A compound is made up of three elements X, Y and Z. X atoms are present at each corner of the unit cell, Y atoms are present on each edge and Z atoms are present in the centre of each face. The formula of the compound is:

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  1. XY6Z3
  2. XY3Z6
  3. XY3Z3
  4. XYZ

Answer (Detailed Solution Below)

Option 3 : XY3Z3
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Detailed Solution

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Concept:

  • Atoms at the edge of a unit cell:
    • In a crystal lattice, an edge of a unit cell is shared by four other unit cells.
    • One is the adjacent unit cell and the other two are on the layer just above the unit cell.
    • So each edge contributes to one-fourth of an atom in a unit cell.
    • There are 12 edges per unit cell, thus a total number of atoms becomes 12 × 1/4 = 3.

  • Atoms at the corners of a unit cell:
    • The corner atoms are shared via 8 cubes, so the contribution from each corner is 1/8.
    • Total number of atoms on the corner of a cube = 8
    • The total number of atoms from corners is 8 × 1/8 = 1.
  • Atoms at the face of a unit cell:
    • ​Each face is shared by two unit cells.
    • the contribution from each face is thus 1/2.
    • There are six faces in a cube, so the total contribution from the faces of the cube = 3

Calculation:

  • The atoms present at the corners are X.
  • Contribution from all the corners = 1.

So, the total number of X atoms = 1.

  • Y atoms are present at the edges.
  • Total number of atoms at edges = 3,

So the number of Y atoms = 3.

  • The total contribution from all 6 faces is 3.
  • Z atoms are present at the faces.

So total number of Z atoms = 3.

Hence, the formula of the compound is XY3Z3.

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