Magnesium oxide has the rock salt crystal structure and a density of \(3.58 \mathrm{~g} / \mathrm{cm}^{3}\). (a) Determine the unit cell edge length. (b) How does this result compare with the edge length as determined from the radii in Table \(12.3\), assuming that the \(\mathrm{Mg}^{2+}\) and \(\mathrm{O}^{2-}\) ions just touch each other along the edges?

In a rock salt structure, one unit cell consists of 4 complete MgO formula units. This is because the structure can be seen as a face-centered cubic (fcc) arrangement of Mg²⁺ ions, with O²⁻ ions occupying the tetrahedral holes and vice versa.

We need to determine the mass of one unit cell. Since there are 4 formula units of MgO, we can compute the mass of one unit cell by multiplying the number of formula units (4) by the molar mass of MgO and dividing by Avogadro's number. Mass of one unit cell = (4 * (24.31 g/mol (Mg) + 16.00 g/mol (O))) / (6.022 x 10²³) Mass of one unit cell = 6.656 x 10⁻²³ g

Now, we can calculate the volume of one unit cell using the given density and the mass of the unit cell. Density = mass/volume Volume = mass/density Volume = (6.656 x 10⁻²³ g) / (3.58 g/cm³) Volume = 1.859 x 10⁻²³ cm³

In a cubic unit cell, the volume can be expressed as the cube of the edge length (a): Volume = a³ So we can calculate the edge length: a = (1.859 x 10⁻²³ cm³)^(1/3) a = 2.224 x 10⁻⁸ cm This is the unit cell edge length for the magnesium oxide crystal.

Now, we compare this calculated edge length with the one determined from the ionic radii provided in the table. Let's denote magnesium ion radius with "r(Mg²⁺)" and oxide ion radius with "r(O²⁻)". In the rock salt structure, the ions are touching each other along the edges, so the edge length (a) can be given as: a = 2[r(Mg²⁺) + r(O²⁻)] Now we need to refer to Table 12.3 to find the ionic radii: r(Mg²⁺) = 0.65 Å r(O²⁻) = 1.40 Å (Note: You may need to look in your specific textbook or another source for these values, as this answer assumes you have access to the correct radii table.) We need to convert these to cm: r(Mg²⁺) = 0.65 x 10⁻⁸ cm r(O²⁻) = 1.40 x 10⁻⁸ cm Calculate the edge length based on these radii: a = 2 x (0.65 x 10⁻⁸ cm + 1.40 x 10⁻⁸ cm) a = 2.210 x 10⁻⁸ cm Now, we compare the calculated and given edge lengths: Calculated edge length: 2.224 x 10⁻⁸ cm Edge length based on ionic radii: 2.210 x 10⁻⁸ cm The calculated edge length is very close to the edge length based on the ionic radii. This confirms that the magnesium and oxide ions are indeed touching each other along the edges in the magnesium oxide crystal with the rock salt structure.