# What is the Wavelength of a Photon Emitted by HCl in a Vibrational Transition?

• Dahaka14
In summary, the question is asking for the wavelength of the emitted photon when HCl de-excites from the first vibrational state. This state corresponds to the n=1 state in the quantum harmonic oscillator model, with an energy level given by E_n = \left(n+\frac{1}{2}\right)h\bar\omega. The de Broglie wavelength equation can be used to solve for the wavelength of the emitted photon.
Dahaka14

## Homework Statement

What  is  the  wavelength  of  the  emitted  photon  when  HCl  de‐excites  from  the  first vibrational  state?

Well, I had to solve for the energy of the first vibrational state in the question before, assuming that it behaved like a harmonic oscillator using the reduced mass, which would be at n=1 (here we are only using Bohr's quantization rules). However, how can one de-excite from the n=1 state? Is this not the ground state that we cannot drop any further from, like for the quantum harmonic oscillator?

## Homework Equations

$$E=n h \nu,~\nu=\frac{\sqrt{\frac{k}{m}}}{2\pi},~E=\frac{h c}{\lambda}$$

## The Attempt at a Solution

I would think that if there is a solution, it would just be using the energy from the problem before, and use the de Broglie wavelength equation to solve.

Dahaka14 said:
However, how can one de-excite from the n=1 state? Is this not the ground state that we cannot drop any further from, like for the quantum harmonic oscillator?
Note that the first excited state is by definition, the first state above the ground state. So if the ground state is n=1, then the first excited state is n=2.

I understand that. I guess the question is: what is the first vibrational state? Is it the ground state, or one above for a diatomic molecule?

Dahaka14 said:
I understand that. I guess the question is: what is the first vibrational state? Is it the ground state, or one above for a diatomic molecule?
As you should know, the vibrational energy levels for a quantum harmonic oscillator obey the relationship:

$$E_n = \left(n+\frac{1}{2}\right)h\bar\omega\;\;\;n=0,1,2,3,\ldots$$

So in this case the ground state is the n=0 state and the first vibrational state is n=1. The reason why the question specifically states vibrational energy levels is that a diatomic molecule may also have translational, rotational and electronic energy levels.

## 1. What is HCl Bohr Quantum Oscillator?

HCl Bohr Quantum Oscillator is a theoretical model that describes the behavior of hydrogen chloride (HCl) molecules in a quantum mechanical system. It combines the principles of quantum mechanics and classical mechanics to explain the oscillatory motion of the molecule.

## 2. How does the HCl Bohr Quantum Oscillator work?

The HCl Bohr Quantum Oscillator model is based on the concept of energy levels, where the energy of the molecule is quantized and can only exist in certain discrete states. The molecule oscillates between these energy states due to the force of attraction between the positively charged hydrogen atom and the negatively charged chlorine atom.

## 3. What are the applications of the HCl Bohr Quantum Oscillator?

The HCl Bohr Quantum Oscillator model has various applications in theoretical chemistry and physics. It can be used to study the infrared spectra of HCl molecules, the behavior of diatomic molecules, and the energy levels of atoms and molecules in a quantum system.

## 4. How is the HCl Bohr Quantum Oscillator different from the classical harmonic oscillator?

The classical harmonic oscillator model assumes that the energy of the system can have any value and is continuous, while the HCl Bohr Quantum Oscillator model considers quantized energy levels. Additionally, the classical model does not take into account the principles of quantum mechanics, while the HCl Bohr Quantum Oscillator does.

## 5. Is the HCl Bohr Quantum Oscillator model accurate?

The HCl Bohr Quantum Oscillator is a simplified model that does not account for all the complexities of a real HCl molecule. However, it is a useful tool for understanding the behavior of diatomic molecules and has been validated through experimental data and calculations. Other, more advanced models have been developed to better describe the behavior of molecules in a quantum system.

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