Question

\(\mathrm{NaF}\) 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 unit cell edge length of NaF is calculated to be \(4.70 \times 10^{-10}\,\mathrm{m}\) and the estimated density of NaF is \(2.526 \times 10^3\,\mathrm{kg/m^3}\).

Step-by-step solution

Determine the unit cell edge length

To calculate the unit cell edge length, we need to know the ionic radii of Na⁺ and F⁻. In the NaCl structure (also known as face-centered cubic or FCC), one ion is surrounded by six ions with the opposite charge, allowing us to assume that the unit cell edge length is equal to twice the sum of their ionic radii. Given the ionic radii of Na⁺ (102 pm) and F⁻ (133 pm), we can calculate the edge length by adding them together and multiplying by 2: Unit cell edge length = \(2\times(102\,\mathrm{pm} + 133\,\mathrm{pm})\)

Calculate the unit cell edge length

Now let's perform the calculation for the unit cell edge length: Unit cell edge length = \(2\times(102\,\mathrm{pm} + 133\,\mathrm{pm}) = 2\times(235\,\mathrm{pm}) = 470\,\mathrm{pm}\) So, the edge length of the NaF unit cell is 470 pm. As we need this edge length in meters for the next step, let's convert it: 470 pm = 470 x \(10^{-12}\) m = \(4.70 \times 10^{-10}\,\mathrm{m}\)

Calculate the unit cell volume

Next, we need to find the volume of the unit cell. The volume of a cubic unit cell is calculated by raising the edge length to the power of 3. Unit cell volume = \(edge\,length^3 = (4.70\times 10^{-10}\,\mathrm{m})^3\)

Calculate the unit cell volume

Perform the calculation for the unit cell volume: Unit cell volume = \((4.70\times 10^{-10}\,\mathrm{m})^3 = 1.039\times 10^{-28}\,\mathrm{m^3}\) So, the unit cell volume of NaF is \(1.039 \times 10^{-28}\,\mathrm{m^3}\).

Calculate the number of formula units per unit cell

In the NaCl structure, each corner ion contributes 1/8 to the unit cell, and each ion on the face contributes 1/2. Knowing that a face-centered cubic unit cell has 8 corner ions and 6 face ions, we can determine the number of formula units per FCC unit cell: Number of formula units = \(\frac{8}{8} + \frac{6}{2} = 1 + 3 = 4\) There are 4 formula units of NaF per unit cell.

Calculate the mass of formula units in the unit cell

To calculate the mass of formula units in the unit cell, we need to know the molar mass of sodium (22.99 g/mol) and fluorine (19.00 g/mol). The mass of one NaF formula unit can be calculated as follows: Mass of one NaF formula unit = (22.99 g/mol + 19.00 g/mol)

Calculate the mass of formula units in the unit cell

Perform the calculation for the mass of one NaF formula unit: Mass of one NaF formula unit = (22.99 g/mol + 19.00 g/mol) = 41.99 g/mol Now, let's find the mass of 4 NaF formula units in the unit cell: Mass of 4 formula units = 4 × 41.99 g/mol = 167.96 g/mol

Calculate the density of NaF

Finally, we can calculate the density of NaF using the mass of formula units in the unit cell, the unit cell volume, and Avogadro's number (\(6.022\times 10^{23}\,\mathrm{mol^{-1}}\)). Density = \(\frac{mass\,of\,formula\,units\,in\,unit\,cell\,\times\,Avogadro's\,number}{unit\,cell\,volume\,\times\,4}\)

Calculate the density of NaF

Perform the calculation for the density of NaF: Density = \(\frac{167.96\,\mathrm{g/mol}\,\times\,6.022\times 10^{23}\,\mathrm{mol^{-1}}}{1.039\times 10^{-28}\,\mathrm{m^3}\times 4} = 2.526\times 10^3\,\mathrm{kg/m^3}\) So, the estimated density of NaF is \(2.526 \times 10^3\,\mathrm{kg/m^3}\).

Key concepts

Ionic Radii

Ionic radii are critical when examining ionic compounds like NaF. They define the size of ions like Na⁺ and F⁻. The ionic radius is a measure of an ion's size, relevant in how ions fit together in a crystal structure.

For NaF, we use the ionic radii of sodium (\(\text{102 pm}\)) and fluorine (\(\text{133 pm}\)). These measurements are crucial in determining the dimensions of the unit cell. We calculate the sum of the ionic radii to get dimensions for the structure. In the face-centered cubic (FCC) arrangement, the ions' size directly influences the edge length of the unit cell.

Understanding ionic radii helps predict how substances combine to form crystal structures.

Unit Cell Calculation

The unit cell is the fundamental building block of a crystal structure. For NaF, calculating the unit cell begins by knowing the ionic radii, where the unit cell edge length is approximately twice the sum of the Na⁺ and F⁻ radii.

The formula is:

• Unit cell edge length = \(2 \times (\text{radii of } \text{Na}^+ + \text{radii of } \text{F}^-)\)
• Given: \(\text{Na}^+ = 102\,\text{pm}, \text{F}^- = 133\,\text{pm}\).
• Edge length = \(2 \times (102 + 133)\,\text{pm} = 470\,\text{pm}\) = \(4.70 \times 10^{-10}\,\text{m}\).

To find the volume, the edge length is cubed: \((4.70 \times 10^{-10} \text{m})^3 = 1.039 \times 10^{-28} \text{m}^3\).

This calculation is essential to determine other properties like density.

Density Estimation

Density estimation involves determining how much mass is contained within a given volume. In the context of NaF, once we know the unit cell volume and mass, we can estimate density.

Here's how it works:

• Calculate the mass of NaF using its molar mass: 41.99 g/mol per formula unit.
• Each unit cell in FCC structures contains four NaF formula units, so: \(4 \times 41.99 \text{ g/mol} = 167.96 \text{ g/mol}\).
• Convert mass to kg and multiply by Avogadro's number: \(m = \frac{167.96 \times 10^{-3}}{6.022 \times 10^{23}}\).
• Calculate density using: \(\text{Density} = \frac{m}{\text{Volume}}\)

For NaF, density is approximately \(2.526 \times 10^3 \text{kg/m}^3\), showing how tightly mass is packed in the crystal structure.