Chap14 solutions

Physical Chemistry

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14 Molecular structure Solutions to exercises Discussion questions E14.1(b) Consider the case of the carbon atom. Mentally we break the process of hybridization into two major steps. The first is promotion, in which we imagine that one of the electrons in the 2 s orbital of carbon (2 s 2 2 p 2 ) is promoted to the empty 2 p orbital giving the configuration 2 s 2 p 3 . In the second step we mathematically mix the four orbitals by way of the specific linear combinations in eqn 14.3 corresponding to the sp 3 hybrid orbitals. There is a principle of conservation of orbitals that enters here. If we mix four unhybridized atomic orbitals we must end up four hybrid orbitals. In the construction of the sp 2 hybrids we start with the 2 s orbital and two of the 2 p orbitals, and after mixing we end up with three sp 2 hybrid orbitals. In the sp case we start with the 2 s orbital and one of the 2 p orbitals. The justification for all of this is in a sense the first law of thermodynamics. Energy is a property and therefore its value is determined only by the final state of the system, not by the path taken to achieve that state, and the path can even be imaginary. E14.2(b) It can be proven that if an arbitrary wavefunction is used to calculate the energy of a system, the value calculated is never less than the true energy. This is the variation principle. This principle allows us an enormous amount of latitude in constructing wavefunctions. We can continue modifying the wavefunctions in any arbitrary manner until we find a set that we feel provide an energy close to the true minimum in energy. Thus we can construct wavefunctions containing many parameters and then minimize the energy with respect to those parameters. These parameters may or may not have some chemical or physical significance. Of course, we might strive to construct trial wavefunctions that provide some chemical and physical insight and interpretation that we can perhaps visualize, but that is not essential. Examples of the mathematical steps involved are illustrated in Sections 14.6(c) and (d), Justification 14.3, and Section 14.7. E14.3(b) These are all terms originally associated with the Huckel approximation used in the treatment of con- jugated π -electron molecules, in which the π -electrons are considered independent of the σ -electrons. π -electron binding energy is the sum the energies of each π -electron in the molecule. The delocaliza- tion energy is the difference in energy between the conjugated molecule with n double bonds and the energy of n ethene molecules, each of which has one double bond. The π -bond formation energy is the energy released when a π bond is formed. It is obtained from the total π -electron binding energy by subtracting the contribution from the Coulomb integrals, α .
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