Animated Solution for Chemistry - States of Matter: The intermolecular interaction that is dependent on the inverse cube of distance between the molecules is

When you encounter a question asking about the distance dependency of an "intermolecular interaction," your first instinct might be to jump straight to the formulas you memorized. However, there is a subtle trap here. In the realm of physics and physical chemistry, the word interaction is notoriously ambiguous. It can refer to either the Interaction Force ($F$) or the Interaction Potential Energy ($U$).

Because force is the negative spatial derivative of potential energy ($F=-\frac{dU}{dr}$), their distance dependencies are always off by one power of $r$. If the energy varies as $1/r^n$, the corresponding force will vary as $1/r^{n+1}$. To safely navigate this question, we must act like detectives and analyze both the force and the energy for every single option provided.

Let's systematically break down the four types of interactions mentioned in the options.

1. Ion-Ion Interaction This is the most fundamental electrostatic interaction, governed by Coulomb's Law. The force between two point charges $q_1$ and $q_2$ separated by a distance $r$ is given by: Thus, the force is proportional to $1/r^2$. The corresponding potential energy is proportional to $1/r$. Neither of these is the inverse cube ($1/r^3$) we are looking for.

2. Ion-Dipole Interaction Imagine a permanent dipole (like a water molecule) and an ion (like a sodium ion, ${\text{Na}}^{+}$). A dipole creates an electric field in the space around it. Unlike a point charge whose field drops off as $1/r^2$, the electric field of a dipole drops off faster because the fields from its positive and negative ends partially cancel each other out. The magnitude of this electric field $E$ at a distance $r$ is proportional to $1/r^3$.

When an ion of charge $q$ is placed in this field, it experiences a force $F=qE$. Therefore, the interaction force is: The corresponding interaction energy, obtained by integrating the force, is proportional to $1/r^2$.

3. Dipole-Dipole Interaction (and Hydrogen Bonds) When two permanent dipoles interact, the situation becomes highly dependent on their orientation. For two stationary dipoles, the interaction energy is proportional to $1/r^3$. Because a hydrogen bond is essentially a very strong, highly directional dipole-dipole interaction, its energy also scales as $1/r^3$.

However, the force between these two dipoles (the derivative of the energy) will scale as $1/r^4$.

4. London Dispersion Forces These are the weakest intermolecular forces, arising from instantaneous, temporary dipoles in otherwise non-polar electron clouds. Quantum mechanics dictates that the interaction energy for London dispersion forces is universally proportional to $1/r^6$. Consequently, the force scales as $1/r^7$.

Force vs. Energy

We have arrived at a crossroads. The question asks for the interaction that depends on $1/r^3$.

- If "interaction" means Force, the correct answer is the Ion-Dipole interaction. - If "interaction" means Energy, the correct answer is the Hydrogen bond (Dipole-Dipole).

So, which path do we take? In the context of this specific JEE Main 2015 question, the examiners intended for "interaction" to mean the fundamental electrostatic force field created between the entities. The official answer key validated (b) ion-dipole interaction as the correct choice.

This serves as a crucial lesson for competitive exams: when faced with an ambiguous term like "interaction" where multiple options could technically be correct depending on the interpretation, prioritize the fundamental force expression unless "energy" or "potential" is explicitly stated.