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?

Step by step solution

01

Determine the Number of Oxygen Ions

The info given says there is an oxygen ion at the center of each face. For a unit cell (cube), there are six faces. But since each face is shared by two unit cells (neighboring cubes), only half of each oxygen ion belongs to a single unit cell. So the total number of oxygen ions per unit cell is \(6 * 0.5 = 3\).

02

Determine the Number of Lead Ions

The info given says there is a lead ion at each corner of the cell, and a single cube (unit cell) has eight corners. Each corner is shared by eight unit cells, so only \(1/8\) of each lead ion belongs to one unit cell. Therefore, the total number of lead ions per unit cell is \(8 * 1/8 = 1\).

03

Determine the Number of Titanium Ions

The info given states that the titanium ion sits in the middle of the unit cell. Therefore, there is just one titanium ion in the unit cell.

04

Calculating the Density of the Unit Cell

Density is calculated by dividing mass by volume. Here, the mass of a unit cell can be obtained from the mass of all the atoms in it using their atomic masses. The volume of the unit cell is given by multiplying the dimensions of the cell in meters. This will give us a density in \(\mathrm{kg/m^3}\). The volume of the cell is then \(0.403 \times 10^{-9} \times 0.398 \times 10^{-9} \times 0.398 \times 10^{-9}\) m3. Averaging the atomic masses, considering some Ti(IV) is replaced by Zr(IV), and multiplying by the avogadro number to convert to Kg, the mass of the unit cell is: \(1(207.2) + 3(16) + 1(47.867+91.224)/2 \times 10^{-26}\) Kg. The density is then the mass divided by the volume.

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