Question

Electron in benzene. Consider an electron that can move freely throughout the aromatic orbitals of benzene. Model the electron as a particle in a two- dimensional box \(4 \AA \times 4 \AA\). (a) Compute \(\Delta \varepsilon\), the energy change from the ground state to the excited state, \(n_{x}=n_{y}=2\). (b) Compute the wavelength \(\lambda\) of light that would be absorbed in this transition, if \(\Delta \varepsilon=h c / \lambda\), where \(h\) is Planck's constant and \(c\) is the speed of light. (c) Will this transition be in the visible part of the electromagnetic spectrum (i.e., is liquid benzene colored or transparent), according to this simple model?

Short answer

The energy change is

Step-by-step solution

Understand the 2D particle-in-a-box model

The electron in benzene is modeled as a particle in a two-dimensional box with dimensions 4 Å × 4 Å. The energy levels for a particle in a 2D box can be expressed as:

Determine the energy levels

For a particle in a 2D box, the energy levels are given by:

Compute the energy for the ground state

For the ground state, both assuming:

Compute the energy for the excited state

For the excited state, given as:

Calculate the energy change ()

Subtract the ground state energy from the excited state energy to find :

Convert the energies to proper units

Convert the energy difference from Joules to electron volts for easier interpretation.

Find the wavelength of absorbed light

Using the relationship : . Solve for the wavelength : .

Determine visibility of the transition

Compare the wavelength found to the visible spectrum range (approximately 400-700 nm) to determine if this transition falls within the visible range.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the behavior of particles on very small scales, such as atoms and subatomic particles. In this realm, particles exhibit both wave-like and particle-like properties. The concept was developed in the early 20th century and revolutionized our understanding of the physical universe. Key principles include:
- **Wave-Particle Duality:** Particles, like electrons, can behave as both waves and particles.
- **Quantization:** Energy levels are discrete, not continuous.
- **Uncertainty Principle:** It is impossible to simultaneously know the exact position and momentum of a particle.
In the context of the given problem, we are applying quantum mechanics to model an electron in benzene as a particle confined in a two-dimensional box.

Energy Levels

The concept of energy levels is central to quantum mechanics. For a particle in a 2D box, the energy levels are discrete and quantized. This means that the electron can only exist in specific energy states. The energy levels for a particle in a 2D box are given by the formula:
\[ E_{n_x, n_y} = \frac{h^2}{8mL^2} (n_x^2 + n_y^2) \]
Here, \(h\) is Planck's constant, \(m\) is the mass of the particle, \(L\) is the length of the box, and \(n_x\) and \(n_y\) are quantum numbers representing the energy levels in the x and y dimensions. For the ground state, both \(n_x\) and \(n_y\) are 1. For the excited state given in the exercise, \(n_x = n_y = 2\). The energy change, \(\Delta \varepsilon\), is the difference between these two energy states.

Wavelength of Light

The wavelength of light absorbed or emitted during a transition between energy levels can be determined using the formula:
\[ \Delta \varepsilon = \frac{hc}{\lambda} \]
Here, \(c\) is the speed of light. Rearranging this relationship gives us the wavelength of the light associated with the energy transition:
\[ \lambda = \frac{hc}{\Delta \varepsilon} \]
Using this formula, once we have calculated \(\Delta \varepsilon\), we can find the wavelength \(\lambda\) of the light that will be absorbed during this transition.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from gamma rays to radio waves. Each type of radiation corresponds to a different wavelength range.
The visible spectrum is the portion of the electromagnetic spectrum that humans can see, ranging approximately from 400 nm to 700 nm. Colors in this range include:
- **Violet:** ~400-450 nm
- **Blue:** ~450-495 nm
- **Green:** ~495-570 nm
- **Yellow:** ~570-590 nm
- **Orange:** ~590-620 nm
- **Red:** ~620-700 nm
In the context of benzene, determining if the electron transition falls within this visible range can help us understand whether benzene would appear colored or transparent.

Planck's Constant

Planck's constant (\(h\)) is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. Its value is approximately \(6.626 \times 10^{-34}\) Js. Planck's constant is crucial in equations that describe quantum phenomena. For instance, in the formula for energy levels and the wavelength of light we have been using, Planck's constant provides the proportionality factor:
\[ E = hf \]
and
\[ \lambda = \frac{hc}{\Delta \varepsilon} \]
This constant connects the scale of quantum effects to measurable phenomena in the macroscopic world.
Understanding Planck's constant helps us grasp how energy quantization occurs, significantly impacting everything from atomic physics to modern electronics.