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

### 10. Describing the Behavior of Liquids: Polar Liquids

### 10.2. Towards a Microscopic Interpretation

So, an applied field *E* introduces into a material a polarization *P* = *ε*_{0}(*ε*_{r} − 1)*E*., and our task is now to interpret the polarization *P* in microscopic terms. In general, we have several molecular contributions to the polarization. First, if the molecules are polar and have a dipole moment^{3)}*μ*, we have the alignment of these permanent dipoles due to the applied field. This process results in a polarization contribution labeled *P _{μ}*. Second, even if we have nonpolar molecules, all molecules have an electronic polarizability

^{4)}*α*

_{ele}and molecular polarizability

*α*

_{mol}, collectively denoted by

*α*, that results in a dipole moment induced by the applied field. This contribution is labeled

*P*. The field experienced by the molecules is

_{α}__not__equal to the applied field

*E*as the surrounding molecules shield the reference molecule from the applied field, and a polar reference molecule will influence the surroundings leading to a different shielding than for a nonpolar molecule. If we have

*N*molecules in a volume

*V*, we may write

*P*= (

_{μ}*N*/

*V*)

*p*and

_{μ}*P*= (

_{α}*N*/

*V*)

*p*. Here,

_{α}*p*represents the (rotational) average dipole moment , which is dependent on the

_{μ}*directional fieldE*

_{dir}experienced by the molecules. For

*p*we have

_{α}*p*=

_{α}*αE*

_{int}, where

*E*

_{int}is the

*internal field*. Although both fields

*E*

_{dir}and

*E*

_{int}are proportional to the applied field

*E*, they are not the same because a proportional increase in all three components of

*E*will affect the magnitude of the induced dipole moment but not the directional effect for the permanent dipole. Hence, our task is to calculate

*E*

_{int},

*E*

_{dir}and so that

*P*and

_{μ}*P*can be determined.

_{α}Altogether, in microscopic terms polarization can be interpreted as the number of molecules *N* with (effective) dipole moment *μ*_{eff} in a volume *V*, that is

(10.9)

Overall, the effective dipole moment is described by *μ*_{eff} = *α*_{eff}*E*_{loc}, where *α*_{eff} is the *effective polarizability* of the molecule and *E*_{loc} is the *local field*, the latter being the appropriate combination of the internal and directional field. We deal first with gases and thereafter with liquids and solutions.

Problem 10.3

The relative permittivity *ε*_{r} can be accurately described by *ε*_{r}(*T*) = *ε*_{r,0}exp(−*LT*), with as parameters *ε*_{r,0} and *L*. Calculate *ε*_{r}(298) for the three liquids indicated (see __Table 10.1__).

** Table 10.1** Parameters

*ε*

_{r,0}and

*L*for H

_{2}O, CH

_{3}OH, and C

_{2}H

_{5}OH.