Deduce the relation \(\left\langle r_{i}^{2}\right\rangle=N l^{2}\) for the mean square distance of a monomer from the origin in a freely jointed chain of N units each of length l. Hint: Use the distribution in eqn 14D.4.

Text Transcription:

langle r_i^2 rangle=N l^2

VSEPR Valence Shell Electron Pair Repulsion Theory (VSEPR) is a qualitative theory that allows us to predict the geometry of molecules based upon their Lewis structures. This model looks at the repulsion of different electron regions (either bonding or non-bonding) and chooses a geometry that minimizes their repulsions. Valence Bond Theory The valence bond model is a quantum mechanical model for understanding bonding in molecules. However, now it is mostly used by chemists as a means to describe the bonding with organic systems qualitatively. The idea is that bonds form between atoms as a result of the overlap of atomic orbitals. This leads to what we would call a localized bonding picture. You can point to a bond in a molecule and say what types of atomic orbitals lead to that particular bond. The only complication is that the atomic orbitals we know and love cannot lead to the geometries that we know molecules have. Thus, we have to invent new "hybrid" orbitals by making combinations of the atomic orbitals we started with (s,p,d,f orbitals). Molecular Orbital Theory Molecular Orbital theory (MO) is the most important quantum mechanical theory for describing bonding in molecules. It is an approximate theory (as any theory that utilizes "orbitals"), but it is a very good approximation of the bonding. The MO perspective on electrons in molecules is very different from that of a localized bonding picture such as valence bond (VB) theory. In VB we describe particular bonds as coming from the overlap of orbitals on atomic centers. In MO this idea is not completely g