Problem 2

Copper(I) oxide

\ce{Cu2O}

is a semiconductor used in solid-state electronics and solar cells. The cubic unit cell has lattice constant

a = 427.0

pm, with oxygen atoms (A) forming a body-centered cubic sub-lattice and copper atoms (B) occupying positions forming a face-centered cubic sub-lattice. (a) What are the coordination numbers of Cu and O? (b) Calculate the shortest

\ce{O-O}

\ce{Cu-O}

, and

\ce{Cu-Cu}

distances. (c) Calculate the theoretical density of stoichiometric

\ce{Cu2O}

in g cm

^{-3}

. (d) A non-stoichiometric sample has

0.2\%

of all Cu atoms oxidized to

\ce{Cu^2+}

. What percentage of Cu sites are vacant, and what is

x

\ce{Cu_{2-x}O}

? (e) Write balanced equations for the reaction of

\ce{Cu2O}

with: (1) atmospheric

\ce{O2}

in humid air; (2) dilute sulfuric acid; (3) hot concentrated sulfuric acid.

Step 4 of 5: Vacancy concentration and non-stoichiometry

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998\,\ce{Cu+} + 2\,\ce{Cu^2+} \Rightarrow \text{charge} = 1002+ = 501\,\ce{O^2-} \Rightarrow \text{vacancies} = \dfrac{2}{1002} \approx 0.20\%, \quad x = 2 \times 0.002 = 0.004

Analysis

For 1000 Cu atoms (998 $\ce{Cu+}$ , 2 $\ce{Cu^2+}$ ), the total positive charge is 1002. This balances 501 $\ce{O^2-}$ anions, which occupy 501 sites and require $2\times 501 = 1002$ Cu sites. Thus 2 out of 1002 Cu sites are vacant ( $0.20\%$ ). The formula is $\ce{Cu_{1000/501}O} = \ce{Cu_{1.996}O}$ , so $x = 0.004$ .

Common pitfall. Each

\ce{Cu^2+}

substitution replaces two

\ce{Cu+}

ions with one

\ce{Cu^2+}

to maintain neutrality, leaving one vacancy — do not assume the vacancy percentage is zero.