Question

Rank each of the molecules below in order of the shortest to the longest sulfur-oxygen bond length. a. \(\mathrm{SO}_{2} \quad\) b. SO_ \(_{3}^{2-} \quad\) c. \(\mathrm{SO}_{4}^{2-}\)

Short answer

Shortest to longest sulfur-oxygen bond length: \(SO_{2} < SO_{4}^{2-} < SO_{3}^{2-}\)

Step-by-step solution

Identify Bond Order and Resonance Structures

In this step, let's analyze the resonance structures and bond order in the given molecules. a. SO2: In this molecule, the sulfur atom forms two double bonds with the two oxygen atoms. The bond order is 2. There are no resonance structures. b. SO3^2-: The sulfur atom in this ion forms a total of three bonds with three oxygen atoms. Due to resonance, the bond order becomes: \( \frac{Total\ number\ of\ bonds}{Number\ of\ O\ atoms} =\frac{3}{3}=1\) c. SO4^2-: The sulfate ion has four oxygen atoms around the sulfur atom, and a total of 4 bonds (1 each) due to resonance. Thus, the bond order becomes: \( \frac{Total\ number\ of\ bonds}{Number\ of\ O\ atoms} =\frac{4}{4}=1\)

Correlate Bond Order with Bond Lengths

Higher bond order generally corresponds to a shorter bond length and a stronger bond. Based on the bond order determined in Step 1, we can identify their relative bond lengths: a. SO2: Bond order 2, shorter sulfur-oxygen bond. b. SO3^2-: Bond order 1, longer sulfur-oxygen bond (as compared to SO2). c. SO4^2-: Bond order 1, but has a higher charge distribution around the oxygen atoms which might make its bond a little more stable and shorter than SO3^2-.

Rank the Molecules in Order of Shortest to Longest Sulfur-Oxygen Bond Length

Based on Steps 1 and 2, we can rank the given molecules in order of shortest to longest sulfur-oxygen bond length as: Shortest: SO2 Middle: SO4^2- Longest: SO3^2-

Key concepts

Bond Order

Bond order is an important concept when discussing molecular bonding and properties such as bond length. It gives insight into the quantity of bonds between a pair of atoms. Higher bond orders typically indicate stronger and shorter bonds. For example, a single bond has a bond order of 1, a double bond has a bond order of 2, and a triple bond has a bond order of 3.

To determine the bond order, you can follow this simple formula:

• Calculate the total number of bonds present between atoms in resonance structures.
• Divide this by the number of bond-sharing groups.

This explains why in the case of sulfur compounds like \(SO_2\), which exhibits no resonance, the bond order is simply equal to the number of double bonds present, which is 2. However, in resonating structures of \(SO_3^{2-}\) and \(SO_4^{2-}\), the bond order is averaged over multiple equivalent structures, arriving at 1 for both based on their distribution of bonds over multiple oxygen atoms.

Resonance Structures

Resonance structures are a way to visualize certain molecules that can't be adequately represented by a single Lewis structure. Instead, these molecules can be drawn in several different ways with atoms connected by varying bond orders. This concept is crucial to understand as it affects electron distribution and the actual observed molecular structure.

In sulfur and oxygen compounds like \(SO_3^{2-}\) and \(SO_4^{2-}\), resonance helps distribute charges evenly across all bonding locations, which affects the bond order and thus the bond length. For instance, consider how the \(SO_3^{2-}\) ion, although it involves multiple resonance structures, each oxygen atom doesn’t have a fixed double or single bond. Instead, the bonds are delocalized due to resonance. This causes all bonds to effectively exist as intermediates, leading to an averaged bond order.

Molecular Ranking

Rankings of molecules based on bond lengths often rely on bond order and resonance. The relationship is generally straightforward:

• Higher bond order leads to shorterdistances because the bonds are stronger.
• But when resonance is involved, an effective bond order results from averaging.

In molecules like \(SO_2\), which has a straightforward calculation of bond order due to no resonance, it maintains the shortest bond length due to its double bonds (bond order of 2). \(SO_4^{2-}\), though exhibiting resonance, has a slight electron charge stabilization keeping it slightly below the more fluctuating \(SO_3^{2-}\), causing subtle differences in bond length that lean \(SO_4^{2-}\) into a closer intermediate position compared to the longest, \(SO_3^{2-}\). Thus, understanding these molecular rankings entails considering both these fundamental theoretical perspectives as applied practically reifies how specific chemical properties emerge.