Question

Sodium oxide \(\left(\mathrm{Na}_{2} \mathrm{O}\right)\) adopts a cubic structure with \(\mathrm{Na}\) atoms represented by green spheres and \(\mathrm{O}\) atoms by red spheres. (a) How many atoms of each type are there in the unit cell? (b) Determine the coordination number and describe the shape of the coordination environment for the sodium ion. (c) The unit cell edge length is \(555 \mathrm{pm}\). Determine the density of \(\mathrm{Na}_{2} \mathrm{O}\)

Short answer

(a) In the unit cell, there are 4 Sodium (Na) atoms and 3 Oxygen (O) atoms. (b) The coordination number for both Sodium (Na) and Oxygen (O) ions is 6, and the shape of the coordination environment for the sodium ion is octahedral. (c) The density of \(\mathrm{Na}_{2} \mathrm{O}\) is 3.498 g/cm³.

Step-by-step solution

(a) Number of each type of atom in the unit cell

For a unit cell, we can determine the number of Na and O atoms based on its cubic structure. Sodium oxide (Na2O) has a rock-salt structure with Sodium ion (Na+) at FCC lattice points and Oxygen ion (O-) at the face center of the cube. In the face-centered cubic (FCC) structure, there are: 1. 8 atoms per unit cell at each corner (sharing with other 8 neighbouring cubes) contributing 1/8 per unit cell. 2. 6 atoms per unit cell at each face center (sharing with one other neighbouring cube) contributing 1/2 per unit cell. So the number of Sodium (Na) atoms in the unit cell is: 8 x (1/8) + 6 x (1/2) = 1 + 3 = 4 Each Oxygen (O) atom occupies the center of each unit cell face. Since there are 6 faces/cube and half of each oxygen atom is inside and half outside the unit cell, we get the number of Oxygen (O) atoms in the unit cell: 6 x (1/2) = 3 Thus, there are 4 Sodium (Na) atoms and 3 Oxygen (O) atoms in the unit cell.

(b) Coordination number and shape of the coordination environment

The coordination number is defined as the number of closest opposite charged ions around an ion of interest. Here, we will investigate the sodium ion's environment and its coordination number. For Sodium ion (Na+), the nearest neighbors are the Oxygen ions (O-) present at the face centers. As we have 6 face-centers containing Oxygen ions, each Sodium ion is surrounded by 6 Oxygen ions. Therefore, the coordination number of Sodium ion (Na+) is 6. Similarly, for each Oxygen ion (O-), the nearest neighbors are the Sodium ions (Na+) present at the corners and face-centers. So, every Oxygen ion has 6 nearest Sodium ions. Hence, the coordination number of Oxygen ion (O-) is also 6. In Sodium oxide (Na2O), both Sodium (Na) and Oxygen (O) ions have a coordination number of 6. The shape of the coordination environment for the sodium ion is octahedral.

(c) Density of Na2O

Now it's time to calculate the density of Sodium oxide (Na2O), considering the unit cell edge length is given as 555 pm. First, let's find the molar mass of \(\mathrm{Na}_{2} \mathrm{O}\): Molar mass of Na = 23 g/mol Molar mass of O = 16 g/mol Molar mass of \(\mathrm{Na}_{2} \mathrm{O}\) = 2(23) + 16 = 46 + 16 = 62 g/mol The volume of the unit cell can be calculated using the given edge length (a) of the cubic cell as: Volume = a³ where a = 555 pm (converting to cm by multiplying with 10⁻¹⁰) Volume = (555 x 10⁻¹⁰)³ = 1.707 x 10⁻²³ cm³ Now we can calculate the number of formula units per unit cell: In a unit cell, there are 4 Na and 3 O atoms, so the number of formula units in a unit cell is: Na₂O = 4:3 or 4 molecules on Na and 3 molecules of O in a ratio of 2:1 Hence, there are 4 formula units of \(\mathrm{Na}_{2} \mathrm{O}\) per unit cell. Now, we can calculate the density of Na2O using molar mass, volume, and Avogadro's number (6.022 x 10²³): Density = (molar mass x number of formula units) / (volume x Avogadro's number) Density = (62 g/mol x 4) / (1.707 x 10⁻²³ cm³ x 6.022 x 10²³ mol⁻¹) Density = 3.498 g/cm³ Hence, the density of \(\mathrm{Na}_{2} \mathrm{O}\) is 3.498 g/cm³.

Key concepts

Coordination Number

In the context of ionic solids like sodium oxide (\(\text{Na}_2\text{O}\)), the coordination number refers to how many ions of opposite charge neighbor a given ion. It is a critical factor in predicting the geometric arrangement in the crystal structure. For sodium oxide, both the sodium ions (\(\text{Na}^+\)) and oxygen ions (\(\text{O}^{2-}\)) have a coordination number of six. This means each sodium ion is closely surrounded by six oxygen ions, and vice versa. This specific arrangement maximizes the attractive interactions between oppositely charged ions to minimize energy, resulting in greater stability of the crystal structure. Understanding coordination numbers helps us grasp the complexity and rationale behind the formation of various ionic compounds, ultimately influencing their chemical and physical properties.

Rock-Salt Structure

The term "rock-salt structure" reflects the famous mineral rock salt (or \(\text{NaCl}\)), which shares the same type of crystal organization as sodium oxide (\(\text{Na}_2\text{O}\)). This structure involves a face-centered cubic (FCC) lattice where the sodium ions are positioned at each corner and face-center of the cube. This arrangement results in high packing efficiency minimizing space within the crystal system.

A characteristic feature of the rock-salt structure is that the same pattern repeats throughout, showing translational symmetry. In the case of sodium oxide, the rock-salt structure conditions ensure each sodium ion at the cube’s face centers is equally surrounded by oxygen ions, yielding a uniform, predictable pattern. This predictability is crucial in fields like materials science, where specific physical properties are desired for practical applications such as electronic devices.

Octahedral Coordination

The octahedral coordination is a geometric arrangement found in various ionic compounds, including sodium oxide (\(\text{Na}_2\text{O}\)). In this formation, each ion is surrounded by six nearest neighbor ions, aligned along the vertices of an octahedron. This spatial configuration stems naturally from the rock-salt structure.

• The sodium ions (\(\text{Na}^+\)) are centrally located within an octahedron formed by coordinating oxygen ions (\(\text{O}^{2-}\)).
• Similarly, the oxygen ions are nestled within an octahedral cluster of sodium ions.

The octahedral coordination plays a crucial role in determining the packing of ions within the crystal lattice. By ensuring maximum packing efficiency, it instills stability into the structure while maintaining equally short and uniform bond lengths. For students exploring crystalline structures, recognizing these geometric patterns provides valuable insight into the underlying principles governing solid-state chemistry.