Problem 75 - Chemistry The Central Science In Si Units

Question

(a) What are the relationships among bond order, bond length, and bond energy? (b) According to molecular orbital theory, would either \(\mathrm{Be}_{2}\) or \(\mathrm{Be}_{2}^{+}\) be expected to exist? Explain.

Step-by-Step Solution

Answer

Higher bond order means shorter and stronger bonds. \\(\mathrm{Be}_{2}\) likely does not exist, but \\(\mathrm{Be}_{2}^{+}\) could exist with weak bonding.

Step 1: Understanding Bond Order

Bond order is the number of chemical bonds between a pair of atoms. Higher bond order means stronger bonds due to more shared electrons. Mathematically, bond order can be calculated as \(( ext{number of bonding electrons} - ext{number of antibonding electrons}) / 2\).

Step 2: Analyzing Bond Length

Bond length is the average distance between the nuclei of two bonded atoms. A higher bond order generally leads to shorter bond lengths because the stronger, multi-bond interactions pull the atoms closer together.

Step 3: Examining Bond Energy

Bond energy is the measure of bond strength in a chemical bond. It is the energy required to break one mole of bonds in the gas phase. Higher bond order typically results in higher bond energy, indicating a more stable and robust bond.

Step 4: Applying Molecular Orbital Theory to Be2

The molecular orbital configuration for \(\mathrm{Be}_{2}\) is \([\sigma_{1s}]^2 [\sigma^*_{1s}]^2 [\sigma_{2s}]^2 [\sigma^*_{2s}]^2\). This results in zero net bonding electrons, giving a bond order of zero \([\frac{4\text{ bonding electrons} - 4\text{ antibonding electrons}}{2}]\), suggesting that \(\mathrm{Be}_{2}\) is not stable.

Step 5: Assessing \\(\mathrm{Be}_{2}^{+}\) Existence

For \(\mathrm{Be}_{2}^{+}\), one electron is removed, giving the orbital configuration \([\sigma_{1s}]^2 [\sigma^*_{1s}]^2 [\sigma_{2s}]^2 [\sigma^*_{2s}]^1\). This results in a bond order of \(\frac{4 - 3}{2} = 0.5\), indicating that \(\mathrm{Be}_{2}^{+}\) might exist, though weakly bonded.

Key Concepts

Bond Order

Bond order is an essential concept when studying the strength and stability of chemical bonds. In simple terms, bond order refers to the number of chemical bonds between a pair of atoms. Imagine each bond as a bridge made out of electrons connecting two atoms.
The higher the bond order, the stronger the connection because more bridges mean more stability.
A bond order of 1 indicates a single bond, 2 indicates a double bond, and so on. To calculate the bond order, use the formula:

• \[\text{Bond Order} = \frac{\text{Number of Bonding Electrons} - \text{Number of Antibonding Electrons}}{2}\]

This formula tells us that with more electrons in bonding molecular orbitals as compared to antibonding ones, the bond order increases. Remember, a higher bond order suggests a stronger bond with more shared electron pairs.

Bond Length

Bond length is the distance between the nuclei of two bonded atoms. Picture it as the length of the bridge of electrons connecting the atoms. Typically, the greater the number of shared electrons (higher bond order), the shorter the bond length because the atoms are more strongly pulled together.

Several factors can affect bond length:

• Higher Bond Order: More electron bridges mean atoms are pulled closer, resulting in shorter bonds.
• Atomic Sizes: Larger atoms have longer bond lengths.
• Electrical Effects: The nature of the atoms involved can alter bond lengths.

A good rule of thumb is: the stronger the bond, the shorter the bond length. This inverse relationship stems from the greater attractive forces pulling the atoms closer together.

Bond Energy

Bond energy is the measurement of the strength of a chemical bond. It's defined as the energy required to break one mole of bonds in the gas phase. Bond energy gives an insight into how much power is needed to pull those atoms apart.

A higher bond energy does two things:

• Indicates a more robust, stable bond: More energy required to break it means a stronger bond.
• Reflects higher bond order: As with bond length, increased bond order often leads to higher bond energy.

Generally, stronger and shorter bonds (due to high bond order) will have high bond energy, indicating a significant amount of energy is needed to break them. Understanding these concepts allows for the prediction of molecular behavior and properties, playing a crucial role in the realm of chemistry.