Problem 5 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Using the one-dimensional particle in a box model, calculate the first four electronic energy levels for 1,3,5 -hexatriene. Assume that a \(2 p\) electron can move freely along the delocalized \(\pi\) system. At what wavelength would the lowest energy electronic transition appear? Assume the average carbon- carbon distance to be \(0.15 \mathrm{nm}\). (Section 10.2 )

Step-by-Step Solution

Answer

The first four energy levels are approximately \(6.02, 24.08, 54.18, \text{ and } 96.32\) times \(10^{-19}\) J. The lowest energy transition appears at 110 nm.

Step 1: Calculating Length of the Box

1,3,5-hexatriene has 6 carbon atoms, creating a conjugated system with 5 bonds. The average carbon-carbon distance is 0.15 nm. Therefore, the length of the box is the total distance spanned by these bonds: \(L = 5 \times 0.15 \text{ nm} = 0.75 \text{ nm}\).

Step 2: Understanding the Energy Levels Formula

In quantum mechanics, the energy levels for a particle in a one-dimensional box are given by: \(E_n = \frac{h^2 n^2}{8mL^2} \), where \(h\) is Planck's constant, \(n\) is the quantum number, \(m\) is the mass of the electron, and \(L\) is the length of the box.

Step 3: Calculate the First Energy Level

Using \(n=1\), the first energy level is \(E_1 = \frac{h^2}{8mL^2}\). Substitute \(h = 6.626 \times 10^{-34}\) J s, \(m = 9.109 \times 10^{-31}\) kg, and \(L = 0.75 \times 10^{-9}\) m to calculate: \(E_1 = \frac{(6.626 \times 10^{-34})^2}{8 \times 9.109 \times 10^{-31} \times (0.75 \times 10^{-9})^2} \approx 6.02 \times 10^{-19}\) J.

Step 4: Calculate the Second Energy Level

For \(n=2\), the energy is \(E_2 = \frac{4h^2}{8mL^2} = 4E_1 \). Thus, \(E_2 = 4 \times 6.02 \times 10^{-19} \approx 24.08 \times 10^{-19}\) J.

Step 5: Calculate the Third Energy Level

For \(n=3\), the energy is \(E_3 = \frac{9h^2}{8mL^2} = 9E_1 \) which gives \(E_3 = 9 \times 6.02 \times 10^{-19} \approx 54.18 \times 10^{-19}\) J.

Step 6: Calculate the Fourth Energy Level

For \(n=4\), it is \(E_4 = \frac{16h^2}{8mL^2} = 16E_1 \). This calculates to \(E_4 = 16 \times 6.02 \times 10^{-19} \approx 96.32 \times 10^{-19}\) J.

Step 7: Calculate the Wavelength for the Lowest Energy Transition

The lowest energy transition is between \(n=1\) to \(n=2\) levels. The energy for this transition is \(\Delta E = E_2 - E_1 = 24.08 \times 10^{-19} - 6.02 \times 10^{-19} = 18.06 \times 10^{-19}\) J. The wavelength \(\lambda\) of this transition can be found using \( \Delta E = \frac{hc}{\lambda} \), rearranging gives \(\lambda = \frac{hc}{\Delta E}\). Substitute \(c = 3.00 \times 10^8\) m/s to find \(\lambda = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{18.06 \times 10^{-19}} \approx 1.10 \times 10^{-7}\) m or 110 nm.

Key Concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy on the smallest scales. It explains how particles such as electrons behave at the atomic and subatomic levels. Unlike classical mechanics, which operates under deterministic laws, quantum mechanics introduces probability and uncertainty.

• Particles are described as wave functions, which provide the probability of finding a particle in a given location.
• Quantum mechanics involves key principles like wave-particle duality, uncertainty principle, and quantization of energy levels.
• The particle in a box model exemplifies the quantization concept, where electrons are confined within a finite region and can only occupy specific energy levels.

The model is widely used to understand how energy levels work in systems like the molecule of 1,3,5-hexatriene, where an electron is assumed to be moving freely along a conjugated system.

Electronic Energy Levels

Electronic energy levels refer to the discrete energy states that electrons can occupy within an atom or molecule. In the context of quantum mechanics, the energy of these states is quantized, meaning electrons can only exist at specific energy levels.

• In a one-dimensional box, energy levels are determined by the equation \(E_n = \frac{h^2 n^2}{8mL^2} \), where \(n\) is the principal quantum number.
• The principal quantum number, \(n\), which ranges from 1 to positive integers, dictates the energy: higher \(n\) means higher energy.
• In the solved exercise, the first four electronic energy levels were calculated, exemplifying how energy levels step up quadratically as \(n^2\).

Such calculations help in determining the properties of molecules and their potential reactions based on electron distribution and energy transitions.

Conjugated Systems

Conjugated systems in chemistry involve alternating double and single bonds between carbon atoms. These systems have delocalized electrons that make the entire system more stable and allow for unique chemical properties.

• The delocalization of π electrons across the carbon atoms allows these systems to absorb specific wavelengths of light, contributing to their color and reactivity.
• In the particle in a box model, the system forms the box, where electrons can freely move, following the limitations imposed by the bonds' spacing.
• Such systems are important in organic molecules like 1,3,5-hexatriene, influencing their electronic structure and optical properties.

Understanding conjugated systems is essential for applications in materials science, organic electronics, and understanding molecular pi-bonding interactions.

Quantum Transitions

Quantum transitions refer to the process where an electron moves from one energy level to another within an atom or molecule. This movement requires energy that typically comes from absorbing or emitting photons of specific wavelengths.

• The energy difference between levels is expressed as \( \Delta E = E_2 - E_1 \) and corresponds to a specific photon energy or wavelength.
• The calculation of the wavelength \( \lambda \) for the lowest transition in 1,3,5-hexatriene demonstrated how this change is quantifiable: \( \lambda = \frac{hc}{\Delta E} \).
• Such transitions are fundamental to spectroscopy, allowing scientists to study the electronic structure and dynamics of molecules.

Quantum transitions are crucial for understanding phenomena like fluorescence, phosphorescence, and the mechanisms of vision as well as harnessing solar energy.