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 1 of 4: Moles of CO2 released

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n(\ce{CO2}) = \dfrac{pV}{RT} = \dfrac{101325\times 3.8\times 10^{-3}}{8.314\times 773} = 0.060\ \text{mol}

Analysis

The gas is collected hot, so the ideal gas law at $T = 500$ °C $= 773$ K gives $n(\ce{CO2}) = 0.060$ mol, corresponding to $0.060/6 = 0.010$ mol of $\ce{Z3O4}$ formed.

Common pitfall. Use T = 773 K (the 500 °C decomposition temperature), not 298 K, when applying pV = nRT.