Calcium metal crystallises in a face centered cubic lattice with edge length of 0.556nm. Calculate the density of the metal. [Atomic mass of calcium 40 g/mol]
`[N_(A) = 6.022 xx 10^(23) " atoms/ mol"]`

Explanation

(a) Let a be the uit cell edge length and be the face diagonal
In `Delta ABC, " "AC^(2) = BC^(2) + AB^(2)`
`b^(2) = a^(2) + a^(2) = 2a^(2)`
`b = sqrt(2)a`
if r is the radius of the sphere, we find
`b = 4r = sqrt(2)a` or `a = (4r)/(sqrt(2)) = 2sqrt(2)r`
f.c.c. structure has 4 spheres per unit cell and their volume is `4 xx (4)/(3) pi r^(3)` and the volume of the cube is `a^(3)` or `(2sqrt(2)r)^(3)`
Therefore, Packing efficiency `= ("Volume occupied by four spheres in the unit cell")/("Total volume of the unit cell") xx 20`
`= (4 xx (4)/(3)pi r^(3))/((2sqrt(2)r)^(3)) xx 100% = ((16)/(3) pi r^(3))/(16sqrt(2)r^(3)) xx 100% = 74%`
(b) `d = (z xx M)/(a^(3) xx N_(A))`
`d = (4 xx 40)/((0.555 xx 10^(-7))^(2)(6.022 xx 10^(23)))`
`= 1.545 g//cm^(3)` or `1.545 xx 10^(3) kg//m^(3)`.