To calculate the packing factor, first calculate the area and volume of unit cell.
Area is calculated as:
![A=6R^{2}\sqrt{3}](https://tex.z-dn.net/?f=A%3D6R%5E%7B2%7D%5Csqrt%7B3%7D)
Here, R is radius and is related to a as follows:
![R=\frac{a}{2}](https://tex.z-dn.net/?f=R%3D%5Cfrac%7Ba%7D%7B2%7D)
Putting the value in expression for area,
![A=6(\frac{a}{2})^{2}\sqrt{3}=1.5a^{2}\sqrt{3}](https://tex.z-dn.net/?f=A%3D6%28%5Cfrac%7Ba%7D%7B2%7D%29%5E%7B2%7D%5Csqrt%7B3%7D%3D1.5a%5E%7B2%7D%5Csqrt%7B3%7D)
The value of a is 0.4961 nm
Since, ![1 nm=10^{-7}cm](https://tex.z-dn.net/?f=1%20nm%3D10%5E%7B-7%7Dcm)
Thus, ![0.4961 nm=4.961\times 10^{-8} cm](https://tex.z-dn.net/?f=0.4961%20nm%3D4.961%5Ctimes%2010%5E%7B-8%7D%20cm)
Putting the value,
![Area=1.5(4.961\times 10^{-8}cm)^{2}\sqrt{3}=6.39\times 10^{-15}cm^{2}](https://tex.z-dn.net/?f=Area%3D1.5%284.961%5Ctimes%2010%5E%7B-8%7Dcm%29%5E%7B2%7D%5Csqrt%7B3%7D%3D6.39%5Ctimes%2010%5E%7B-15%7Dcm%5E%7B2%7D)
Now, volume can be calculated as follows:
![V=Area\times c](https://tex.z-dn.net/?f=V%3DArea%5Ctimes%20c)
The value of c is 1.360 nm or ![1.360\times 10^{-7} cm](https://tex.z-dn.net/?f=1.360%5Ctimes%2010%5E%7B-7%7D%20cm)
Putting the value,
![V=(6.39\times 10^{-15}cm^{2})\times (1.360\times 10^{-7} cm)=8.7\times 10^{-22}cm^{3}](https://tex.z-dn.net/?f=V%3D%286.39%5Ctimes%2010%5E%7B-15%7Dcm%5E%7B2%7D%29%5Ctimes%20%281.360%5Ctimes%2010%5E%7B-7%7D%20cm%29%3D8.7%5Ctimes%2010%5E%7B-22%7Dcm%5E%7B3%7D)
now, number of atom in unit cell can be calculated by using the following formula:
![n=\frac{\rho N_{A}V_{c}}{A}](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B%5Crho%20N_%7BA%7DV_%7Bc%7D%7D%7BA%7D)
Here, A is atomic mass of
is 151.99 g/mol.
Putting all the values,
![n=\frac{(5.22 g/cm^{3})(6.023\times 10^{23} mol^{-1})(8.7\times 10^{-22}cm^{3})}{(151.99 g/mol)}\approx 18](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B%285.22%20g%2Fcm%5E%7B3%7D%29%286.023%5Ctimes%2010%5E%7B23%7D%20mol%5E%7B-1%7D%29%288.7%5Ctimes%2010%5E%7B-22%7Dcm%5E%7B3%7D%29%7D%7B%28151.99%20g%2Fmol%29%7D%5Capprox%2018)
Thus, there will be 18
units in 1 unit cell.
Since, there are 2 Cr atoms and 3 oxygen atoms thus, units of chromium and oxygen will be 2×18=36 and 3×18=54 respectively.
The atomic radii of
and
is 62 pm and 140 pm respectively.
Converting them into cm:
![1 pm=10^{-10}cm](https://tex.z-dn.net/?f=1%20pm%3D10%5E%7B-10%7Dcm)
Thus,
![r_{Cr^{3+}}=6.2\times 10^{-9}cm](https://tex.z-dn.net/?f=r_%7BCr%5E%7B3%2B%7D%7D%3D6.2%5Ctimes%2010%5E%7B-9%7Dcm)
and,
![r_{O^{2-}}=1.4\times 10^{-8}cm](https://tex.z-dn.net/?f=r_%7BO%5E%7B2-%7D%7D%3D1.4%5Ctimes%2010%5E%7B-8%7Dcm)
Volume of sphere will be sum of volume of total number of cations and anions thus,
![V_{S}=V_{Cr^{3+}}+V_{O^{2-}}](https://tex.z-dn.net/?f=V_%7BS%7D%3DV_%7BCr%5E%7B3%2B%7D%7D%2BV_%7BO%5E%7B2-%7D%7D)
Since, volume of sphere is
,
![V_{S}=36\left ( \frac{4}{3}\pi (r_{Cr^{3+})^{3}} \right )+54\left ( \frac{4}{3}\pi (r_{O^{2-})^{3}} \right )](https://tex.z-dn.net/?f=V_%7BS%7D%3D36%5Cleft%20%28%20%5Cfrac%7B4%7D%7B3%7D%5Cpi%20%28r_%7BCr%5E%7B3%2B%7D%29%5E%7B3%7D%7D%20%5Cright%20%29%2B54%5Cleft%20%28%20%5Cfrac%7B4%7D%7B3%7D%5Cpi%20%28r_%7BO%5E%7B2-%7D%29%5E%7B3%7D%7D%20%5Cright%20%29)
Putting the values,
![V_{S}=36\left ( \frac{4}{3}(3.14) (6.2\times 10^{-9} cm)^{3}} \right )+54\left ( \frac{4}{3}(3.14) (1.4\times 10^{-8} cm)^{3}} \right )=6.6\times 10^{-22}\times 10^{-8}cm^{3}](https://tex.z-dn.net/?f=V_%7BS%7D%3D36%5Cleft%20%28%20%5Cfrac%7B4%7D%7B3%7D%283.14%29%20%286.2%5Ctimes%2010%5E%7B-9%7D%20cm%29%5E%7B3%7D%7D%20%5Cright%20%29%2B54%5Cleft%20%28%20%5Cfrac%7B4%7D%7B3%7D%283.14%29%20%281.4%5Ctimes%2010%5E%7B-8%7D%20cm%29%5E%7B3%7D%7D%20%5Cright%20%29%3D6.6%5Ctimes%2010%5E%7B-22%7D%5Ctimes%2010%5E%7B-8%7Dcm%5E%7B3%7D)
The atomic packing factor is ratio of volume of sphere and volume of crystal, thus,
![packing factor=\frac{V_{S}}{V_{C}}=\frac{6.6\times 10^{-22}cm^{3}}{8.7\times 10^{-22}cm^{3}}=0.758](https://tex.z-dn.net/?f=packing%20factor%3D%5Cfrac%7BV_%7BS%7D%7D%7BV_%7BC%7D%7D%3D%5Cfrac%7B6.6%5Ctimes%2010%5E%7B-22%7Dcm%5E%7B3%7D%7D%7B8.7%5Ctimes%2010%5E%7B-22%7Dcm%5E%7B3%7D%7D%3D0.758)
Thus, atomic packing factor is 0.758.