Question

Consider the hypothetical metallic element \(\mathrm{M},\) which is capable of forming stable simple cations that have charges of \(1+, 2+,\) and \(3+,\) respectively. Consider also the nonmetallic elements \(\mathrm{D}, \mathrm{E},\) and \(\mathrm{F},\) which form anions that have charges of \(1-, 2-\), and \(3-\) respectively. Write the formulas of all possible compounds between metal \(M\) and nonmetals \(D, E\) and E.

Short answer

The possible compounds between the elements M and D, E, and F are: \(MD\), \(M_2E\), \(M_3F\), \(MD_2\), \(ME\), \(M_3F_2\), \(MD_3\), \(M_2E_3\), and \(MF\).

Step-by-step solution

List the possible ions

Before writing the chemical formulas for the compounds, we list the possible ions that can be formed by each element: - For element M: \(M^{1+}, M^{2+}, M^{3+}\), - For element D: \(D^{1-}\), - For element E: \(E^{2-}\), - For element F: \(F^{3-}\).

Combine the ions to form neutral compounds

Now, we will consider all possible combinations of cations and anions, always making sure that the total positive charge is balanced by the total negative charge: 1. \(M^{1+}\) and \(D^{1-}\) - The compound formed is \(MD\) as the positive and negative charges balance each other. 2. \(M^{1+}\) and \(E^{2-}\) - To balance the charges, we need two \(M^{1+}\) ions for each \(E^{2-}\) ion. The compound formed is \(M_2E\). 3. \(M^{1+}\) and \(F^{3-}\) - To balance the charges, we need three \(M^{1+}\) ions for each \(F^{3-}\) ion. The compound formed is \(M_3F\). 4. \(M^{2+}\) and \(D^{1-}\) - To balance the charges, we need one \(M^{2+}\) ion for each two \(D^{1-}\) ions. The compound formed is \(MD_2\). 5. \(M^{2+}\) and \(E^{2-}\) - The compound formed is \(ME\) as the positive and negative charges balance each other. 6. \(M^{2+}\) and \(F^{3-}\) - To balance the charges, we need three \(M^{2+}\) ions for each two \(F^{3-}\) ions. The compound formed is \(M_3F_2\). 7. \(M^{3+}\) and \(D^{1-}\) - To balance the charges, we need one \(M^{3+}\) ion for each three \(D^{1-}\) ions. The compound formed is \(MD_3\). 8. \(M^{3+}\) and \(E^{2-}\) - To balance the charges, we need two \(M^{3+}\) ions for each three \(E^{2-}\) ions. The compound formed is \(M_2E_3\). 9. \(M^{3+}\) and \(F^{3-}\) - The compound formed is \(MF\) as the positive and negative charges balance each other.

List all the possible compounds

Based on the combinations that we found in Step 2, all possible compounds between the elements M and D, E, and F are: - \(MD\), - \(M_2E\), - \(M_3F\), - \(MD_2\), - \(ME\), - \(M_3F_2\), - \(MD_3\), - \(M_2E_3\), - \(MF\).

Key concepts

Ions and Ionic Charge

The concept of ions and ionic charge is crucial in the field of chemistry, as it relates to the electrical charge of an atom when it gains or loses electrons. Ions are atoms, or groups of atoms, that have a net positive or negative charge due to the loss or gain of one or more electrons, respectively. Cations are positively charged ions that have lost electrons, and anions are negatively charged ions that have gained electrons.

For example, in the given exercise, the metal element M can form cations with charges of \(1+\), \(2+\), and \(3+\) by losing 1, 2, or 3 electrons, respectively. Conversely, the nonmetals D, E, and F can form anions with charges of \(1-\), \(2-\), and \(3-\) by gaining 1, 2, or 3 electrons, respectively. Understanding the concept of ions and their charges is essential when predicting the formation of chemical compounds and writing their formulas.

Chemical Formula Writing

Writing a chemical formula involves using the symbols of the elements and numerical subscripts to represent the number of atoms of each element in a compound. The goal is to create a neutral compound in which the total positive charge from the cations equals the total negative charge from the anions.

As per our exercise, when combining metal M with nonmetals D, E, and F, we arrange the metal cations and nonmetal anions in a way that achieves charge neutrality. For instance, for \(M^{1+}\) and \(E^{2-}\), we require two \(M^{1+}\) ions to balance the charge of one \(E^{2-}\) ion, leading to the formula \(M_2E\). It's important to understand that the subscript in a chemical formula indicates the number of atoms present, allowing us to quote the proper proportions to maintain charge balance.

Charge Balance in Compounds

Ensuring charge balance in compounds is a fundamental principle in chemistry that states the total positive charge must equal the total negative charge in a neutral compound. This principle allows us to predict the correct stoichiometry of ionic compounds. The crux lies in combining cations and anions in a ratio that results in a neutral compound, i.e., the sum of the charges should be zero.

In our example, when writing the formula for a compound formed by \(M^{3+}\) and \(F^{3-}\), the charges are already balanced (\(+3\) from \(M^{3+}\) and \(-3\) from \(F^{3-}\)), producing the compound \(MF\) with no subscripts necessary. However, for \(M^{2+}\) and \(F^{3-}\), we need to find the lowest common multiple of the charges, which is 6, thus requiring the combination of three \(M^{2+}\) ions and two \(F^{3-}\) ions to achieve charge neutrality, resulting in the formula \(M_3F_2\). This approach ensures the creation of compounds adhering to the principle of charge balance.