Understanding Diatomic Molecule Potential Energy Relation

The potential energy of a diatomic molecule describes the energy associated with the interaction between its two constituent atoms as a function of the distance ($r$) separating them.

This interaction energy is generally modeled as a sum of two main components:

• Attractive Term: This term arises from forces like van der Waals interactions and becomes significant at intermediate and larger distances. It leads to a decrease in potential energy as atoms approach, stabilizing the molecule. It is typically represented as a negative inverse power function of distance: $ U_{attr} = -\frac{a}{r^m} $, where $a$ and $m$ are positive constants.
• Repulsive Term: This term becomes dominant at very short distances due to the overlap of electron clouds and the Pauli exclusion principle, preventing the atoms from merging. It leads to a sharp increase in potential energy as atoms get too close. It is typically represented as a positive inverse power function of distance: $ U_{rep} = +\frac{b}{r^n} $, where $b$ and $n$ are positive constants.

The total potential energy ($U$) is the sum of these two terms:

This mathematical form captures the essential features of the interaction potential: attraction at longer ranges and repulsion at shorter ranges, leading to a minimum potential energy at an equilibrium bond distance.

Selecting the Correct Potential Energy Relation

We examine the given options against the established model:

• Option 1: $ -\frac{a}{r^m} + \frac{b}{r^n} $ - This option correctly combines a negative (attractive) term with a positive (repulsive) term, consistent with the physical model of diatomic molecular interactions.
• Option 2: $ ar^m - br^n $ - This form does not represent the typical attractive and repulsive interactions.
• Option 3: $ ar^m + br^n $ - This represents purely repulsive interactions, which is incorrect for describing the stable bonding in a diatomic molecule.
• Option 4: $ ar^m - \frac{b}{r^n} $ - This combines a positive term with a negative inverse power term, not matching the standard model.

Based on the physical principles governing interatomic forces in diatomic molecules, the potential energy relation is given by the sum of an attractive and a repulsive term, matching Option 1.