Answer :
(a) The volume of unit cell is, ![9.91\times 10^{-23}cm^3/\text{ unit cell}](https://tex.z-dn.net/?f=9.91%5Ctimes%2010%5E%7B-23%7Dcm%5E3%2F%5Ctext%7B%20unit%20cell%7D)
(b) The theoretical density of Ti is, ![4.81g/cm^3](https://tex.z-dn.net/?f=4.81g%2Fcm%5E3)
Explanation :
(a) First we have to calculate the volume of the unit cell.
Formula used :
![V=6r^2c\sqrt {3}](https://tex.z-dn.net/?f=V%3D6r%5E2c%5Csqrt%20%7B3%7D)
where,
V = volume of unit cell = ?
r = atomic radius = ![0.1445nm=1.445\times 10^{-8}cm](https://tex.z-dn.net/?f=0.1445nm%3D1.445%5Ctimes%2010%5E%7B-8%7Dcm)
conversion used : ![(1nm=10^{-7}cm)](https://tex.z-dn.net/?f=%281nm%3D10%5E%7B-7%7Dcm%29)
Ratio of lattice parameter = c : a = 1.58 : 1
So, c = 1.58 a
And, a = 2r
c = 1.58 × 2r
Now put all the given values in this formula, we get:
![V=6\times r^2\times (1.58\times 2r)\sqrt {3}](https://tex.z-dn.net/?f=V%3D6%5Ctimes%20r%5E2%5Ctimes%20%281.58%5Ctimes%202r%29%5Csqrt%20%7B3%7D)
![V=6\times r^3\times (1.58\times 2)\sqrt {3}](https://tex.z-dn.net/?f=V%3D6%5Ctimes%20r%5E3%5Ctimes%20%281.58%5Ctimes%202%29%5Csqrt%20%7B3%7D)
![V=6\times (1.445\times 10^{-8}cm)^3\times (1.58\times 2)\sqrt {3}](https://tex.z-dn.net/?f=V%3D6%5Ctimes%20%281.445%5Ctimes%2010%5E%7B-8%7Dcm%29%5E3%5Ctimes%20%281.58%5Ctimes%202%29%5Csqrt%20%7B3%7D)
![V=9.91\times 10^{-23}cm^3/\text{ unit cell}](https://tex.z-dn.net/?f=V%3D9.91%5Ctimes%2010%5E%7B-23%7Dcm%5E3%2F%5Ctext%7B%20unit%20cell%7D)
The volume of unit cell is, ![9.91\times 10^{-23}cm^3/\text{ unit cell}](https://tex.z-dn.net/?f=9.91%5Ctimes%2010%5E%7B-23%7Dcm%5E3%2F%5Ctext%7B%20unit%20cell%7D)
(b) Now we have to calculate the density of Ti.
Formula used for density :
![\rho=\frac{Z\times M}{N_{A}\times a^{3}}](https://tex.z-dn.net/?f=%5Crho%3D%5Cfrac%7BZ%5Ctimes%20M%7D%7BN_%7BA%7D%5Ctimes%20a%5E%7B3%7D%7D)
..........(1)
where,
= density of Ti = ?
Z = number of atom in unit cell = 6 atoms/unit cell (for HCP)
M = atomic mass = 47.87 g/mol
= Avogadro's number = ![6.022\times 10^{23}atoms/mole](https://tex.z-dn.net/?f=6.022%5Ctimes%2010%5E%7B23%7Datoms%2Fmole)
= volume of unit cell = ![9.91\times 10^{-23}cm^3/\text{ unit cell}](https://tex.z-dn.net/?f=9.91%5Ctimes%2010%5E%7B-23%7Dcm%5E3%2F%5Ctext%7B%20unit%20cell%7D)
Now put all the values in above formula (1), we get:
![\rho=\frac{6\times 47.87}{(6.022\times 10^{23})\times (9.91\times 10^{-23})}](https://tex.z-dn.net/?f=%5Crho%3D%5Cfrac%7B6%5Ctimes%2047.87%7D%7B%286.022%5Ctimes%2010%5E%7B23%7D%29%5Ctimes%20%289.91%5Ctimes%2010%5E%7B-23%7D%29%7D)
![\rho=4.81g/cm^3](https://tex.z-dn.net/?f=%5Crho%3D4.81g%2Fcm%5E3)
The theoretical density of Ti is, ![4.81g/cm^3](https://tex.z-dn.net/?f=4.81g%2Fcm%5E3)