Question

Among the following, which has the shortest bond and which has the longest: \(\mathrm{Li}_{2}, \mathrm{B}_{2}, \mathrm{C}_{2}, \mathrm{N}_{2}, \mathrm{O}_{2} ?\)

Short answer

\(\mathrm{N}_2\) has the shortest bond; \(\mathrm{Li}_2\) has the longest.

Step-by-step solution

Understanding Bond Lengths

Bond length is the distance between the nuclei of two bonded atoms. In molecular orbitals, the bond order determines the bond length; a higher bond order results in a shorter bond.

Determine Bond Order

Calculate the bond order for each molecule. The bond order is given by the formula: \( \text{Bond Order} = \frac{( ext{Number of bonding electrons}) - ( ext{Number of antibonding electrons})}{2} \).

Li₂ Bond Order

For \(\mathrm{Li}_2\), the electrons fill as \((\sigma_{1s})^2(\sigma^*_{1s})^2(\sigma_{2s})^2\). There are 2 bonding and 0 antibonding electrons in the 2s orbital giving a bond order of 1.

B₂ Bond Order

For \(\mathrm{B}_2\), the electrons fill as \((\sigma_{1s})^2(\sigma^*_{1s})^2(\sigma_{2s})^2(\sigma^*_{2s})^2(\pi_{2p})^2\). There are 6 bonding and 4 antibonding electrons resulting in a bond order of 1.

C₂ Bond Order

For \(\mathrm{C}_2\), the electrons fill as \((\sigma_{1s})^2(\sigma^*_{1s})^2(\sigma_{2s})^2(\sigma^*_{2s})^2(\pi_{2p})^4\). There are 8 bonding and 4 antibonding electrons resulting in a bond order of 2.

N₂ Bond Order

For \(\mathrm{N}_2\), the electrons fill as \((\sigma_{1s})^2(\sigma^*_{1s})^2(\sigma_{2s})^2(\sigma^*_{2s})^2(\pi_{2p})^4(\sigma_{2p})^2\). There are 10 bonding and 4 antibonding electrons, leading to a bond order of 3.

O₂ Bond Order

For \(\mathrm{O}_2\), the electrons fill as \((\sigma_{1s})^2(\sigma^*_{1s})^2(\sigma_{2s})^2(\sigma^*_{2s})^2(\pi_{2p})^4(\sigma_{2p})^2(\pi^*_{2p})^2\). There are 10 bonding and 6 antibonding electrons, resulting in a bond order of 2.

Determine Shortest and Longest Bond

The shortest bond is in \(\mathrm{N}_2\), which has the highest bond order of 3. The longest bond is in either \(\mathrm{Li}_2\) or \(\mathrm{B}_2\), both having a bond order of 1. However, typically, larger atoms like Li have longer bonds compared to smaller atoms like B.

Key concepts

Bond Order

Bond order is a fundamental concept in molecular chemistry. It measures the stability of a bond between two atoms in a molecule. To calculate bond order, use the formula: \[ \text{Bond Order} = \frac{\text{(Number of Bonding Electrons)} - \text{(Number of Antibonding Electrons)}}{2} \] This calculation shows how many electron pairs are shared between a pair of atoms. When the bond order is higher, the bond is shorter and stronger. A zero bond order indicates that a molecule is not stable. Calculating bond order helps predict the geometry and size of molecules. For instance, in diatomic nitrogen \((\text{N}_2)\), ten bonding electrons and four antibonding electrons lead to a bond order of three. This results in a triple bond, making it very short and strong. Remember: the higher the bond order, the shorter the bond length and the stronger the bond between atoms.

Molecular Orbitals

Molecular orbitals are formed when atomic orbitals from bonding atoms overlap and combine. This concept is crucial in understanding how bonds form in molecules. There are two main types of molecular orbitals: bonding and antibonding orbitals. When atomic orbitals combine constructively, they create bonding molecular orbitals, which lead to an increase in electron density between the two nuclei, effectively holding the atoms together. Conversely, antibonding molecular orbitals result from destructive interference, decreasing electron density between nuclei and making the bond weaker. Electron filling in molecular orbitals follows Aufbau's principle, filling lower energy orbitals first. Consider \(\text{C}_{2}\): its electrons fill in bonding configurations such as \((\pi_{2p})^4\), giving it a bond order of two. Understanding these distributions helps explain the properties of molecules like different bond lengths and strengths.

Bonding and Antibonding Electrons

Electrons play a vital role in determining molecular stability and bond characteristics. They occupy either bonding or antibonding orbitals, classifying them as bonding or antibonding electrons. Bonding electrons reside in lower energy molecular orbitals. They stabilize the molecule by increasing electron density between the atomic nuclei. In contrast, antibonding electrons occupy higher energy molecular orbitals, decreasing electron density between the nuclei and thus destabilizing the molecule. The balance of bonding and antibonding electrons directly influences the bond order and, therefore, the overall stability and length of a bond. Consider \(\text{O}_{2}\): ten electrons occupy bonding orbitals, while six fill antibonding orbitals, resulting in a bond order of two. Appreciating the role of these electrons is pivotal for predicting molecular behavior and properties.