Question

1. Discuss the change in potential energy when a molecule is formed from two atoms with the help of a diagram.

1. Explain the formation of bonding and antibonding molecular orbitals in the hydrogen molecule.

2. What are sp, sp² and sp³ hybridization? Use carbon-based molecules to explain.

3. Discuss the Fermi energy level of metals at absolute zero. Why carrier occupancy change above absolute zero?

Answer 1

Initially, when the two H atoms are far apart, there is no interaction. We say that the potential energy of this system (that is, the two H atoms) is zero. We recall that an object has potential energy by virtue of its position. As the atoms approach each other, each electron is attracted by the nucleus of the other atom; at the same time, the electrons repel each other, similarly the nuclei also repel each other. While the atoms are still separated, attraction is stronger than repulsion, so that the potential energy of the system decreases (that is, it becomes negative) as the atoms approach each other.

                        

4 1 0 1 Energy Energy absorbed released when bond when bond -100 forms -Bond Energy) breaks (+Bond Energy) -200 2 -300 4 -400 -432 -500 100 200 74 (H2 bond length) Internuclear distance (pm) Potential energy (kJ/mol)

Fig: The potential energy curve for the formation of H₂ molecule as a function of inters nuclear distance of the H atoms. The minimum in the curve corresponds to the most stable state of H₂.

This trend continues until the potential energy reaches a minimum value. At this point, when the system has the lowest potential energy, it is most stable. This condition corresponds to substantial overlap of the 1s orbitals and the formation of a stable H₂ molecule. If the distance between nuclei were to decrease further, the potential energy would rise steeply and finally becomes positive as a result of the increased electron-electron and nucleus-nucleus repulsions.

In accord with the law of conservation of energy, the decrease in potential energy as a result of H₂ formation must be accompanied by a release of energy. Experiments show that as a H₂ molecule is formed from two H atoms, heat is given off. The converse is also true. To break a H – H bond, energy must be supplied to the molecule.

Thus, valence bond theory gives a clearer picture of chemical bond formation than the Lewis theory does. Valence bond theory states that a stable molecule forms from reacting atoms when the potential energy of the system has decreased to a minimum; but the Lewis theory ignores energy changes in chemical bond formation.

a)  

Bonding and Antibonding Molecular Orbitals:

Let us consider H₂ molecule. According to MO theory, the overlap of the 1s orbitals of two hydrogen atoms leads to the formation of two molecular orbitals, one is bonding molecular orbital and another one is antibonding molecular orbital. A bonding molecular orbital has lower energy and greater stability than the atomic orbitals from which it was formed. An antibonding molecular orbital has higher energy and lower stability than the atomic orbitals. Placing electrons in a bonding molecular orbital yields a stable covalent bond, whereas placing electrons in an antibonding molecular orbital results an unstable bond.

In the bonding molecular orbital, the electron density is greatest between the nuclei of the bonding atoms. In the antibonding molecular orbital, on the other hand, the electron density decreases to zero between the nuclei. We can understand this distinction in the background that electrons in orbitals have wave characteristics. Waves of the same type or different type can interact in such a way that the resultant wave has either enhanced amplitude or diminished amplitude. In the former case, we call the interaction constructive interference; in the later case, it is destructive interference.

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Fig: constructive and destructive interferences

The formation of bonding molecular orbitals corresponds to constructive interference (buildup of electron density between the two nuclei). The formation of antibonding molecular orbitals corresponds to destructive interference (decrease in electron density between the two nuclei). In the

formation of bonding molecular orbital, the two electron waves of the bonding atoms reinforce each other due to constructive interference while in the formation of antibonding molecular orbital, the electron waves cancel each other due to destructive interference. As a result, the electron density in a bonding molecular orbital is located between the nuclei of the bonded atoms because of which the repulsion between the nuclei is very less while in case of an antibonding molecular orbital, most of the electron density is located away from the space between the nuclei. Infact, there is a nodal plane (on which the electron density is zero) between the nuclei and hence the repulsion between the nuclei is high. Electrons placed in a bonding molecular orbital tend to hold the nuclei together and stabilize a molecule. Therefore, a bonding molecular orbital always possesses lower energy than either of the atomic orbitals that have combined to form it. In contrast, the electrons placed in the antibonding molecular orbital destabilize the molecule. This is because the mutual repulsion of the electrons in this orbital is more than the attraction between the electrons and the nuclei, which causes a net increase in energy.

The constructive and the destructive interactions between the two 1s orbitals in the H₂ molecule lead to the formation of a bonding molecular orbital σ_1s and an antibonding molecular orbital , here the star denotes an antibonding molecular orbital. In a sigma molecular orbital (bonding or antibonding) the electron density is concentrated symmetrically around a line between the two nuclei of the bonding atoms. Two electrons in a sigma molecular orbital form a sigma bond. We remember that a single covalent bond (such as H – H or F – F) is almost always a sigma bond.

Figure below shows the molecular orbital energy level diagram that is, the relative energy levels of the orbitals produced in the formation of the H₂ molecule and the constructive and destructive interference between the two 1s orbitals.

Destructive Antibonding sigma (o) Molecule interference Molecular orbital Atom Atom 1s 1s Constructive Bonding sigma (a interference Molecular orbital н-atom H-atom (а) (b) H2-molecule Molecular orbital energy diagram Energy