## 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 (isotropic__ ^{5)}__)

*polarizability*

^{6)}*α*

_{2}of molecule 2, that is,

*μ*_{2}=

*α*

_{2}

*E*_{1}.

The field of a charge *q*_{1} 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 *μ*_{1} 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**r*

^{2}

**]**

*μ*^{2}= (1 + 3cos

^{2}

*θ*)

*μ*

^{2}

*r*

^{4}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 = (3*b*/16π*N*_{A})^{1/3}, where *σ*/2 is the vdW radius, and use vdW and polarizability data as given in __Appendix E__, for example, for CH_{4}, N_{2}, and Xe.

Problem 3.6

Verify the expressions for the energy *W*_{c-id} and the energy *W*_{d-id}.