Question

In solid \(\mathrm{KCl}\) the smallest distance between the centers of a potassium ion and a chloride ion is 314 pm. Calculate the length of the edge of the unit cell and the density of \(\mathrm{KCl}\), assuming it has the same structure as sodium chloride.

Short answer

In solid KCl with the same structure as NaCl, the length of the edge of the unit cell is approximately 222 pm, and the density is approximately 2.73 x 10^6 g/m³.

Step-by-step solution

Identify the structure of the unit cell

Since KCl has the same structure as NaCl, it is a face-centered cubic (fcc) lattice. In an fcc lattice, the atoms are located at the corners and the centers of the faces of the cube. The smallest distance between ions in KCl is given as 314 pm.

Determine the edge length of the unit cell

In an fcc lattice, the shortest distance between ions lies along the face diagonal. Let 'a' be the edge length of the unit cell, and 'd' be the shortest distance between the ions. Using Pythagorean theorem for right triangles, we can write: \(a^2 + a^2 = d^2\) Since there are two K+ ions and two Cl- ions lying along the face diagonal: \(2a^2 = d^2\) Now we can use the given shortest distance (d = 314 pm) to calculate the edge length (a): \(2a^2 = (314 \, \text{pm})^2\) Solve for 'a': \(a^2 = \frac{(314 \, \text{pm})^2}{2}\) \(a = \sqrt{\frac{(314 \, \text{pm})^2}{2}}\) \(a \approx 222 \, \text{pm}\)

Calculate the density of KCl

To calculate the density of KCl, we need to find the mass of KCl in the unit cell and divide it by the volume of the unit cell. In an fcc lattice, there are 4 formula units of KCl per unit cell (8 corner ions and 6 face-centered ions with each of them contributing 1/2). The molar mass of KCl is approximately 39.1 g/mol (for K+) and 35.5 g/mol (for Cl-). Therefore, the mass of 4 formula units of KCl is: \(m_{\text{KCl}} = 4 \times (39.1 \, \text{g/mol} + 35.5 \, \text{g/mol}) = 298.4 \, \text{g/mol}\) Now we can find the volume of the unit cell by using the cube formula and the edge length: \(V = a^3 \approx (222 \, \text{pm})^3 = 1.093 \times 10^{-29} \, \text{m}^3\) Finally, we can find the density (ρ) of KCl by dividing the mass by the volume: \(\rho = \frac{m_{\text{KCl}}}{V} = \frac{298.4 \, \text{g/mol}}{1.093 \times 10^{-29} \, \text{m}^3}\) \(\rho \approx 2.73 \times 10^6 \, \text{g/m}^3\) So, the density of KCl is approximately 2.73 x 10^6 g/m³.

Key concepts

Unit Cell

A unit cell is a basic building block that describes the structure and arrangement of a crystal lattice. It's like a 3D brick that repeats to form the entire crystal. In the case of KCl (potassium chloride), it is structured as a face-centered cubic (fcc) lattice. This means that each unit cell has atoms positioned at each corner and at the center of each face of the cube.

Each corner of the cube in an fcc lattice has a fraction of an atom, typically shared with many other unit cells, while the face-centered positions contain atoms that each belong to only two cells. This arrangement results in each unit cell effectively containing four full atoms, contributing to the overall composition and properties of the solid state structure. Understanding this allows scientists to predict other properties, like the shortest distance between ions, which in KCl is given as 314 pm (picometers).

Density

Density is a physical property defined as mass per unit volume. For crystals like KCl, determining the density involves calculating the mass of the formula units within the unit cell and dividing it by the volume of that cell. This gives one a measure of how tightly packed the ions are in the crystal.

In the case of KCl with an fcc structure, we know a single unit cell contains four KCl formula units. By calculating the molar mass of KCl (summing 39.1 g/mol for K+ and 35.5 g/mol for Cl-), we find that the total mass of those four units is 298.4 g/mol. The volume is derived from the calculated edge length, 222 pm, by cubing it to find 1.093 x 10⁻²⁹ m³. Plug this information into the density formula \[\rho = \frac{m_{KCl}}{V}\]and you find the density. For KCl, the density calculates out to approximately 2.73 million grams per cubic meter, showing just how densely the ions are packed together in its fcc structure.

Potassium Chloride

Potassium chloride, commonly known as KCl, is an ionic compound composed of potassium cations (K+) and chloride anions (Cl-). It's widely used in various applications including medicine, food processing, and industrial processes. In solid form, it adopts the same crystal structure as sodium chloride, famously known as halite.

KCl's face-centered cubic (fcc) structure is characterized by its efficient ion packing. Each K+ ion is surrounded by six chloride ions and vice versa, leading to its distinctive and stable lattice configuration. The simplicity of its cubic symmetry greatly aids in studying its physical properties, such as solubility and electrical conductivity, as well as understanding how it interacts with other compounds in solutions and processes.

• Medical Uses: KCl is crucial in healthcare for supplementing low potassium levels in patients.
• Industrial Applications: It's used in fertilizer production, a key ingredient to provide essential nutrients for plant growth.

This summarizes the versatility and importance of KCl in both scientific study and practical application.