(a) Suppose that \(\mathrm{Li}_{2} \mathrm{O}\) is added as an impurity to \(\mathrm{CaO}\). If the \(\mathrm{Li}^{+}\)substitutes for \(\mathrm{Ca}^{2+}\), what kind of vacancies would you expect to form? How many of these vacancies are created for every \(\mathrm{Li}^{+}\)added? (b) Suppose that \(\mathrm{CaCl}_{2}\) is added as an impurity to \(\mathrm{CaO}\). If the \(\mathrm{Cl}^{-}\)substitutes for \(\mathrm{O}^{2-}\), what kind of vacancies would you expect to form? How many of the vacancies are created for every \(\mathrm{Cl}^{-}\)added?

Impurity doping in materials, like semiconductors and ceramics, often leads to the creation of vacancies to maintain overall charge neutrality. In the case where a lower-valence cation replaces a higher-valence cation, cation vacancies are generated. For example, when Li_2O is added to CaO, and Li^+ (which has a +1 charge) substitutes for Ca^2+ (which has a +2 charge), there are insufficient positive charges to balance the charges from O^2-. To maintain charge neutrality, cation vacancies will form.

The crucial aspect in the formation of vacancies is maintaining electrical neutrality. In the given exercise, the addition of each Li^+ ion effectively removes one positive charge that would have been contributed by a Ca^2+ ion. Therefore, for every two Li^+ ions added, a single Ca^2+ vacancy must be created to counterbalance the deficit of positive charge. This ensures that the overall material remains electrically neutral, which is a fundamental requirement for the stability of the structure.

The principle of charge neutrality in solids states that the total number of positive charges must equal the total number of negative charges within a crystal lattice or any part of it. Charge neutrality is critical for maintaining the structural and electrical properties of materials. When impurities are introduced into a lattice, like in doping processes, the overall charge balance can be disturbed. To maintain charge neutrality, the material compensates by creating vacancies or adding extra ions.

Improving upon the textbook solution, it's helpful to understand that charge neutrality applies at the microscale as well as the macroscale. In every tiny region of the compound, the number of positive and negative charges must be equal for the material to be stable. This is why, in the exercise under discussion, for every two Li^+ ions added, we need one Ca^2+ vacancy because the Li^+ only brings one positive charge instead of the two positive charges that Ca^2+ would have provided.

Similarly, when doping involves substituting a higher-valence anion with a lower-valence anion, anion vacancies are generated to preserve charge neutrality. Consider the substitution of CaCl2 into CaO, where Cl^- (with a -1 charge) replaces O^2- (with a -2 charge). This substitution introduces a relative deficiency in negative charge. Hence, for every two Cl^- ions that enter the lattice, one O^2- vacancy is created to rebalance the charge.

In this scenario, the anion vacancy allows the material to adjust to the change in charge without significantly altering its structural framework. Anion vacancies, like cation vacancies, play an integral role in the electrical conductivity and chemical properties of the material. By understanding both cation and anion vacancies, students can gain deeper insights into how materials can be engineered for specific applications by manipulating their crystal structure and electronic properties.