# Harmonic Potentials and energy state populations

• DanPhysChem
In summary, the conversation discusses a potential energy surface with two minima, labeled A and B. Each well has a corresponding force constant, kA and kB respectively. The question at hand is how small kB must be compared to kA for the higher energy state (B) to be more likely at a given temperature. The speaker is seeking help and has not received a response on the site.

#### DanPhysChem

Hi All - Let's say I have a potential energy surface with two minima, one higher than the other. We will label the higher minima B and the lower minima A. Each potential well can be approximated as a harmonic potential with force constants kA and kB for wells A and B respectively. How small must kB be in terms of kA in order for that state (higher energy) to be the more likely state at temperature T?

Sry to bump my own thread. I'm really dying on this one here, any help is much appreciated.

Is anyone out there? I've yet to get a response to anything I've posted on this site :-(

## 1. What is a harmonic potential?

A harmonic potential is a mathematical function used to describe the potential energy of a system, such as a molecule or an atom. It is characterized by a parabolic shape, and it represents the restoring force of a system, which brings it back to equilibrium when it is disturbed.

## 2. How do harmonic potentials affect energy state populations?

Harmonic potentials play a significant role in determining the energy state populations of a system. As the potential energy increases, the energy states become more closely spaced, causing the population of higher energy states to decrease. This relationship is described by the Boltzmann distribution, which states that the population of a given energy state is proportional to e^(-E/kT), where E is the energy of the state, k is the Boltzmann constant, and T is the temperature.

## 3. What are some real-world examples of systems with harmonic potentials?

Harmonic potentials can be observed in various physical systems, such as a mass-spring system, a vibrating diatomic molecule, or a single atom trapped in a laser beam. They are also used to model the energy levels of electrons in a crystal lattice, known as phonons, and the vibrational modes of molecules.

## 4. How do temperature and potential energy affect the shape of a harmonic potential?

The shape of a harmonic potential is dependent on both temperature and potential energy. As temperature increases, the potential becomes wider and shallower, while as potential energy increases, the potential becomes narrower and steeper. This is because temperature and potential energy affect the average distance between particles in a system, which in turn affects the restoring force and shape of the potential.

## 5. Can harmonic potentials be used to study phase transitions?

Yes, harmonic potentials can be used to study phase transitions, such as melting or boiling, in a system. As the potential energy of the system increases, the particles become more energetic and can overcome the intermolecular forces holding them together, resulting in a change of phase. The shape of the potential also changes at the phase transition point, indicating a change in the system's behavior.