Question

Assuming the olivine series of minerals consists of crystal in which $F{e^{2 + }}$ and $M{g^{2 + }}$ ions may substitute for each other causing substitutional impurity defect without changing the volume of the unit cell. In the olivine series of minerals, oxide ion exits as fcc with $S{i^{4 + }}$occupying $\dfrac{1}{4}th$ of octahedral void and divalent ion occupying $\dfrac{1}{4}th$ of the tetrahedral void. The density of Forsterite (Magnesium silicate) is $3.21\dfrac{g}{{cc}}$ and that of Fayalite (Ferrous Silicate) is$4.34\dfrac{g}{{cc}}$. If the density of olivine is$3.88\dfrac{g}{{cc}}$, then which of the following statements is /are correct?
A. Olivine contains $40.71\%$Forsterite and $59.29\%$Fayalite.
B. Fayalite is $F{e_2}Si{O_4}$and Forsterite is$MgSi{O_4}$.
C. Percentage of Forsterite$MgSi{O_4}$is$59.29$.
D. Both options A and B.

Answer 1

Hint : Chemical composition of olivine falls somewhere close to pure Forsterite ($MgSi{O_4}$) and pure Fayalite($F{e_2}Si{O_4}$). Around there $Mg$and can elective uninhibitedly for one another in the mineral’s nuclear design-in any proportion, known as “solid arrangement”.

Complete step-by-step solution:
According to the question we get-
Number of ${O^{ - 2}}$atoms

$$= 8 \times \dfrac{1}{8} + 6 \times \dfrac{1}{2} \\\ = 1 + 3 \\\ = 4 \\\$$

Number of $S{i^{4 + }}$atoms $= \dfrac{1}{4} \times$octahedral void

$$= \dfrac{1}{4} \times 4 \\\ = 1 \\\$$

Number of $M{g^{2 + }}$ atoms $\dfrac{1}{4} \times$tetrahedral void

$$= \dfrac{1}{4} \times 8 \\\ = 2 \\\$$

We know the formula of Forsterite $= MgSi{O_4}$ and Fayallite$= F{e_2}Si{O_4}$
Let us assume the Forterite is $x\%$ and Fayalite is $(100 - x)\%$
Therefore,

$$\dfrac{{x \times 3.21 + (100 - x) \times 4.34}}{{100}} = 3.88 \\\ \therefore x = 40.71\% - Forsterite \\\ \\\$$

And the percentage of Fayalite $\left( {100 - 40.71} \right) = 59.29$.
Therefore, the right option is D.

Note: Olivine has additionally been utilized as a hard–headed material. It is utilized to make a headstrong block and utilized as projecting sand. It is the primary mineral found on the earth's surface.