Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

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 E_c of molecule 1 and the induced dipole moment μ_id in molecule 2 can be calculated as

(3.11)

and is a contribution to the total intermolecular interaction.

The field of a dipole moment μ₁ is given by

The interaction between the field E_d of molecule 1 and the induced dipole moment μ_id of molecule 2 can be calculated as

(3.12)

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.

Problem 3.5

Show that the molecular radius r_pol, 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/16πN_A)^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.

Problem 3.6

Verify the expressions for the energy W_c-id and the energy W_d-id.