Question

If the unit cell of a mineral has cubic packed (c.c.p) array of oxygen atoms with m fraction of octahedral holes occupied by aluminium ions and n fraction of tetrahedral holes occupied by magnesium ions, m and n respectively are:
(A) $\dfrac{1}{2}$, $\dfrac{1}{8}$
(B) 1, $\dfrac{1}{4}$
(C) $\dfrac{1}{2}$, $\dfrac{1}{2}$
(D) $\dfrac{1}{4}$, $\dfrac{1}{8}$

Answer 1

Hint: Recollect the concept of packing in crystal lattices. Think about the number of octahedral voids occupied by the atoms forming the CCP array. The number of tetrahedral voids is twice the number of octahedral voids. Solve and reduce algebraic expressions and then input the values in options and accordingly determine the answer.

Complete answer:
- In a CCP lattice, the number of octahedral holes is equal to the number of oxygen atoms and number of tetrahedral holes is equal to twice the number of anions.
- The number of oxygen atoms present in the lattice is 4.
- Therefore, the total number of octahedral holes is 4.
- The total number of tetrahedral holes is 8.
- Aluminium ions, $A{{l}^{3+}}$ occupy ‘m’ fraction of the octahedral holes and therefore, the number of aluminium ions is $4\times m=4m$.
- Magnesium ions, $M{{g}^{2+}}$ occupy ‘n’ fraction of the tetrahedral holes and therefore, the number of magnesium ions is $8\times n=8n$.
- A mineral of the metal will exist in a neutral state. Aluminium ion has +3 oxidation state and Magnesium ion has +2 oxidation state. Oxygen atoms will be present as oxides, so they will have -2 oxidation state. Therefore,
\[4\left( -2 \right)+4m(+3)+8n\left( +2 \right)=0\]
\[12m+16n=8\]
\[3m+4n=2\]……. (1)
- Now if we substitute the value of $m=\dfrac{1}{2}$ and $n=\dfrac{1}{8}$ in equation (1) then the equation is balanced.
R.H.S=2
L.H.S=$3\left( \dfrac{1}{2} \right)+4\left( \dfrac{1}{8} \right)=\dfrac{3}{2}+\dfrac{4}{8}=\dfrac{3}{2}+\dfrac{1}{2}=\dfrac{4}{2}=2$
- Therefore, the value of m and n respectively is $\dfrac{1}{2}$, $\dfrac{1}{8}$.
- Therefore, aluminium ions occupy half of octahedral holes and magnesium ions occupy one-eighth of tetrahedral holes.

Therefore, the correct answer is option (A).

Note:
Remember the total number of octahedral voids will be equal to the number of anions forming the cubic close packed array. The total number of tetrahedral voids will be twice the number of octahedral voids. A mineral will always exist in a stable neutral state.