If the edge length of an NaH unit cell is 488 pm, what is the length in picometers of an Na¬H bond? (See Problem 12.50.)

Verified step by step guidance

1. Identify the type of crystal structure for NaH. Sodium hydride (NaH) typically crystallizes in the face-centered cubic (fcc) structure.
2. Understand the relationship between the edge length of the unit cell and the atomic positions in an fcc lattice. In an fcc lattice, atoms are located at each corner and the centers of all the faces of the cube.
3. Determine the position of Na and H in the unit cell. In NaH, Na and H atoms alternate positions at the corners and face centers of the cube.
4. Calculate the distance between a corner atom and a face-centered atom along the face diagonal. This distance represents half the face diagonal, since the face-centered atom is midway along the diagonal.
5. Use the Pythagorean theorem to find half the face diagonal: half the face diagonal = \(\sqrt{\left(\frac{a}{2}\right)^2 + \left(\frac{a}{2}\right)^2}\), where \(a\) is the edge length of the unit cell. This distance is the length of the Na-H bond.

Key Concepts

Unit Cell Structure

A unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. In ionic compounds like NaH, the unit cell can be cubic, and its edge length is crucial for determining the distances between ions within the lattice. Understanding the geometry of the unit cell helps in calculating bond lengths and other structural properties.

Ionic Bonding

Ionic bonding occurs when atoms transfer electrons, resulting in the formation of positively and negatively charged ions. In NaH, sodium (Na) donates an electron to hydrogen (H), creating Na⁺ and H⁻ ions. The bond length in ionic compounds is influenced by the size of the ions and their arrangement in the crystal lattice, which is essential for calculating the Na-H bond length.

Bond Length Calculation

Bond length is the distance between the nuclei of two bonded atoms, typically measured in picometers (pm). In a simple cubic unit cell, the bond length can often be derived from the edge length of the unit cell, as it relates to the distance between ions. For NaH, the bond length can be approximated as half the edge length of the unit cell, assuming a direct relationship between the unit cell dimensions and the ionic distances.