The question asks about the intermolecular interaction that depends on the inverse cube of distance (1/r³) between molecules. Let's analyze the options.

Intermolecular forces are the attractive forces between molecules. Their strength often depends on the distance (r) between them, following a power law. The key is to identify which force has a 1/r³ dependence.

Let's evaluate the options:

1. Ion-ion interaction: This is the force between two ions (e.g., in an ionic crystal like NaCl). It is governed by Coulomb's law. The potential energy (U) for two ions with charges q₁ and q₂ is given by:

The force (F) is the negative derivative of potential energy with respect to distance (F = -dU/dr).

This shows a 1/r² dependence for the force, not 1/r³.

2. Ion-dipole interaction: This is the force between an ion and a polar molecule (a molecule with a permanent dipole moment, μ). The potential energy for this interaction is:

Taking the derivative to find the force:

This shows a clear 1/r³ dependence for the force.

3. London force (Dispersion force): These are weak, temporary attractive forces between all molecules, including non-polar ones, caused by instantaneous dipoles. The potential energy for London dispersion forces is:

The force, therefore, has a 1/r⁷ dependence, which is a much stronger dependence on distance than 1/r³.

4. Hydrogen bond: This is a special type of dipole-dipole interaction. While it is a strong force, its distance dependence is complex but is generally considered to be similar to other dipole-related interactions, often approximated with a 1/r⁴ to 1/r⁶ dependence for the potential energy, leading to a force dependence stronger than 1/r³.

Conclusion: The mathematical derivation shows that the force in an ion-dipole interaction is directly proportional to the inverse cube of the distance (1/r³). This makes it the correct answer.

Related Topics & Formulae

Intermolecular Forces: These are forces of attraction between molecules. They are weaker than intramolecular forces (like covalent bonds) but determine many physical properties like boiling point, melting point, and solubility. The major types are ion-dipole, dipole-dipole, hydrogen bonding, and London dispersion forces.

Key Formulae:

• Ion-Ion Force (Coulomb's Law): $F=\frac{1}{4\pi {ϵ}_0}\frac{q_1{q}_2}{r^2}$ (F ∝ 1/r²)
• Ion-Dipole Force: $F\propto \frac{1}{r^3}$
• Dipole-Dipole Force: Potential Energy $U\propto \frac{1}{r^3}$, leading to F ∝ 1/r⁴.
• London Force: Potential Energy $U\propto \frac{1}{r^6}$, leading to F ∝ 1/r⁷.

Understanding the distance dependence of these forces helps predict the strength of interactions in solutions, especially in chemistry involving solvents and electrolytes.