Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
3. Basic Energetics: Intermolecular Interactions
3.3. Induction Interaction
Apart from the electrostatic interactions between permanent charges (c) and dipoles (d), a charge in molecule 1 will also induce a dipole moment in molecule 2 (as will a dipole in molecule 1). The induced dipole (id) in molecule 2 is proportional to the field generated by the charge (or the dipole) of molecule 1 with as proportionality factor the (isotropic5)) polarizability6) α2 of molecule 2, that is, μ2 = α2E1.
The field of a charge q1 is given by
The interaction between the field Ec of molecule 1 and the induced dipole moment μid in molecule 2 can be calculated as
and is a contribution to the total intermolecular interaction.
The field of a dipole moment μ1 is given by
The interaction between the field Ed of molecule 1 and the induced dipole moment μid of molecule 2 can be calculated as
using [3(μ·r)r−r2μ]2 = (1 + 3cos2θ)μ2r4 and . This interaction is denoted as the Debye interaction and is always attractive. Since the Helmholtz energy W is independent of temperature (at least as derived here for an isotropic polarizability α), it also represents the internal energy U. Both interactions are due to the field of molecule 1 (originating either from its charge or dipole moment) and the polarizability of molecule 2. For the total interaction we have to add the interaction between the field of molecule 2 and polarizability of molecule 1.
Show that the molecular radius rpol, as estimated from the polarization volume α′ using the information of footnote 6, equals about 0.85(σ/2) as estimated from the van der Waals constant b. Derive first the relation σ/2 = (3b/16πNA)1/3, where σ/2 is the vdW radius, and use vdW and polarizability data as given in Appendix E, for example, for CH4, N2, and Xe.
Verify the expressions for the energy Wc-id and the energy Wd-id.