Problem 2 - Chemistry The Central Science In Si Units

Question

Illustrated are four ions \(-\mathrm{A}, \mathrm{B}, \mathrm{X},\) and \(\mathrm{Y}\) - showing their relative ionic radii. The ions shown in red carry positive charges: a \(2+\) charge for \(\mathrm{A}\) and a \(1+\) charge for \(\mathrm{B}\). Ions shown in blue carry negative charges: a \(1-\) charge for \(X\) and a \(2-\) charge for \(Y\). (a) Which combinations of these ions produce ionic compounds where there is a 1:1 ratio of cations and anions? (b) Among the combinations in part (a), which leads to the ionic compound having the largest lattice energy? [Section 8.2]

Step-by-Step Solution

Answer

1:1 compounds are \(B^{1+}X^{1-}\) and \(A^{2+}Y^{2-}\). \(A^{2+}Y^{2-}\) has the largest lattice energy.

Step 1: Understand Ionic Charges

The ions have the following charges: \(A^{2+}\), \(B^{1+}\), \(X^{1-}\), and \(Y^{2-}\). We need combinations of these to have a neutral overall charge.

Step 2: Determine 1:1 Ratio of Cations and Anions

To achieve a 1:1 ratio, the total positive charge from cations must equal the total negative charge from anions. Possible pairs include: \(B^{1+}\) and \(X^{1-}\) as well as \(A^{2+}\) and \(Y^{2-}\).

Step 3: Analyze Lattice Energy Criteria

Lattice energy depends on the product of the charges of the ions. Higher charge products result in higher lattice energy. Therefore, the pair \(A^{2+}\) and \(Y^{2-}\) (charge product of 4) will have a higher lattice energy than \(B^{1+}\) and \(X^{1-}\) (charge product of 1).

Step 4: Conclude

The combinations that produce a 1:1 ratio are \(B^{1+}X^{1-}\) and \(A^{2+}Y^{2-}\). Among these, \(A^{2+}Y^{2-}\) has the largest lattice energy.

Key Concepts

Ionic Charges: How They Determine Ionic Compounds

Ionic charges are essentially the electrical charges that ions carry. These are critical in the formation of ionic compounds, which are types of chemical compounds.

• The ions are atoms that have gained or lost electrons.
• When an atom loses electrons, it becomes positively charged, forming a cation. Examples include \(A^{2+}\) and \(B^{1+}\) from the exercise.
• When an atom gains electrons, it becomes negatively charged, forming an anion. In the exercise, \(X^{1-}\) and \(Y^{2-}\) are anions.

For ionic compounds to be stable, the total charges must balance to zero. This means that the total positive charges from the cations must equal the total negative charges from the anions.
Thus, in the exercise, pairings like \(B^{1+}\) with \(X^{1-}\) result in a stable compound because their charges cancel each other out to zero. Similarly, \(A^{2+}\) pairs well with \(Y^{2-}\) since their charges also balance each other out.

Lattice Energy: The Strength Behind Ionic Bonds

Lattice energy is a critical concept when examining the strength of ionic compounds. It is the energy released when ions in the gaseous state form an ionic solid. It determines the stability and physical properties of the compound.

• Lattice energy is directly dependent on the charges of the ions and inversely on the distance between them.
• Higher charges on the ions lead to stronger attractions, hence larger lattice energies.

The exercise shows that \(A^{2+}\) and \(Y^{2-}\), having a charge product of 4, release more energy when forming a compound compared to \(B^{1+}\) and \(X^{1-}\), which have a charge product of 1. Thus, \(A^{2+}Y^{2-}\) will have a much stronger ionic bond due to its greater lattice energy.
This means such compounds require more energy to break apart compared to those with lower lattice energy, making them more stable.

Ionic Radii: Size Matters in Ionic Compounds

Ionic radii refer to the size of ions and play a significant role when considering the properties of ionic compounds. The size of an ion affects the distance between ions in a crystal lattice, impacting the compound's properties.

• Cations typically have smaller radii than the original atom since they lose electrons. For instance, \(A^{2+}\) and \(B^{1+}\) are smaller than their corresponding neutral atoms.
• Anions usually have larger radii than their neutral atoms because they gain electrons. Examples include \(X^{1-}\) and \(Y^{2-}\).

In the context of lattice energy, the ionic radii influence because shorter distances between ions usually mean stronger interactions, hence higher lattice energies.
Recall, however, that while larger charges greatly increase lattice energy, smaller ionic radii also contribute by allowing ions to pack more closely together, giving rise to stronger electrostatic forces within the lattice.