Question

When \(\mathrm{NaCl}\) is dopped with \(10^{-5}\) mole \(\%\) of \(\operatorname{SrCl}_{2}\), what is the no. of cationic vacancies? (a) \(10^{-5} \times N_{A}\) (b) \(10^{-7} \times N_{A}\) (c) \(2 \times 10^{-7} \mathrm{~N}_{\mathrm{A}}\) (d) None of these

Short answer

cationic vacancies are created when is dopped with mole of leading to vacancies.

Step-by-step solution

Understand the Concept of Doping

Doping takes place in an ionic solid through the introduction of a small amount of a foreign substance. In this case, each two moles of taken out are replaced by one mole of to maintain electrical neutrality. Thus, the number of cationic vacancies will be equal to the number of to be added.

Calculate the Number of Moles of Replacing

The mole percent of to is given as . Thus, for every moles of added, one mole of is removed. Consequently, moles of create moles of vacancies.

Convert Moles of Vacancies to Number of Vacancies

To find the actual number of cationic vacancies, multiply the number of moles of vacancies by where is Avogadro's number ().

Key concepts

Doping in Ionic Solids

Doping in ionic solids is the intentional addition of impurities into an ionic crystal to alter its electrical or optical properties. When foreign atoms or molecules, often referred to as dopants, are incorporated into the crystal lattice of an ionic solid, they can create vacancies or interstitial sites that affect the overall behavior of the material.

In the context of the exercise, \textbf{SrCl}\(_2\) is used to dope \textbf{NaCl}, substituting the \textbf{Na}\(^+\) ions with larger \textbf{Sr}\(^{2+}\) ions. Because the strontium ion has a double positive charge (\textbf{Sr}\(^{2+}\)), replacing two single-charged sodium ions (\textbf{Na}\(^+\)) with one double-charged strontium ion maintains charge balance but results in the formation of cationic vacancies. This is important for maintaining the electrical neutrality of the solid.

Electrical Neutrality in Solids

Electrical neutrality is a state where the number of positive charges equals the number of negative charges in a material. For ionic solids, this is crucial for the stability and integrity of their crystal lattice structure. In the doping example, for every \textbf{Sr}\(^{2+}\) ion that replaces two \textbf{Na}\(^+\) ions, the electrical neutrality is preserved by the creation of a cationic vacancy.

This implies that the ionic solid must maintain a zero overall charge, thus ensuring the crystalline structure remains stable despite the dopant-induced disruptions. Without electrical neutrality, the ionic solid would have Coulombic forces that could disrupt the crystal lattice, leading to potential material failure.

Avogadro's Number

Avogadro's number, denoted as \textbf{N}\(_{A}\), is a fundamental constant in chemistry and physics that provides the relationship between the number of particles and the amount of substance in moles. It is defined as the number of atoms in exactly 12 grams of carbon-12 (\textbf{C}-12), and its value is approximately 6.022 × 10\(^{23}\) particles per mole.

When encountered with problems involving moles and particles, Avogadro's number becomes a key factor in the conversion. As illustrated in the problem, discerning the number of cationic vacancies involves converting the moles of vacancies to actual particles, utilizing Avogadro's number.

Mole Concept

The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we interact with daily. One mole refers to a quantity of substance that contains as many particles, such as atoms or ions, as there are in 12 grams of pure carbon-12—isotopically defined as Avogadro’s number (\textbf{N}\(_{A}\)).

When calculating the number of cationic vacancies during doping, we use the mole concept to determine how many atoms, in this case, sodium ions, have been replaced. Understanding the mole concept facilitates the quantification of chemical reactions and, in the context of the problem, allows us to calculate the number of vacancies formed as a direct result of the doping process.