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.

Short answer

(a) Bond order is the number of chemical bonds between a pair of atoms, and as it increases, the bond length decreases, and the bond energy increases. There is an inverse relationship between bond order and bond length, while both bond order and bond energy are directly proportional. (b) Using molecular orbital theory, neither Be₂ nor Be₂⁺ are expected to exist. Be₂ has a bond order of 0, indicating no bond and making it unstable. Be₂⁺ has a weak bond with a bond order of 0.5, not strong enough to overcome the electrostatic repulsion between nuclei, making it non-existent as well.

Step-by-step solution

Part (a): Relationship among Bond Order, Bond Length, and Bond Energy

1. Bond Order: Bond order is the number of chemical bonds between a pair of atoms in a molecule. It is calculated as the difference between the number of bonding electrons and the number of antibonding electrons, divided by 2. A higher bond order implies a stronger bond. 2. Bond Length: Bond length is the distance between the nuclei of two bonded atoms in a molecule. As the bond order increases, the bond becomes stronger, and the two nuclei are pulled closer to each other. Therefore, there is an inverse relationship between bond order and bond length - as bond order increases, bond length decreases. 3. Bond Energy: Bond energy is the energy required to break a chemical bond and form neutral atoms. A higher bond order corresponds to a stronger bond, which means more energy will be required to break it. So, bond energy and bond order are directly proportional. In summary, as bond order increases, the bond length decreases, and the bond energy increases.

Part (b): Predicting the Existence of Be₂ and Be₂⁺ using Molecular Orbital Theory

1. Be₂ Electronic Configuration: Be has an electronic configuration of [He] 2s². When two Be atoms come together to form a Be₂ molecule, the 2s orbitals of both Be atoms will combine to form bonding (σ) and antibonding (σ*) molecular orbitals. Thus, the molecular orbital configuration for Be₂ is (σ₂s)²(σ₂s*)². 2. Be₂ Bond Order Calculation: The bond order for Be₂ can be calculated as (number of electrons in bonding orbitals - number of electrons in antibonding orbitals) / 2 = (2 - 2) / 2 = 0 3. Existence of Be₂: Since the bond order for Be₂ is 0, it means that there's essentially no bond, and hence Be₂ is not stable and not expected to exist. 4. Be₂⁺ Electronic Configuration: For Be₂⁺, one electron is removed from the antibonding orbital. So, the molecular orbital configuration becomes (σ₂s)²(σ₂s*)¹. 5. Be₂⁺ Bond Order Calculation: The bond order for Be₂⁺ is (2 - 1) / 2 = 0.5 6. Existence of Be₂⁺: In this case, the bond order for Be₂⁺ is 0.5, meaning that there's a weak bond. However, this weak bond is not enough to overcome the electrostatic repulsion between the two Be nuclei. Therefore, Be₂⁺ is also not expected to exist. In conclusion, according to molecular orbital theory, neither Be₂ nor Be₂⁺ are expected to exist due to their bond orders.

Key concepts

Bond Order

Bond order refers to the number of chemical bonds between a pair of atoms. It can be defined by subtracting the number of antibonding electrons from the number of bonding electrons, then dividing by two. Imagine it like this: the more pairs of electrons holding two atoms together, the stronger that bond is.

Higher bond orders imply stronger, more stable bonds. Why? Because there are more electron pairs pulling the atoms closer. This is very useful when predicting the stability of a molecule or its chemical properties.

It's like getting a bigger band to keep two beach balls together. A single band might do the trick, but two or three bands will keep them glued much more firmly. As bond order increases, it directly impacts both bond length and bond energy, which we'll cover next.

Bond Length

Bond length is simply the distance between the nuclei of two bonded atoms. A useful analogy is a seesaw game – higher bond order causes stronger attraction, pulling the nuclei closer for a shorter bond length.

Generally, as the bond order increases, the atoms are held more tightly together, reducing the bond length. This can be seen experimentally: a triple bond, which has a bond order of 3, is shorter than a double bond of the same type of atoms, which itself is shorter than a single bond.

In summary, understanding bond length helps chemists predict how molecules will interact physically with each other, and can be crucial in understanding reaction dynamics and physical properties of substances.

Bond Energy

Bond energy represents the energy required to break a chemical bond and separate atoms in a molecule. It is intimately related to bond order, as higher bond orders generally mean greater bond energy.

Think of it as the effort needed to pull apart glued pieces of paper: the stronger the glue (or the more glue you use), the harder it is to separate. Similarly, higher bond orders result in higher bond energies, making bonds harder to break.

From a practical standpoint, understanding bond energy is vital for any chemical reaction, as it directly affects reaction conditions, such as temperature and catalysts, required to break bonds and form new ones.

In essence, bond energy gives insight into the strength and stability of chemical bonds, crucial for everything from cooking to constructing materials. Understanding how bond energy works helps chemists design and predict reactions efficiently.