%20%5C%3A%20%20sphere%20%20%7D%5C%3A%20%20%5C%5C%20%5Crm%20and%20%5C%3A%20%5Cbf%7Ba%7D%20%5C%3A%20%20%5Crm%20%7Bbe%20%5C%3A%20%20the%20%5C%3A%20%20edge%20%5C%3A%20%20of%20%20%5C%3A%20unit%20%5C%3A%20%20cell.%7D%20%5C%5C%20%20%5C%5C%20%20%5C%3B%20%20%5Cbigstar%20%5Cboxed%7B%20%5Csf%7BVolume_%7B%28sphere%29%7D%20%3D%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20r%5E3%7D%7D%20%5Cbigstar)
❒ Since, simple cubic unit cell contain 2 atom (sphere). So, the total volume of sphere will be:
![: : \implies \rm 4 \times \dfrac{4}{3}\pi r^3 = \dfrac{16}{3}\pi r^3](https://tex.z-dn.net/?f=%20%20%20%3A%20%3A%20%5Cimplies%20%5Crm%204%20%5Ctimes%20%5Cdfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3%20%3D%20%5Cdfrac%7B16%7D%7B3%7D%5Cpi%20r%5E3)
Volume of unit cell = (2√2r)³
Volume of unit cell = 16√2r³
❒ <u>Now, we know that</u>,❒
![\bigstar \boxed{ \sf \: \bigg(Packing \; effeciency = \dfrac{Total \; value \; of \; sphere}{Volume \; of \; unit \; cell} \times 100 \bigg)}](https://tex.z-dn.net/?f=%5Cbigstar%20%5Cboxed%7B%20%5Csf%20%5C%3A%20%20%5Cbigg%28Packing%20%5C%3B%20effeciency%20%3D%20%5Cdfrac%7BTotal%20%5C%3B%20value%20%5C%3B%20of%20%5C%3B%20sphere%7D%7BVolume%20%5C%3B%20of%20%5C%3B%20unit%20%5C%3B%20cell%7D%20%5Ctimes%20100%20%5Cbigg%29%7D)
➪Substituting the known values in the formula, we get the following results:
![: : \implies \rm {\dfrac{\dfrac{16}{3}\pi r^3}{16\sqrt{2}r^3} \times 100 }\\\\ \implies \rm \dfrac{\pi}{3\sqrt{2}} \times 100 \\\\\ \implies\bf52.4 \%](https://tex.z-dn.net/?f=%20%3A%20%20%20%3A%20%5Cimplies%20%5Crm%20%7B%5Cdfrac%7B%5Cdfrac%7B16%7D%7B3%7D%5Cpi%20r%5E3%7D%7B16%5Csqrt%7B2%7Dr%5E3%7D%20%5Ctimes%20100%20%7D%5C%5C%5C%5C%20%20%5Cimplies%20%5Crm%20%5Cdfrac%7B%5Cpi%7D%7B3%5Csqrt%7B2%7D%7D%20%5Ctimes%20100%20%5C%5C%5C%5C%5C%20%20%5Cimplies%5Cbf52.4%20%5C%25)
❖ Hence, the packing efficiency of face centred cubic structure is 74%.
![\\ \qquad{\rule{200pt}{3pt}}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Cqquad%7B%5Crule%7B200pt%7D%7B3pt%7D%7D%20)
❖ Packing efficiency of body cubic structure (BCC).
![\rm \: {Let \: } \bf{r } \: \rm{ be \: the \: radius \: of \: a \: sphere }\: \\ \rm and \: \bf{a} \: \rm {be \: the \: edge \: of \: unit \: cell.} \\ \\ \; \bigstar \boxed{ \sf{Volume_{(sphere)} = \dfrac{4}{3} \pi r^3}} \bigstar](https://tex.z-dn.net/?f=%20%5Crm%20%5C%3A%20%7BLet%20%5C%3A%20%7D%20%5Cbf%7Br%20%7D%20%5C%3A%20%5Crm%7B%20be%20%20%5C%3A%20the%20%5C%3A%20%20radius%20%20%5C%3A%20of%20%20%5C%3A%20a%20%5C%3A%20%20sphere%20%20%7D%5C%3A%20%20%5C%5C%20%5Crm%20and%20%5C%3A%20%5Cbf%7Ba%7D%20%5C%3A%20%20%5Crm%20%7Bbe%20%5C%3A%20%20the%20%5C%3A%20%20edge%20%5C%3A%20%20of%20%20%5C%3A%20unit%20%5C%3A%20%20cell.%7D%20%5C%5C%20%20%5C%5C%20%20%5C%3B%20%20%5Cbigstar%20%5Cboxed%7B%20%5Csf%7BVolume_%7B%28sphere%29%7D%20%3D%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20r%5E3%7D%7D%20%5Cbigstar)
❒ Since, simple cubic unit cell contain 2 atom (sphere). So, the total volume of sphere will be:
![: : \implies \rm \: 2 \times \dfrac{4}{3}\pi r^3 = \dfrac{8}{3}\pi r^3](https://tex.z-dn.net/?f=%20%20%20%3A%20%3A%20%5Cimplies%20%20%5Crm%20%5C%3A%202%20%5Ctimes%20%5Cdfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3%20%3D%20%5Cdfrac%7B8%7D%7B3%7D%5Cpi%20r%5E3)
Volume of unit cell = (4r/√3)³
Volume of unit cell = 64r³/3√3
❒ <u>Now, we know that</u>,❒
![\bigstar \boxed{ \sf \: \bigg(Packing \; effeciency = \dfrac{Total \; value \; of \; sphere}{Volume \; of \; unit \; cell} \times 100 \bigg)}](https://tex.z-dn.net/?f=%5Cbigstar%20%5Cboxed%7B%20%5Csf%20%5C%3A%20%20%5Cbigg%28Packing%20%5C%3B%20effeciency%20%3D%20%5Cdfrac%7BTotal%20%5C%3B%20value%20%5C%3B%20of%20%5C%3B%20sphere%7D%7BVolume%20%5C%3B%20of%20%5C%3B%20unit%20%5C%3B%20cell%7D%20%5Ctimes%20100%20%5Cbigg%29%7D)
➪Substituting the known values in the formula, we get the following results:
![: : \implies \rm {\dfrac{\dfrac{8}{3}\pi r^3}{\dfrac{64r^3}{3\sqrt{3}}} \times 100} \\\\ \implies \dfrac{3\pi}{8} \times 100 \\\ \\ \bf \implies \: 68 \%](https://tex.z-dn.net/?f=%20%20%20%3A%20%3A%20%5Cimplies%20%5Crm%20%7B%5Cdfrac%7B%5Cdfrac%7B8%7D%7B3%7D%5Cpi%20r%5E3%7D%7B%5Cdfrac%7B64r%5E3%7D%7B3%5Csqrt%7B3%7D%7D%7D%20%5Ctimes%20100%7D%20%5C%5C%5C%5C%20%20%5Cimplies%20%5Cdfrac%7B3%5Cpi%7D%7B8%7D%20%5Ctimes%20100%20%5C%5C%5C%20%5C%5C%20%20%5Cbf%20%20%5Cimplies%20%5C%3A%2068%20%5C%25)
❒ Hence, the packing efficiency of body centred cubic structure is 68%.
![\\ \qquad{\rule{200pt}{3pt}}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Cqquad%7B%5Crule%7B200pt%7D%7B3pt%7D%7D%20)