In some barbecue grills the electric lighter consists of a small hammer-like
device striking a small crystal, which generates voltage and causes a spark
between wires that are attached to opposite surfaces of the crystal. The
phenomenon of causing an electric potential through mechanical stress is known
as the piezoelectric effect. One type of crystal that exhibits the
piezoelectric effect is lead zirconate titanate. In this perovskite crystal
structure, a titanium(IV) ion sits in the middle of a tetragonal unit cell
with dimensions of \(0.403 \mathrm{nm} \times 0.398 \mathrm{nm} \times 0.398
\mathrm{nm} .\) At each corner is a
lead(II) ion, and at the center of each face is an oxygen anion. Some of the
Ti(IV) are replaced by Zr(IV). This substitution, along with
\(\mathrm{Pb}(\mathrm{II}),\) results in the piezoelectic behavior.
(a) How many oxygen ions are in the unit cell?
(b) How many lead(II) ions are in the unit cell?
(c) How many titanium(IV) ions are in the unit cell?
(d) What is the density of the unit cell?