Question

NaF has the same structure as \(\mathrm{NaCl}\). (a) Use ionic radii from Chapter 7 to estimate the length of the unit cell edge for NaF. (b) Use the unit cell size calculated in part (a) to estimate the density of NaF.

Short answer

The edge length of the NaF unit cell is found to be 470 pm. Using this value and the known molar mass of NaF (41.99 g/mol), the estimated density of NaF is calculated to be 2.68 g/cm³.

Step-by-step solution

Collecting Required Data

To estimate the edge length of the NaF unit cell, we need to know the ionic radii of both Na+ and F- ions. From Chapter 7, the ionic radii of Na+ and F- are: Ionic radius of Na+ = 102 pm Ionic radius of F- = 133 pm

Calculating the Edge Length of the NaF Unit Cell

NaCl has a face-centered cubic crystal structure, so the NaF unit cell will have the same structure. In this Fcc structure, the edge length is equal to the sum of ionic radii of both ions multiplied by a factor of 2, because it is the distance between two opposite vertices along the face. Edge length (a) = 2 * (Ionic radius of Na+ + Ionic radius of F-) Using the values of ionic radii from step 1: a = 2 * (102 pm + 133 pm) a = 2 * (235 pm) a = 470 pm So, the edge length of the NaF unit cell is 470 pm.

Calculating the Mass of One Unit Cell

To find the mass of one NaF unit cell, we need to find the number of formula units per unit cell and then multiply it by the molar mass of NaF. In an Fcc structure, there are 4 formula units per unit cell. Molar mass of NaF = Molar mass of Na + Molar mass of F Molar mass of NaF = 22.99 g/mol + 19.00 g/mol Molar mass of NaF = 41.99 g/mol Mass of one unit cell = (Number of formula units per unit cell) * (Molar mass of NaF) / Avogadro's number Mass of one unit cell = (4 * 41.99 g/mol) / (6.022 x 10^23 mol^(-1)) Mass of one unit cell = 2.79 x 10^(-22) g

Calculating the Volume of One Unit Cell

The volume of one unit cell is given by the cube of edge length. Volume of one unit cell (V) = a^3 Converting edge length from pm to cm: a = 470 pm = 4.70 x 10^(-8) cm V = (4.70 x 10^(-8) cm)^3 V = 1.04 x 10^(-22) cm^3

Calculating the Density of NaF

The density of NaF can be calculated by dividing the mass of one unit cell by the volume of one unit cell. Density (ρ) = Mass of one unit cell / Volume of one unit cell ρ = (2.79 x 10^(-22) g) / (1.04 x 10^(-22) cm^3) ρ = 2.68 g/cm^3 So, the estimated density of NaF is 2.68 g/cm³.

Key concepts

Ionic Radii Estimation

Understanding the estimation of ionic radii is crucial when examining the structural properties of ionic compounds. Ionic radii refer to the approximate size of an ion in a crystal lattice. It is measured from the center of an ion to its outermost electron cloud. The size may vary depending on the ion's charge and the surrounding ions.

For instance, in the provided exercise, the radii of Na+ and F- ions are used to estimate the edge length of the NaF unit cell. Since we're dealing with a face-centered cubic (Fcc) crystal structure, similar to NaCl, the distance across the face of the cube, or the edge length, is simply twice the sum of the ionic radii of the cations and anions that line this face.

To enhance understanding, it's essential to remember that the ionic radius can change based on the crystal environment the ion is in, as electron clouds can be pulled closer or pushed away by neighboring ions. A visual representation or a 3D model can often aid in grasping how ionic radii determine the spacing in a crystal lattice.

Crystal Structure Analysis

Analyzing the crystal structure of an ionic compound like NaF involves looking at the arrangement of ions in the solid state. The crystal structure is what gives a material its physical properties, such as brittleness and melting point. NaF, as highlighted in the original exercise, adopts a Fcc structure where each ion is octahedrally coordinated; that means each Na+ is surrounded by six F- ions and each F- by six Na+ ions.

Looking deeper, the structure can be envisioned as layers of ions stacking upon each other, with each sodium ion sitting in the gap between fluoride ions, and vice versa. In the Fcc lattice, the unit cell—a repeating unit that constructs the entire crystal—is defined by the edge length calculated from the ionic radii. It's beneficial to study models or diagrams of Fcc lattices to fully appreciate how ions pack together to minimize repulsive forces and maximize attraction.

Density Calculation

The density of a substance is a measure of how much mass is contained within a unit volume. In the context of crystallography, the density tells us how tightly the ions are packed within the crystal lattice. The exercise solution applied the concept of density calculation to NaF by first determining the mass of a single unit cell, taking into account the number of formula units within that cell and the molar mass of the compound.

Next, the volume of the unit cell is found by cubing the edge length—directly related to the ionic radii. The density is then simply the mass divided by the volume. Remember, this calculated density is an estimate, as real-world factors may cause deviations. To aid comprehension, it helps to visualize this by considering that the greater the mass in a given volume, or the smaller the volume for a given mass, the higher the density of the material. This can lead to discussions on the material's properties, as denser materials generally have different mechanical and thermal behaviors.