How to Find the Final Energy Level: A Comprehensive Guide

When it comes to understanding the behavior of atoms and subatomic particles, the concept of energy levels plays a crucial role. The final energy level of a system is a fundamental parameter that determines the stability, reactivity, and various other properties of the system. In this comprehensive guide, we will delve into the step-by-step process of finding the final energy level, covering both the theoretical foundations and practical applications.

Understanding the Bohr Model and Energy Levels

The Bohr model, proposed by Danish physicist Niels Bohr, provides a framework for understanding the behavior of electrons in an atom. According to this model, electrons can only occupy specific, discrete energy levels, known as the principal quantum numbers (n). These energy levels are represented by positive integers, with n = 1 being the lowest energy level (ground state) and higher values of n corresponding to higher energy levels.

The energy of an electron in a hydrogen atom can be calculated using the following formula:

$E = -\frac{13.6\,\text{eV}}{n^2}$

where $E$ is the energy of the electron in electronvolts (eV) and $n$ is the principal quantum number.

Calculating the Change in Energy (ΔE)

To find the final energy level, we need to know the initial energy level and the change in energy (ΔE) that occurs during a transition or reaction. The formula for calculating the change in energy is:

$\Delta E = -13.6\,\text{eV}\left(\frac{1}{n_f^2} – \frac{1}{n_i^2}\right)$

where $n_i$ and $n_f$ are the initial and final energy levels, respectively.

Example 1: Electron Transition in a Hydrogen Atom

Let’s consider an example where an electron in a hydrogen atom jumps from the second energy level (n = 2) to the third energy level (n = 3).

Using the formula for the change in energy:

$\Delta E = -13.6\,\text{eV}\left(\frac{1}{3^2} – \frac{1}{2^2}\right)$
$\Delta E = -13.6\,\text{eV}\left(\frac{1}{9} – \frac{1}{4}\right)$
$\Delta E = -13.6\,\text{eV}\left(\frac{4 – 9}{36}\right)$
$\Delta E = 1.89\,\text{eV}$

To convert the energy to Joules, we can use the conversion factor:

$1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}$

Therefore, the energy absorbed by the electron during the transition is:

$\Delta E = 1.89\,\text{eV} \times 1.6 \times 10^{-19}\,\text{J/eV} = 3.02 \times 10^{-19}\,\text{J}$

Example 2: Photon Emission in a Hydrogen Atom

When an electron in a hydrogen atom transitions from a higher energy level to a lower energy level, it emits a photon with a specific energy. Let’s consider an example where an electron jumps from the third energy level (n = 3) to the second energy level (n = 2).

Using the formula for the change in energy:

$\Delta E = -13.6\,\text{eV}\left(\frac{1}{2^2} – \frac{1}{3^2}\right)$
$\Delta E = -13.6\,\text{eV}\left(\frac{1}{4} – \frac{1}{9}\right)$
$\Delta E = -13.6\,\text{eV}\left(\frac{9 – 4}{36}\right)$
$\Delta E = 2.86\,\text{eV}$

The energy of the emitted photon can be calculated using Planck’s constant (h) and the wavelength (λ) of the photon:

$E = \frac{hc}{\lambda}$

where $h$ is Planck’s constant ($6.626 \times 10^{-34}\,\text{J}\cdot\text{s}$) and $c$ is the speed of light ($3.00 \times 10^8\,\text{m/s}$).

Substituting the values, we get:

$E = \frac{(6.626 \times 10^{-34}\,\text{J}\cdot\text{s})(3.00 \times 10^8\,\text{m/s})}{656 \times 10^{-9}\,\text{m}}$
$E = 2.86 \times 10^{-19}\,\text{J}$

Therefore, the energy of the emitted photon is 2.86 × 10^-19 J.

Practical Applications and Considerations

The ability to calculate the final energy level has numerous practical applications in various fields, including:

1. Atomic and Molecular Spectroscopy: Understanding the energy transitions in atoms and molecules can be used to analyze and interpret the spectra of these systems, which is crucial for identifying their composition and structure.

2. Laser Technology: The principles of energy level transitions are fundamental to the operation of lasers, where the stimulated emission of photons is used to amplify light.

3. Quantum Mechanics and Quantum Computing: The concept of discrete energy levels is a cornerstone of quantum mechanics, which is the foundation for the development of quantum computing and other quantum technologies.

4. Astrophysics and Cosmology: Analyzing the spectra of celestial objects, such as stars and galaxies, can provide insights into their composition, temperature, and other physical properties, which is essential for understanding the evolution of the universe.

5. Materials Science and Engineering: The energy levels of electrons in materials can influence their electrical, optical, and chemical properties, which is crucial for the design and development of new materials and devices.

When working with energy level calculations, it’s important to consider the following factors:

• The accuracy of the initial energy level and the change in energy (ΔE) values
• The influence of external factors, such as electric and magnetic fields, on the energy levels
• The limitations of the Bohr model and the need to consider more advanced quantum mechanical theories for complex systems

Conclusion

Mastering the ability to find the final energy level is a crucial skill for any student or researcher in the field of physics. By understanding the underlying principles, formulas, and practical applications, you can gain a deeper understanding of the behavior of atoms and subatomic particles, and apply this knowledge to a wide range of scientific and technological advancements.

References

1. Finding final energy level for Beryllium ion using the Bohr Model: https://www.youtube.com/watch?v=Go5s4PuWS68
2. How to Calculate the Energy Absorbed by Electrons Jumping Energy Levels: https://study.com/skill/learn/how-to-calculate-the-energy-absorbed-by-electrons-jumping-energy-levels-explanation.html
3. Line Spectra and the Bohr Model – Chemistry LibreTexts: https://chem.libretexts.org/Courses/University_of_Missouri/MU:1330H%28Keller%29/06._Electronic_Structure_of_Atoms/6.3:_Line_Spectra_and_the_Bohr_Model