Calculate the number of Frenkel defects per cubic meter in zinc oxide at \(1000^{\circ} \mathrm{C}\). The energy for defect formation is \(2.51 \mathrm{eV}\), whereas the density for \(\mathrm{ZnO}\) is \(5.55 \mathrm{~g} / \mathrm{cm}^{3}\) at \(\left(1000^{\circ} \mathrm{C}\right)\)

Defect formation energy is a crucial concept in the study of materials science, particularly when it comes to understanding the intrinsic properties of solid materials like zinc oxide. It refers to the amount of energy required to create a point defect in a crystal structure. A point defect is an irregularity within the crystal lattice, where the ideal arrangement of atoms is disrupted.

In zinc oxide, Frenkel defects are one of the possible point defects that can occur. They consist of a lattice site vacancy and an interstitial atom, which is an atom placed into a normally unoccupied position. These defects are critical in determining the electrical and thermal properties of semiconductors and other materials. For example, Frenkel defects in zinc oxide can affect its ability to conduct electricity by altering the concentration of charge carriers.

The calculation of the defect concentration in a material requires the consideration of its defect formation energy. Generally, a higher defect formation energy implies that defects are less likely to form spontaneously at a given temperature, suggesting that the material will have fewer defects. Hence, understanding and being able to calculate defect formation energy gives us valuable insights into the behavior and qualities of the material under various conditions.

The Arrhenius equation is a mathematical expression that provides a way to calculate the rate of a chemical reaction or a process such as defect formation in materials. It links the rate constant to the temperature and activation energy of the reaction or process.

Applied to the study of defects in materials like zinc oxide, the equation allows us to estimate the concentration of defects at different temperatures. The general form of the Arrhenius equation is: \[ k = A \times \text{exp}\bigg(-\frac{E_a}{kT}\bigg) \]
where:

• \(k\) is the rate constant or in the context of defects, the concentration of defects.
• \(A\) is the pre-exponential factor, which represents the number of times that reactants or defect-forming events approach the activation energy per unit time.
• \(E_a\) is the activation energy or the defect formation energy.
• \(T\) is the absolute temperature (in kelvin).
• \(k\) is Boltzmann's constant, which relates the average kinetic energy in a gas with the temperature of the gas.

The equation explains why at higher temperatures, the formation of defects increases, as the thermal energy available to overcome the activation barrier becomes greater. When solving problems related to defect concentrations, one often uses a version of the Arrhenius equation that is suited for the specific type of defect, as seen in the provided exercise solution.

The calculation of molar mass is a fundamental skill in chemistry, important for a variety of applications ranging from stoichiometric calculations to the determination of the number of molecules in a given sample. Molar mass is defined as the mass of one mole of a substance, and it is usually expressed in units of grams per mole (g/mol).

For compounds, the molar mass is calculated by adding the molar masses of the individual elements that compose the compound, proportional to their stoichiometric coefficients. For example, in zinc oxide (\(ZnO\)), the molar mass is the sum of the molar mass of zinc (\(65.38 \text{ g/mol}\)) and oxygen (\(16.00 \text{ g/mol}\)), resulting in \(81.38 \text{ g/mol}\) for zinc oxide.

Understanding how to calculate molar mass allows students to work out the number of molecules or formula units present in a known mass of a substance. This is crucial when determining concentrations or the number of defects per unit volume in solid materials, such as the number of Frenkel defects in zinc oxide, which is directly related to the density and the molar mass of the substance.

Boltzmann's constant (\(k\) or \(k_B\)) is a fundamental constant in physics that appears in various fields, notably in thermodynamics and statistical mechanics. The constant relates the average kinetic energy of particles in a gas with the temperature of the gas, embodying the connection between microscopic and macroscopic states.

Its value is approximately \(8.617 \times 10^{-5} \text{ eV/K}\), which is used when expressing energy in electron volts (\(\text{eV}\)) and temperature in kelvin (\(\text{K}\)). In the context of semiconductor physics and materials science, Boltzmann's constant is utilized primarily in calculations involving temperature dependence of properties like electrical resistance or defect concentrations.

The use of Boltzmann's constant is evident in the Arrhenius equation for estimating the number of defects in materials at a given temperature, which reflects the microscopic movements and energies of atoms and their macroscopic implications. In such calculations, it's critical to ensure that the units are consistent to avoid errors, especially when energies are given in electron volts and temperatures in kelvin, as seen in the exercise.