Problem 2

Resistive gas sensors using semiconducting metal oxides (SMOX) detect tiny amounts of impurities. The mixed oxide X, which has the normal spinel structure

\mathrm{A^{2+}B^{3+}_2O^{2-}_4}

, is obtained by decomposing in air the crystalline hydrated metal oxalate

\ce{ZC2O4*\mathit{k}H2O}

of metal Z. When heated to 140 °C the hydrate loses 19.7 % of its mass; further heating in air at 500 °C yields 2.407 g of black X together with 3.8 dm

^3

\ce{CO2}

(at 101325 Pa, 500 °C). (a) Determine the formula of X and the value of

k

. (b) In the spinel structure the

\ce{O^{2-}}

ions form a face-centred cubic lattice;

\mathrm{A^{2+}}

cations occupy part of the tetrahedral voids and

\mathrm{B^{3+}}

cations part of the octahedral voids. Calculate the percentage of tetrahedral sites occupied in a normal spinel

\mathrm{AB_2O_4}

Step 4 of 4: Tetrahedral site occupancy

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\mathrm{occupancy} = \dfrac{1\ \mathrm{A}}{8\ \text{tetrahedral sites}} = 12.5\%

Analysis

An FCC lattice of 4 O atoms per unit cell contains 8 tetrahedral voids and 4 octahedral voids. A normal spinel $\mathrm{AB_2O_4}$ places its single A cation in a tetrahedral site, so $1/8 = 12.5\%$ of tetrahedral voids are filled (and $2/4 = 50\%$ of octahedral voids).