Question

Illustrated are four ions - \(A, B, X\), and \(Y-\) showing their relative ionic radii. The ions shown in red carry positive charges: a \(2+\) charge for A and a \(1+\) charge for 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]

Short answer

The ionic compounds that achieve a 1:1 cation/anion ratio are AX₂, AY, and BX. Among these, the ionic compound AY has the largest lattice energy.

Step-by-step solution

Identify possible ion pairs with 1:1 ratio of cations and anions

To achieve a 1:1 ratio of cations and anions, the overall charge of the compound should be zero. We have the following ions: A (2+), B (1+), X (1-), Y (2-). Let's test the possible combinations of these ions: 1. A (2+) and X (1-): AX₂ → overall charge is zero, it satisfies the 1:1 ratio condition. 2. A (2+) and Y (2-): AY → overall charge is zero, it satisfies the 1:1 ratio condition. 3. B (1+) and X (1-): BX → overall charge is zero, it satisfies the 1:1 ratio condition. 4. B (1+) and Y (2-): BY₂ → overall charge is not zero, so it doesn't satisfy the 1:1 ratio condition. Hence, AX₂, AY, and BX are the ionic compounds that achieve a 1:1 cation/anion ratio.

Determine which ionic compound has the largest lattice energy

To find out which compound has the largest lattice energy, we will apply the lattice energy formula: Lattice energy = k * (Q₁ * Q₂) / r Here, k is a constant, Q₁ and Q₂ are the charges of ions, and r is the sum of the ionic radii of the two ions. Since k is a constant, we only need to compare the (Q₁ * Q₂) / r part. 1. For AX₂: Charge part: Q₁ = 2, Q₂ = -1, Q₁ * Q₂ = -2 Radius part: r = r(A) + r(X) 2. For AY: Charge part: Q₁ = 2, Q₂ = -2, Q₁ * Q₂ = -4 Radius part: r = r(A) + r(Y) 3. For BX: Charge part: Q₁ = 1, Q₂ = -1, Q₁ * Q₂ = -1 Radius part: r = r(B) + r(X) Now, let's compare the values for charge and radius parts of each compound, remembering that the highest charge-to-radius ratio will lead to the largest lattice energy. For AX₂, AY, and BX, we have: 1) AX₂: charge (-2), radius (r(A) + r(X)) 2) AY: charge (-4), radius (r(A) + r(Y)) 3) BX: charge (-1), radius (r(B) + r(X)) Comparing the charge/radius ratio for compounds, we have: \: (-2)/(r(A) + r(X)) < (-1)/(r(B) + r(X)) < (-4)/(r(A) + r(Y)) The largest value is for AY, (-4)/(r(A) + r(Y)). Therefore, ionic compound AY has the largest lattice energy among the ionic compounds satisfying the 1:1 ratio of cations and anions.

Key concepts

Lattice Energy

Lattice energy is a measure of the strength of the forces holding ionic compounds together. In more technical terms, it's the energy released when gaseous ions come together to form a solid ionic compound. This might sound a bit abstract, so think of it like a measure of how much energy would be given off if you could watch tiny charged ions crash together to build a crystal. The stronger the forces that attract the ions to each other (which has a lot to do with their charges and sizes), the higher the lattice energy.

So when we talk about finding the compound with the largest lattice energy, we're looking for the duo of ions that really want to be near to each other, so much that they give off a bunch of energy when they do finally get together in a solid. An important point to remember, as highlighted by our textbook exercise, is that generally, the greater the charges and the smaller the ionic radii, the larger the lattice energy. This is because smaller ions can get closer together, increasing the attraction, and higher charges also mean stronger attractions. But remember, it's all about the combo—both charge and size matter.

Ionic Radii

Ionic radii refer to the measured size of an ion in a crystal lattice. Think of it as the space an ion occupies in the ionic sea of a compound. When ions form, they either lose electrons and become smaller, like the cations (positive ions), or gain electrons and become larger, like the anions (negative ions). These changes in size directly affect how packed together ions can get in a solid, and in turn, affect the lattice energy.

Returning to our exercise, understanding the significance of ionic radii helps us comprehend why certain combinations yield the largest lattice energy. Smaller ionic radii mean ions can get up close and personal, intensifying the attractions between them. This results in a stronger bond and, consequently, a higher lattice energy. It's like having a tight group hug—everyone's close together, and it takes more effort (energy) to pull them apart.

Cation-Anion Ratio

The cation-anion ratio is essentially the proportion of positively charged ions (cations) to negatively charged ions (anions) in an ionic compound. It's important because it determines the overall electric neutrality of the compound — the total charge must balance out to zero. In our exercise, we've seen that different ratios (like 1:1, 1:2) give us different compounds. A 1:1 ratio means there's a cation for every anion, and this balance is crucial for forming an ionic compound in the first place.

However, it's not just about having the right number of charges to balance out. The sizes of the ions (remember ionic radii?) and the magnitudes of their charges affect the overall stability and lattice energy of the compound, as a higher charge and an optimal ion size ratio contribute to a stronger force of attraction. To maximize lattice energy, we are ideally looking for a combination of high charge and small size — this leads to tightly bonded ionic compounds.