Question

Consider the nitrogen-oxygen bond lengths in \(\mathrm{NO}_{2}^{+}, \mathrm{NO}_{2}^{-},\) and \(\mathrm{NO}_{3}^{-} .\) In which ion is the bond predicted to be longest? In which is it predicted to be the shortest? Explain briefly.

Short answer

NO₂⁻ has the longest bond; NO₂⁺ has the shortest bond.

Step-by-step solution

Determine Total Valence Electrons

For each ion, calculate the total number of valence electrons.- **\(\text{NO}_2^+\):** Nitrogen has 5, each oxygen has 6. One positive charge means losing an electron: \(5 + 6 \times 2 - 1 = 16\) electrons.- **\(\text{NO}_2^-\):** One extra electron due to the negative charge: \(5 + 6 \times 2 + 1 = 18\) electrons.- **\(\text{NO}_3^-\):** Nitrogen has 5, each oxygen has 6, plus one more electron for the negative charge: \(5 + 6 \times 3 + 1 = 24\) electrons.

Sketch Lewis Structures

Draw the Lewis structures for each of the ions based on the valence electrons calculated:- **\(\text{NO}_2^+\):** Has an \(N\) double-bonded to each \(O\), and no lone pairs on \(N\).- **\(\text{NO}_2^-\):** One \(O\) double-bonded and one \(O\) single-bonded to \(N\), with a lone pair on \(N\).- **\(\text{NO}_3^-\):** Delocalized bonding with resonance; one \(O\) double-bonded at a time and two single-bonded, all with lone pairs.

Predict Bond Lengths

Bond length is inversely related to bond order. Determine bond order for each ion.- **\(\text{NO}_2^+\):** Has a bond order of 2 (as both \(NO\) bonds are double bonds).- **\(\text{NO}_2^-\):** Average bond order of 1.5 (one double and one single bond).- **\(\text{NO}_3^-\):** Resonance leads to a bond order of \(\frac{4}{3}\).Higher bond order results in shorter bonds.

Compare Bond Orders

Compare the bond orders to determine the length:- **Shortest Bond:** \(\text{NO}_2^+\) (higher bond order).- **Longest Bond:** \(\text{NO}_2^-\) (lowest bond order).

Key concepts

Valence Electrons

Valence electrons are the outermost electrons of an atom. They play a vital role in chemical bonding.
These electrons determine how atoms interact with each other to form chemical compounds. For nitrogen and oxygen, the electrons in the outermost

• Nitrogen has five valence electrons.
• Oxygen has six valence electrons.

In ions like NO₂⁺, NO₂^-, and NO₃^-, charges affect the total number of valence electrons. A positive charge means the loss of an electron, while a negative charge indicates an extra electron. Understanding and calculating the number of valence electrons helps us in predicting the way atoms bond together and ultimately affects the properties of the compound formed.

Lewis Structures

Lewis structures are diagrams that represent the bonding between atoms of a molecule, and the lone pairs of electrons that may exist. They serve as a visual aid to see how atoms connect in molecules.
To draw a Lewis structure, follow these steps:

• Calculate the total number of valence electrons in the molecule.
• Choose a central atom (usually the least electronegative force, except for hydrogen).
• Connect each atom via single bonds initially.
• Assign remaining electrons to fulfill each atom's octet, if possible.
• Use double or triple bonds if necessary to satisfy octets.

For example, NO₂⁺ involves nitrogen double-bonded to oxygen with no lone pairs on nitrogen. This helps predict bond order and bond lengths for molecules.

Bond Length

Bond length is the distance between the nuclei of two bonded atoms. It is affected by the types of bonds and the sizes of the bonded atoms.
Factors influencing bond length include:

• The bond order: higher bond order means shorter bond length.
• Atomic size: larger atoms create longer bonds.
• The presence of resonance structures also averages the length.

By understanding these concepts, predicting bond lengths in molecules like NO₂⁺, NO₂^-, and NO₃^- becomes more straightforward. Usually, a higher bond order signifies a stronger attraction, pulling atoms closer together and shortening the bond length.

Resonance

Resonance occurs when two or more valid Lewis structures can be sketched for a molecule.
It is a concept that helps display the delocalization of electrons in certain molecules or ions. Resonance is crucial because:

• It helps stabilize molecules because electrons can be spread over more than one atom and bond.
• It provides a more accurate description than a single Lewis structure by showing hybrids.
• Enhances our understanding of molecular shape and chemical properties.

For example, in NO₃^-, resonance allows the structure to change between different forms, which contributes to an averaged bond order.

Bond Order

Bond order is the number of chemical bonds between a pair of atoms. It indicates the stability of a bond and is calculated as the difference between the number of bonding electrons and antibonding electrons, divided by two (for simple diatomic molecules).
The bond order affects:

• Bond length: higher bond order usually means shorter bonds.
• Bond energy: more bonds mean stronger bonds.

In molecules such as NO₂⁺, NO₂^-, and NO₃^-, bond order helps in predicting how long the bond could be. With calculations similar to those in resonance, it determines that the higher the bond order, the shorter the bond itself, helping to predict properties like bond length and bond stability.