Question

Potassium chloride has the same crystal structure as NaCl. Careful measurement of the internuclear distance between \(K^{+}\) and \(C l^{-}\) ions gave a value of 314.54 pm. The density of \(\mathrm{KCl}\) is \(1.9893 \mathrm{g} / \mathrm{cm}^{3}\). Use these data to evaluate the Avogadro constant, \(N_{\mathrm{A}}\)

Short answer

The calculated Avogadro's number is \(6.022 \times 10^{23}\) mol^-1.

Step-by-step solution

Calculate the volume of the unit cell

The volume \(V\) of the unit cell can be calculated from the internuclear distance \(d\). For face-centered cubic structures, this is given by the formula \(V = d^3\). Substituting \(d = 314.54 \times 10^{-12}\) m into this formula, we can calculate \(V\) to be \(31.254 \times 10^{-28}\) m^3.

Calculate the mass present in the unit cell

The second step is to calculate the mass of potassium chloride (KCl) present in a unit cell according to the given density. The mass can be calculated from the density \(\rho\) and volume of the unit cell \(V\) using the formula \(mass = \rho \times V\). The given density of KCl is 1.9893 g/cm^3, so converting to kg/m^3 we find the density is 1989.3 kg/m^3. Substituting the value of V and \(\rho\) into the formula gives a mass of \(62.36 \times 10^{-24}\) kg.

Calculate the number of ions per unit cell

In a face-centered cubic structure, there are four formula units (i.e., four KCl pairs) per unit cell. Therefore, in each unit cell of the KCl crystal, there are four K+ ions and four Cl- ions, making a total of eight ions.

Calculate Avogadro's number

Avogadro's number \(N_A\) can be found from the formula \(N_A = mass / (number \times molar \, mass)\). The number is the number of ions per unit cell, which is eight. The molar mass of KCl is the sum of the molar masses of K (39.0983 g/mol) and Cl (35.453 g/mol), which gives 74.5513 g/mol, or 0.0745513 kg/mol when converted. Substituting these values results in \(N_A = 6.022 \times 10^{23}\) mol^-1.

Key concepts

Face-Centered Cubic Structure

When studying crystals, it's crucial to understand their structures. The face-centered cubic (FCC) structure is one of the most common arrangements of atoms in a crystal.

In a face-centered cubic structure, each face of the cube has an atom at its center, in addition to the atoms located at the corners of the cube.
This structure is very efficient and allows for a high packing density.

Here is a simple breakdown of the face-centered cubic unit cell:

• There are 8 atoms or ions at the corners of the cube.
• Each face of the cube contains an additional atom or ion.
• Only 1/8 of each corner atom is inside any given unit cell.
• Only half of each face atom is inside the unit cell.

In terms of potassium chloride (KCl), this structure means that each unit cell contains four full pairs of KCl.
This understanding helps explain the basis for many calculations in crystal chemistry and solid-state physics.

Unit Cell Calculation

Calculating the characteristics of a unit cell involves several steps, but it all begins with the internuclear distance between atoms.

For potassium chloride, the given internuclear distance between the potassium and chlorine ions is 314.54 pm.
This distance is crucial as it determines the volume of the unit cell.

To find the volume of the unit cell, we use the formula for face-centered cubic structures:

• The formula is: \[ V = d^3 \]
• Here, the distance \(d\) is converted into meters to match the units of volume calculated as cubic meters.

By substituting \(d = 314.54 \times 10^{-12}\) m into the formula, the volume \(V\) of the unit cell can be calculated as approximately \(31.254 \times 10^{-28}\) m³.

Understanding this step is crucial for later calculations involving density and mass.

Molar Mass of KCl

The molar mass is essential when dealing with chemical substances as it standardizes the mass of atoms or molecules to a mole.

To find the molar mass of KCl, you need the individual molar masses of its constituent elements.
These values are usually available on a periodic table:

• Potassium (K) has a molar mass of approximately 39.0983 g/mol.
• Chlorine (Cl) has a molar mass of approximately 35.453 g/mol.

The molar mass of KCl is simply the sum of these two values: - 39.0983 g/mol (for K) + 35.453 g/mol (for Cl) = 74.5513 g/mol. To use this value in calculations involving physical quantities like density and volume, it must often be converted into kilograms per mole, resulting in 0.0745513 kg/mol.

Having a solid grasp on how to calculate molar mass empowers you to tackle more complex problems in chemistry, such as those involving Avogadro's number.