Problem 2 - Chemistry The Central Science

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: \(2+\) charge for \(A\) and a \(1+\) charge for \(B\). Ions shown in blue carry negative charges: a 1 - charge for \(\mathrm{X}\) and a 2 - charge for \(\mathrm{Y}\). (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? (c) Which combination of ions leads to the ionic compound having the smallest lattice energy? [Section 8.2\(]\)

Step-by-Step Solution

Answer

(a) The combinations of ions that produce ionic compounds in a 1:1 ratio of cations to anions are AY and BX. (b) The combination with the largest lattice energy is AY, as it has a higher product of charges. (c) The combination with the smallest lattice energy is BX, as it has a smaller product of charges.

Step 1: (a) Finding ionic compounds in 1:1 ratio

To find the combinations of ions that produce a 1:1 ratio, we need to match the positive charges of the cations with the negative charges of the anions, resulting in a net zero charge. This can be done as follows: 1. Ion A has a charge of +2, so it can be paired with ion Y, which has a charge of -2. Thus, one possible combination is: AY 2. Ion B has a charge of +1, so it can be paired with ion X, which has a charge of -1. The second possible combination is: BX These combinations result in a net charge of zero and a 1:1 ratio of cations to anions.

Step 2: (b) Finding the combination with the largest lattice energy

Recall that lattice energy increases as the ionic charges increase and as the ionic radii decrease. To find the combination with the largest lattice energy among AY and BX, let's consider their charges and radii. 1. AY: Ion A has a charge of +2 and ion Y has a charge of -2, giving a product of charges of +4. 2. BX: Ion B has a charge of +1 and ion X has a charge of -1, giving a product of charges of -1. Comparing these, the combination AY has a higher product of charges and so it has the largest lattice energy among the two ionic compounds.

Step 3: (c) Finding the combination with the smallest lattice energy

Since we already know that the ionic compound AY has the largest lattice energy, it implies that the ionic compound BX will have the smallest lattice energy, as it has the smaller product of charges between the two.

Key Concepts

Ionic Radii

Ionic radii refer to the size of the ions in an ionic compound and this affects several properties of the compound. They help us understand how close ions can pack in the lattice. A larger ionic radius means the ion takes up more space, which can make the lattice less tight. Conversely, a smaller ionic radius allows ions to pack tightly and more closely.

Ionic radii vary depending on whether we're looking at cations or anions. Cations, which are positively charged ions, generally have smaller radii than their corresponding neutral atoms. This size reduction is due to the loss of one or more electrons, resulting in a stronger effective nuclear pull on the remaining electrons. Anions, on the other hand, gain electrons and are larger than their parent atoms because the extra electrons cause increased electron-electron repulsion.

• Smaller ionic radii: Cations (e.g., Ion A = +2, Ion B = +1)
• Larger ionic radii: Anions (e.g., Ion X = -1, Ion Y = -2)

Lattice Energy

Lattice energy is a measure of the strength of the forces holding the ions together in an ionic solid. It is the energy released when ions are combined to form an ionic compound, and it provides insights into the compound’s stability. The higher the lattice energy, the more stable the ionic compound.

Two main factors influence lattice energy:

• Charge of the ions: Lattice energy increases with the increase in ionic charges. For example, in our exercise, AY has higher lattice energy because it involves +2 and -2 charged ions, leading to a greater product of charges than BX, which only involves +1 and -1 charges.
• Ionic radii: Lattice energy generally decreases as ionic radii increase. Smaller ions pack closer together and form a stronger electrostatic attraction, thereby increasing the lattice energy. If ion sizes are large, like in BX, the lattice energy would be comparatively lower than if smaller ions were involved.

Cation-Anion Ratio

The cation-anion ratio is the proportion of positively charged ions (cations) to negatively charged ions (anions) in an ionic compound. This ratio is crucial for maintaining electrical neutrality in the compound. For example, if an ionic compound has a 1:1 cation-anion ratio, it means there is one cation for every anion, ensuring the net charge of the compound is zero.

In our exercise, two valid combinations meet this 1:1 ratio requirement:

• Ion A (+2) combined with Ion Y (-2), resulting in a net neutral charge and a stable compound, AY.
• Ion B (+1) paired with Ion X (-1), forming the compound, BX, which is also electrically neutral.

Achieving the correct cation-anion ratio is fundamental to forming stable ionic compounds, as it ensures that the electrostatic balance is maintained within the crystal lattice.