A crystalline solid contains three types of ions, \(\mathrm{Na}^{+}, \mathrm{O}^{2-},\) and \(\mathrm{Cl}^{-}\). The solid is made up of cubic unit cells that have \(\mathrm{O}^{2-}\) ions at each corner, \(\mathrm{Na}^{+}\) ions at the center of each face, and \(\mathrm{Cl}^{-}\) ions at the center of the cells. What is the chemical formula of the compound? What are the coordination numbers for the \(\mathrm{O}^{2-}\) and \(\mathrm{Cl}^{-}\) ions? If the length of one edge of the unit cell is \(a,\) what is the shortest distance from the center of a \(\mathrm{Na}^{+}\) ion to the center of an \(\mathrm{O}^{2-}\) ion? Similarly, what is the shortest distance from the center of a \(\mathrm{Cl}^{-}\) ion to the center of an \(\mathrm{O}^{2-}\) ion?

Step by step solution

01

Identify the Ions at Different Positions

Each corner of the unit cell contains an \(O^{2-}\) ion, there is a total of 8 corners in a cubic cell, but since each corner ion is shared by 8 cells, we have totally 1 \(O^{2-}\) ion in a unit cell. Each face of the unit cell contains a \(Na^{+}\) ion, and there are a total of 6 faces, since each face ion is shared by 2 cells, we have totally 3 \(Na^{+}\) ions in a unit cell. The unit cell contains one \(Cl^{-}\) ion at the center which is completely within the cell. Therefore the chemical formula of the compound is \(Na_3OCl\).

02

Determine the Coordination Numbers

The \(Cl^{-}\) in the center is surrounded by \(Na^{+}\) on each face of the cube so coordination number for \(Cl^{-}\) is 6. For the \(O^{2-}\) ions, they are each in the corner of the cube surrounded by four \(Cl^{-}\) in the center of cubes. Hence, the coordination number for \(O^{2-}\) ions is 4.

03

Calculate the Distances between Ions

The shortest distance from the center of a \(Na^{+}\) ion to the center of an \(O^{2-}\) ion is the distance from the center of a face of the cube to one of its corners. This can be calculated using the Pythagorean theorem as one-half of the edge length, or \(a/2\). To calculate the shortest distance from the center of a \(Cl^{-}\) ion to the center of an \(O^{2-}\) ion, we consider the distance from the center of the cube to one of its corners. This is equal to the diagonal of the cube and can be calculated as \(\sqrt{3}/2 * a\).

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