Understanding Intermolecular Interactions

Intermolecular interactions are forces that exist between molecules. These forces determine many physical properties of substances, such as boiling points, melting points, and solubility. The strength of these interactions depends on various factors, including the distance between the interacting species. Let's examine the distance dependency for three types of interactions:

1. Ion-pair interaction
2. Ion-dipole interaction
3. Dipole-dipole interaction

Ion-pair Interaction Energy

An ion-pair interaction occurs between two oppositely charged ions. The energy of interaction for an ion pair is described by Coulomb's Law. The potential energy ($E$) between two ions with charges $q_1$ and $q_2$ separated by a distance $r$ in a medium with permittivity $\epsilon$ is given by:

\( E \propto \frac{q_1 q_2}{\epsilon r} \)

This equation shows that the energy of interaction for an ion pair is inversely proportional to the distance \(r\) between the ions. A shorter distance leads to a stronger (more negative) interaction energy.

\( E_{\text{ion-pair}} \propto \frac{1}{r} \)

Ion-dipole Interaction Energy

An ion-dipole interaction occurs between a charged ion and a molecule with a permanent dipole moment. The energy of this interaction depends on the charge of the ion, the magnitude of the dipole moment, and the distance and orientation between the ion and the dipole. For a fixed orientation, the electric field ($E'$) produced by the ion at a distance \(r\) is proportional to \(1/r^2\).

\( E' \propto \frac{|q|}{r^2} \)

The potential energy of a dipole (\(\mu\)) in an electric field ($E'$) is proportional to $E'$. Therefore, the energy of the ion-dipole interaction is generally considered to be inversely proportional to the square of the distance \(r^2\), especially when considering the electric field produced by the ion at the dipole's location.

\( E_{\text{ion-dipole}} \propto \frac{1}{r^2} \)

Dipole-dipole Interaction Energy

A dipole-dipole interaction occurs between two molecules with permanent dipole moments. The energy of interaction depends on the magnitude of the dipole moments and their relative orientation. For two fixed, interacting dipoles, the energy is generally considered to be inversely proportional to the cube of the distance \(r^3\) between them.

\( E_{\text{dipole-dipole (fixed)}} \propto \frac{1}{r^3} \)

It is worth noting that for freely rotating dipoles in liquids or gases (where the orientation averages out), the interaction energy is inversely proportional to \(r^6\) (Keesom forces). However, based on the provided options, the dependency \(1/r^3\) is being considered for dipole-dipole interaction.

Summary of Distance Dependency

Let's summarize the inverse proportionality of interaction energies with distance \(r\):

Comparing these dependencies with the given options, the interaction energies are inversely proportional to \(r\), \(r^2\), and \(r^3\) for ion-pair, ion-dipole, and dipole-dipole interactions respectively.