Helium Potential Energy Curve — Worksheet 1.7 Notes

Potential Energy Curve: Key Ideas

Topic: Worksheet 1.7 discussion of a helium potential energy curve (V(r)) for two atoms as they approach, interact, and separate.
Axes: x-axis = internuclear distance (r); y-axis = potential energy (V).
Reading direction: view the graph from right to left as the internuclear distance decreases (atoms move closer together).

At very large internuclear distance (far apart): the electrostatic attraction is the strongest force between the atoms. This is the initial driving force that brings the atoms closer.
When the atoms reach the potential energy minimum (the bottom of the well): the attractive and repulsive electrostatic forces are equal in magnitude (the forces balance).
If the atoms are moved apart from the minimum: the force becomes attractive again, pulling them back toward the minimum.
If the atoms are pushed closer together than the minimum: the force becomes repulsive, pushing them apart.
At the bottom of the well:
- There will be some vibration about the equilibrium distance (bond length), but the atoms stay bound.
- It takes energy to break the interaction and allow the system of two atoms to escape from the potential well (dissociation).
The best answer for the force balance at the minimum is that the attractive and repulsive forces are equal (net force = 0).

As distance decreases toward the minimum, the overall potential energy decreases (stabilization) due to attractive interactions.
At the bottom (r = r0, the equilibrium distance), the forces balance and the total force is zero.
As distance decreases further beyond r0 (atoms pushed very close together), the potential energy increases because electron-electron and nucleus-nucleus repulsions dominate.
When distance is large, the attractive forces dominate enough to pull atoms toward each other; as r approaches r0, the gradient of V(r) approaches zero, and F = -dV/dr becomes zero at the minimum.

The x-axis (internuclear distance) and y-axis (potential energy) together describe a potential energy well that binds the two atoms.

Practice question: What can the depth tell you? The depth tells you how much energy is involved in the interaction (how strong the interaction is) and how much energy is needed to remove the atoms from the well (dissociation energy).
Key takeaway: The deeper the well, the stronger the interaction and the higher the energy required to dissociate; lower energy corresponds to a more stable bonded state.
If V(∞) is chosen as the zero reference, the well depth is ΔV = V(r0) - V(∞). Often V(∞) = 0, so ΔV = V(r0) (with V(r0) < 0 for a bound state).
Dissociation energy: E_diss = |ΔV| = |V(r0) - V(∞)|.

Practice question: What can the position on the x-axis tell you? It tells you the distance between the atom centers at the most stable point, i.e., the equilibrium internuclear distance r0 (the bond length).
The minimum position r0 indicates the bond length where the two atoms are most stable together.
The x-axis is labeled as the internuclear distance, so any point along it corresponds to a distance between atomic centers.
At r0, the system is in its most stable configuration; disrupting it requires a large amount of energy.

At the potential minimum (r = r0):
- The net force is zero: F(r0) = 0.
- This occurs because dV/dr|_{r0} = 0 and F(r) = -dV/dr.
General relation: the total force between the atoms is the negative gradient of the potential energy: F(r) = -\frac{dV}{dr}.
Near the equilibrium distance, one can approximate the potential with a harmonic (parabolic) potential:
- V(r) \approx V(r0) + \frac{1}{2} k (r - r0)^2,\quad k = \left.\frac{d^2 V}{dr^2}\right|{r0} > 0.
- This describes small vibrations about the bond length (bonding oscillations).

The depth of the well relates to bond strength and the energy required to break a bond; deeper wells correspond to stronger bonds and higher dissociation energy.
The equilibrium internuclear distance r0 corresponds to the typical bond length observed for that diatomic system.
Understanding these concepts helps explain molecular stability, reaction energetics, and why certain bonds form or break under different conditions.

This discussion ties into the general idea that chemical bonds are a balance of attractive and repulsive forces at the electronic level, represented by a potential energy surface.
The concept of a potential well with a minimum is foundational to vibrational spectroscopy, where small oscillations about r0 give rise to quantized vibrational energy levels.
The location and depth of the well are related to bond strength, reaction energetics, and material properties in chemistry and materials science.

Question: What can the depth of the potential energy well tell you?
- Answer: b) The depth indicates the energy scale of the interaction; a deeper well means a stronger interaction and a larger energy required to dissociate the atoms, i.e., greater stability.
Question: What can the position on the x-axis tell you?
- Answer: a) The position of the well minimum on the x-axis gives the internuclear distance between atom centers at the most stable point (the bond length).
Note:
- At the minimum, the attractive and repulsive forces balance, so the total force is zero.
- The bottom of the well corresponds to the most stable configuration with vibrational motion possible around r0.