Electrical Properties of Molecules

Electrical properties can arise from the interactions between positively charged nuclei and negatively charged electrons with each other and with externally applied electric fields.  The former gives rise to electric dipoles, etc., while the latter gives us properties such as refractive index, optical activity, etc.

Electrical Dipole Moments

An electric dipole is created when two differing charges interact with each other.  If the two charges are +q and -q, and the vector between them is R then the dipole moment, given by vector μ, is

This direction of R  is quite arbitrary and some people prefer to define it from positive to negative, as I have done in the past in first-year lectures.   To keep consistent with the text, we will go with the direction from the negative charge to the positive charge as the + direction.

 The SI unit for a dipole moment is Cm, for Coulomb metres but we commonly use Debye.   1D = 3.33564×10^-30 Cm.

1D is the dipole from two charges +e and -e, separated by a distance of 1 Å.  Dipole moments of small molecules are typically about 1 D, hence, the unit is useful as a convenience when discussing molecular dipoles.

For simple diatomic molecules, the dipole of the molecule is simply the same as the dipole of the one bond.  However, in polyatomic molecules, the molecular dipole is calculated as the vector sum of all the individual bond dipoles (as a first approximation).  In order to be able to complete this type of analysis, we need to know the full geometry of the molecule, in three dimensions.  VSEPR gives us a first-order estimate of molecular geometries, from which we can make preliminary estimates of molecular dipoles.  For example, water is V-shaped and has bonds that are polar (since O and H have different charges).  Thus, the two bond-dipoles will add up to an overall molecular dipole.

Carbon dioxide, however has a linear geometry O = C = O.  In this case, the two bond dipoles that arise because the O and the C had different charges will cancel each other out, upon vector addition. 

In general, for a molecule with only two dipoles, we can add the two vectors using the following equation.

In general, it is better to re-define the way we calculate the overall (residual) dipole moment.  We begin by defining a Cartesian coordinate system with a centre at the mass-centre of the molecule.  Then, we can define a single component of the residual vector as:

where index i runs over all the centres of charge (generally, the atoms) in the molecule.  Then, we can derive the overall molecular dipole by taking the sum of the three component vectors according to Pythagoras.

Some questions to consider?

1. What would happen if we defined the centre of our co-ordinate system at some location not at the mass centre? 

2. This system involves the addition of point charges.  How would we take into account the fact that the charges are generally distributed in 3-d space about each atom, and even about the whole molecule?

3. The existence of zero-point vibrational energies tell us that the centre of charge will never stay in the one spot. 

   1. How can we take this into account?

   2. What nomenclature might we give to the dipole moment as calculated using the fixed positions? 

Macroscopic properties generally involve large numbers of such polar molecules.  We define the term polarization of the medium to refer to a property of an assembly of such molecules such that polarization P is the product of the mean electric dipole moment of the molecules <μ> and the number density N .

In an isotropic fluid, all the molecular vectors assume random orientations and thus, the average molecular dipole <μ>  is zero.  Thus, the polarization of an isotropic fluid is zero.  However,  in an non-isotropic environment, the fluid be given a non-zero polarization.  If we impose an external electric field on the fluid, the molecular dipoles will tend to align in certain orientations to achieve a minimum in energy.  Thus, the fluid will be no longer isotropic and it will have a non-zero average molecular dipole <μ>.  The amount of orientational effects that the external field exerts on the individual dipole moments μ = |μ| depends on the strength of the field E and on the thermal energy of the molecules as follows:

where z is the direction of the applied external field.  Thus, we see that the induced average dipole moment of the fluid points along the direction of the external electric field.

In addition to aligning the molecules, an applied electric field can also distort the molecules, thus changing their individual residual dipole moments.   The induced dipole μ^* can be determined from the field strength E and the polarizability α of the molecules.

we often express the polarizability volume α' according to:

where ε₀ is the permittivity of a vacuum.  The units of a' will be volume and the magnitude is comparable to the volumes of the molecules themselves. 

Such external field considerations as we just did all assumed a constant or at least slowly changing electric field, whereby the molecules were able to adjust to the field.   If the electric field were to change rapidly, we might loose these effects.  For example, if we apply an oscillating electric field greater than about 10¹¹ Hz (microwaves) then the molecules will not be able to rotate themselves into alignment fast enough.  Hence, the polarization from permanent dipoles becomes zero at frequencies higher than that.  Similarly, molecular distortions happen at different speeds, depending on the molecular vibrational frequencies.  Thus, as we increase the frequency of the external field through the IR band, the individual polarizations due to molecular distortion will also vanish, one vibrational mode at a time.  At very high frequencies, if there is any polarization remaining, it will be due to motion of the electrons themselves and is called the electronic polarizability.  The electronic polarizability will depend on the type of molecular orbitals within which the particular electrons are found.

One last complication involves the interaction of the individual molecular dipoles with the medium itself.  The potential energy of two charges separated by a distance r in a vacuum is given by

and the same two charges, in a medium (such as air) will be 

where e is the permittivity of the medium. 

We often express the relative permittivity of the medium (formerly called the dielectric constant) as

The relative permittivity of a substance is large if it is composed of polar or highly polarizable molecules.  It has a large effect on the strength of ionic interactions.  Water has a relative permittivity of 78 at room temp and the Coulombic interaction energy is reduced by about two orders of magnitude, compared to similar interactions in free space.  We saw one effect of the relative permittivity in equation 4.31, in the Gibbs energy of mixing.

We can determine the molar polarization of the medium, now taking into account both the molecular polarizibility and the medium as follows

and from this, we can get the Debye equation, which gives us the relationship between the relative permittivity and the molecular properties. 

For a  system of non-polar (permanent dipole of zero) molecules, we can simplify this to the Clausius-Mossotti equation.

We can derive a relation between the refractive index of a substance and it's relative permittivity using Maxwell equations.  The derived relation is n_r = ε_r^1/2.  Thus, we can measure refractive indices using visible light and then use the equations above  to determine the molar polarization P_m and the molecular polarizibility, α.