Calculate the fraction of lattice sites that are Schottky defects for sodium chloride at its melting temperature \(\left(801^{\circ} \mathrm{C}\right)\). Assume an energy for defect formation of \(2.3 \mathrm{eV}\).

In crystalline materials, such as sodium chloride (commonly known as table salt), the atoms or ions are arranged in a well-ordered pattern that extends throughout the material. This arrangement is known as a lattice. Occasionally, there are irregularities in the lattice structure, and these irregularities are termed as lattice site defects.

There are several types of lattice site defects, but one of the most important is the Schottky defect. A Schottky defect occurs when a pair of oppositely charged ions are missing from their lattice sites, leading to vacant sites in the crystal. This kind of defect is significant because it affects the properties of the material, such as ionic conductivity and reactivity. During the melting process, the number of these defects increases because higher temperatures allow atoms or ions to overcome energy barriers needed to leave their lattice positions.

Understanding Schottky defects is crucial for materials science, as it can be related to the mechanical strength, diffusion, and electrical properties of the material.

The melting temperature of a substance is the temperature at which it changes from a solid to a liquid at atmospheric pressure. For ionic solids like sodium chloride, the melting temperature is particularly high due to the strong electrostatic forces between the oppositely charged ions in the lattice.

At the melting temperature, the kinetic energy of the ions increases to a point where the ordered lattice begins to break down. This increase in defects like Schottky defects becomes more favorable energetically. Understanding the behavior of materials at their melting temperature is essential in disciplines like materials science and engineering, where the thermal stability of a material heavily influences its applications and performance in real-world conditions.

Defect formation energy is the amount of energy required to create a defect in a crystal lattice. In the case of a Schottky defect, it's the energy needed to remove a pair of ions from their lattice sites to create a vacancy. The value of this energy, often measured in electron volts (eV), determines how easily a defect can form.

Generally, a higher defect formation energy means that fewer defects will be present at a given temperature, assuming all other factors remain constant. These energy considerations are pivotal when predicting the behavior of materials under different temperature conditions—such as at their melting temperature where defects are more prevalent due to increased kinetic energy. Materials with lower defect formation energies will have more defects at a given temperature and vice versa.

The Boltzmann constant (k) is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It plays a critical role in statistical mechanics and thermodynamics. In the context of defect calculations, the Boltzmann constant is used to establish the relationship between thermal energy and defect formation.

Its value is approximately 8.617 x 10^{-5} electron volts per Kelvin (eV/K), which allows us to convert energy into temperature and vice versa when evaluating the energetics of processes like defect formation. The Boltzmann constant is instrumental in calculating the fraction of lattice sites that contain Schottky defects, providing a bridge between the macroscopic observable (temperature) and microscopic energetics (defect formation energy).

Energy can be expressed in various units, and unit conversion is an essential aspect when working with different energy scales in physics and chemistry. One common unit of energy is the electron volt (eV), which is particularly convenient for describing atomic and molecular scale interactions, such as the energy involved in defect formation in a crystal lattice.

However, when dealing with thermal processes and the Boltzmann constant, energy is often considered in terms of temperature, which requires a conversion from eV to Joules (J). The conversion is crucial since values like the Boltzmann constant are typically given in units that relate energy to temperature (eV/K or J/K). In this way, properly converting units ensures accuracy in calculations and helps students to coherently follow the problem-solving process from the energy needed to create a defect to the fraction of defects at a certain temperature.