Question regarding Energy of a Particle in a Box is Quantized

In summary, the conversation is discussing the application of the particle in a one-dimensional box model to pi electrons in a linear conjugated hydrocarbon. The specific example of butadiene is used, with a total of four pi electrons. The Pauli Exclusion Principle is mentioned, explaining that each state can only hold two electrons, leading to the first two levels being filled by the four pi electrons. The energy of the first excited state is also discussed, with the equation for calculating it provided. The conversation then moves on to questions about the calculation of the four pi electrons and the electron elevated from n=2 to n=3 state.
  • #1
luckymango
7
0
I'm not sure how should I ask the question, so I'm going to go ahead and quote from the book.

"The model of a particle in a one-dimensional box has been applied to the pi electons in a linear conjugated hydrocarbons. Consider butadiene, H2C=CHCH=CH2, which has four pi electrons. Although butadiene, like all polyenes, is not a linear molecule, we will assume for simplicity that the pi electrons in butadiene move along line who length can be estimated as equal to two C=C bond lengths (2 x 135pm) plus one C-C bond (154pm) plus the distance of a carbon atom radius at each end (2 x 77pm = 154pm), giving a total distance of 578pm. According to equation E = h^2 n^2 /(8m a^2), where n=1,2...

But the Pauli Exclusion Principle says that each of these states can hold only two electrons and so the four pi electrons fill the first two levels. The energy of the first excited state of this system of four pi electrons is that which has one electron elevated to the n=3 state and the energy to make a transition from n=2 state to the n=3 state is Delta E = h^2 (3^2 - 2^2) /(8m a^2)."

My first question is how did they calculate the four pi electrons in the first place? Also, how did they know that the electron elevated from n=2 state to the n=3 state?

Hopefully someone could help me with my question. I'm stuck here reading the book and I can't move on...:cry: Thanks in advance!
 
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  • #2
pi orbitals have 3-fold degeneracy, px, py and pz. Thus 4 electrons will fill these orbitals, none being the same state.
 

What is a particle in a box and how is its energy quantized?

A particle in a box refers to a theoretical model in quantum mechanics where a particle is confined to a one-dimensional space. The energy of the particle in this model is quantized, meaning it can only take on certain discrete values.

What is the significance of quantized energy in a particle in a box?

The quantized energy levels in a particle in a box model have significant implications in understanding the behavior of particles at the atomic and subatomic level. It helps to explain phenomena such as the stability of atoms, the emission of light, and the properties of materials.

How is the energy of a particle in a box calculated?

The energy of a particle in a box is calculated using the Schrödinger equation, which is a fundamental equation in quantum mechanics. The solution to this equation gives the allowed energy levels of the particle, which are quantized.

How does the size of the box affect the energy of the particle?

The size of the box has a direct impact on the energy levels of the particle. As the size of the box decreases, the energy levels become more closely spaced. This is because the particle is more confined, and therefore, its energy is more restricted.

Are there real-world applications of the particle in a box model?

Although the particle in a box model is a simplified theoretical concept, it has several real-world applications. It is used in fields such as quantum computing, material science, and semiconductor technology. The model also helps in understanding the behavior of electrons in a crystal lattice and the properties of nanomaterials.

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