A compound formed by elements `X` and `Y` has a cubic structure in which `X` atoms are at the face centres. `Y` atoms are present at the body center and also at the alternate edge center of the cube.
a. Calculate: (i) `Z_(eff')` (ii) total number of atoms in a cube, and (iii) formula of the compound.
b. If all the atoms are removed from a single axis passing through the centre of the cube, calculate
(i) `Z_(eff')` (ii) total number of atoms in a cube, and (iii) formula of the compound.

Explanation

a. i. Number of effective `X` atoms
`= 7` corners ` xx (1)/(8)` per corner atom share
`= (7)/(8)` atoms/unit cell
Number of effective `Y` atoms `= 6` faces `xx (1)/(2)` per face atoms share
`= 3` atoms/units cell
`Z_(eff(X + Y)) = (7)/(8) + (3)/(1) = (31)/(8)`
ii. Total atoms in a cube `= 7 ("corners") + 6` faces
`= 13` atoms units cube
iii. Formula of the compound:
`Z_(eff(X)) = (7)/(8), Z_(eff(Y)) = 3`
Thus, formula of the compound is `X_((7)/(8)) Y_(3)`.
Simplifying: `X_(7) Y_(24)`.
b. i. Atoms removed from one face of the cube, containing all atoms at corners `= 4` corners `+ 1` face-centred atom
`= 5` atoms/units cell
Corner atom left `= 7 - 4 = 3` atoms at corner
Face-centre atom left `= 6 - 1 = 5` atoms at face centre.
`Z_(eff(X)) = 3` corners ` xx (1)/(8)` per corner atom share
`= (3)/(8)` atoms/unit cell
`= Z_(eff(Y)) = 5` face-centred atom `xx (1)/(2)` per face atom share
`=(5)/(2)` atoms/unit cell
ii. Total atoms in a cube `= 3("corner") + 5` (faces)
`= 8` atoms/unit cube
iii. Formula of compound:
`Z_(eff(X)) = (3)/(8), Z_(eff(Y)) = (5)/(2)`.
Formula: `X_((3)/(8)) Y_(5)/(2)`
Simplifying `= X_(3) Y_(20)`