Understanding Atomic Energy Levels and Line Spectra

Atoms behave in extraordinary manners when they encounter energy. A concept of energy levels, which are distinct states that electrons within an atom might occupy, is essential to explain atomic behavior. A crucial component of atomic and molecular spectroscopy, i.e., line spectra, can be interpreted using these energy levels as a basis.

The distinct regions surrounding an atom’s nucleus where electrons are prominent to be found are known as energy levels. In accordance with the quantization of energy levels, electrons can only belong to specific discrete energy states and not between the spaces. Since electrons behave like waves according to quantum theory, the concept of quantization arises. The configuration of electrons inside an atom is made apparent by its atomic energy levels. These energy levels assist improvements in researches, and technologies by offering a structure for interpreting diverse atomic properties and phenomena. The notion of atomic energy levels has resolved the issues of atomic stability and electronic configurations raised by classical mechanics.

Historical Development of Atomic Models

Modern atomic theory began in 1803 when John Dalton proposed the concept of atoms based on experimental results. Following Dalton, J.J. Thomsom in 1904 suggested the early atomic model, after his discovery of electrons. In his proposal, the atom is regarded as an evenly distributed sphere of positive charges. Analog to the seeds in the plums (or dates) in a pudding, negatively charged electrons dwell within this positively charged sphere. The atom remains electrically neutral ultimately because its positive and negative charges cancel out each other.

However, the plum pudding model lacked a nucleus. Instead of centered in a tiny central area, the positive charge was supposed to be distributed across the entire atom. It failed to explain certain experimental results, like the dispersed nature shown by alpha particles in Rutherford’s gold foil experiment. The distinct spectral lines seen in atomic emission and absorption spectra were not taken into consideration. Subsequent observations demonstrating the presence of a firm, positively charged nucleus contradicted the assumption of an evenly distributed positive charge.

Ernest Rutherford in 1911 conducted the gold foil experiment, by bombarding thin gold foil sheets with the positively charged alpha particles. A significant portion of alpha particles passed through the foil unobstructed.

A tiny percentage of particles were deviated with small angles. Even fewer were deviated at sharp angles, particularly ones that bounced back directly. The theory was revolutionary as it pointed towards the existence of a small, compact, positively charged nucleus which was responsible for those deviations. The atom was concluded to remain mostly vacant as the majority of particles were undeviated in the experiment. Once more, the distinct spectral lines seen in atomic emission and absorption spectra—especially in hydrogen atoms—could not be explained by this paradigm. Furthermore, the arrangement of electrons surrounding the nucleus was not specified.

Neil Bohr in 1913 employed quantum theory to shed light on the emission of distinct spectral lines and the stability of atoms. This concept states that electrons traverse in fixed, precise orbits around the nucleus known as energy levels or shells. Since these orbits are quantized, electrons are limited to occupying specific discrete energy levels. 

Quantum numbers (n = 1, 2, 3…) are used to represent the energy levels. The lowest possible energy level is represented by n = 1. Higher energy levels (excited states) are denoted by n>1. The line spectra of hydrogen, where electrons change energy levels and either emit or absorb photons at particular wavelengths, were explained by Bohr’s model. Spectral series are formed by these transitions. An electron advances to a higher energy level when it absorbs a photon. Similarly, a photon is released when an electron drops to a lower energy level. The photon’s energy is equal to the difference between the two levels.

ΔE = Ehigher ​− Elower ​= hν =  hc​/λ (Equation 1)

Here, 𝜈 is the frequency of the light that is emitted or absorbed, and ℎ is Planck’s constant (4.135667696×10^-15 eV), λ is the wavelength of light and c is the speed of light.

Discrete Energy Levels in Isolated Atoms

The term “discrete energy levels” describes the particular, fixed energies that electrons in an atom can have. These levels originate from the quantum mechanical nature of electrons and are specific to each element. In accordance with the quantum mechanical laws that govern atoms, particularly the Schrödinger equation, electrons are limited to specific values and cannot have random energies. Thus, the ground state is the lowest attainable energy level of an atom, where electrons are most stable whereas excited states are higher energy levels that electrons can momentarily hold when the atom absorbs energy. When discussing nucleus binding, electrostatic force is taken into account.

The energy states are quantized as a result of the potential energy well created. Electrons serve like waves since standing waves are observed at certain distinct energy levels.

A set of quantum numbers define each energy level

• The principal quantum number (n) gives the orbital’s size and energy.
• The azimuthal quantum number (l) decides the orbital’s shape.
• The magnetic quantum number (m) represents the orbital’s direction in space.
• The spin quantum number (s) explains electrons’ angular momentum.

The following formula describes the energy levels in atomic hydrogen:

En = -13.6/n² eV (Equation 2)

En ​represents the energy of the electron in the n-^th level.

n is the principal quantum number (n=1,2,3,…).

The ionization energy of hydrogen, or the energy needed to extract an electron from its ground state to infinity, is 13.6 eV. A negative indication represents the nucleus’ binding nature. The energy is more stable and negative the closer the electron is to the nucleus. For hydrogen, the ground state is the most stable state. Less binding energy is shown by higher energy levels being less negative.

Formation of Emission and Absorption Line Spectra

Electron transitions between distinct energy levels create the line spectra seen in atomic and molecular systems. Depending on whether energy is produced or absorbed during the transitions, these spectra are classified as either emission spectra or absorption spectra. 

Emission Line Spectra

When electrons make a shift from a higher energy state to a lower energy state, they release photons with particular wavelengths, which results in an emission spectrum. Electrons shift in energy from a lower energy level to a higher energy level as an atom absorbs energy. Hence the atom is said to be in an excited state which is still temporary and the electrons are highly unstable. Ultimately, their energy levels drop again. Different lines appear in the emission spectrum as a result of each transition, which is associated with a photon of a particular wavelength. The energy of emitted photons is given by equation (1). 

Absorption Line Spectra

Electrons in atoms or molecules absorb photons with energies that correspond to the variations between their energy levels when light with a continuous spectrum travels through a gas or other material. Electrons proceed to higher energy levels as a result of this absorbed energy. Absorption occurs only for photons with energies equivalent to the energy discrepancy between levels. The remaining light leaves through. Since certain particular wavelengths are absent, the absorbed wavelengths show up in the continuous spectrum as black lines.

Understanding the Relationship Between Energy and Light

The energy difference between the two levels is equal to the energy of the photon that is emitted or absorbed as given by equation (1). When an electron shifts from a higher energy level (E₁) to a lower energy level (E₂) or shifts inversely, it releases a photon with energy equal to the difference in the energy levels: 

hf = E1​−E2​ (i)

These energy levels are distinct and hence quantized. The difference between E1 and E2 determines the energy and frequency ‘f’ of the emitted or absorbed photon. The formula emphasizes the connection between light and matter as well as the quantum nature of energy transfers in atoms.

The only permissible energy levels shows that the energy levels are quantized. Since energy is quantized, the discrete nature of the atomic line spectra provides crucial support for the quantum theory of atoms. The atomic spectrum is made up of discrete lines instead of a continuous range of wavelengths because of this quantization. The energy difference between atomic energy levels can be observed by the frequency (or wavelength) of light that escapes or gets received. The simplest way of establishing the link between energy and light involves examining atomic spectra. The link comes from the way electrons behave, including how they absorb and release energy.

In conclusion, the atomic spectra illustrate the basic connection between energy and light as outlined by the equations of quantum physics, in addition to demonstrating the quantized character of atomic energy levels.

Applications of Line Spectra

The quantized energy levels of atoms and molecules give rise to line spectra, which are used in a wide range of scientific and technical domains. Important details regarding the makeup, structure, and behavior of matter are revealed by these spectra. 

Spectroscopy in identifying elements

An individual collection of spectral lines produced by each element serves as its signature and thus used to identify elements in a sample. The property is employed in methodologies such as atomic emission spectroscopy (AES) and atomic absorption spectroscopy (AAS).

Astrophysics and Astronomy

Line spectra help in determining the composition, temperature, density, and motion of astronomical objects. For instance, it helps to identify elements present in the stars, galaxies and other celestial bodies. Moreover, the redshift and blueshift of spectral lines are used to measure the velocity of stars and galaxies.

• In lab and industrial settings, line spectra are used to investigate and evaluate plasma states of materials.
• Line spectra are essential for examining atomic structure and confirm that atomic energy levels are quantized. Their role is inevitable in quantum and atomic physics.
• Diagnostic tools employ spectroscopic methods based on line spectra.

Sample Calculations Using  hf=E1−E2

For an electron making an atomic transition from E1 to E2, we can use equation (i) to calculate the frequency of the photon. If an electron is transiting from E1 = – 20.00 eV and E2 = -25.00 eV, then the energy difference is calculated as:

∆E = E1 – E2 = -20.00 – (-25.00) = 5.00 eV

Now, the photon frequency ‘f’ is calculated as: f = ∆E/h.

Using h = 4.136×10⁻¹⁵eV,

f = 5.00/4.136×10⁻¹⁵ = 1.21×10¹⁵ Hz

Thus, we get the frequency of the photon to be 1.21×10¹⁵ Hz.

When an electron in a hydrogen atom transitions from n=2 to n=1(Lyman series), the atom emits a photon. Let’s calculate the energy of the photon and its wavelength.

For transitions in hydrogen atom:

En = −13.6 /n² eV

• For n=1:

E1 = −13.6 /1² = −13.6 eV, 

• For n=2:

E2= −13.6 /2² 

= −13.6 /4 

=−3.4 eV

Now, the energy difference (ΔE) is:

ΔE = E2−E1 

= −3.4 − (−13.6)

=13.6−3.4

 =10.2 eV 

Thus, energy difference ΔE=10.2 eV

Again we have the formula for energy:

E=hc/λ

Hence, λ = hc/E

Keeping h = 4.136×10⁻¹⁵eV, c = 3.00×10⁸m/s and E = 10.2 eV,

Or, λ = (4.136×10⁻¹⁵) (3.00×10⁸)/ 10.2

= 10.21.24×10⁻⁶​/10.2

= 1.216×10⁻⁷ m

Or, λ = 121.6 nm

Thus, the energy of the emitted photon is found to be 10.2 eV and the wavelength is found as 121.6 nm.

Quantum Mechanics and Energy Levels

The idea of energy levels in atoms and molecules—where a particle like electron occupies definite energy states rather than a continuum of values—are essentially explained by quantum mechanics. By providing accurate explanations of physical phenomena like line spectra and chemical bonding, this theory transformed our knowledge of atomic structure. Bohr’s model presented the concept of electron quantized orbits, which quantum mechanics subsequently improved into wavefunction-described energy levels. The discrete spectral lines seen in emission and absorption spectra are examples of phenomena that can be explained by energy quantization.

According to the de Broglie theory of duality, electrons behave as waves, thus wavefunctions (𝜓) characterize the electrons in atoms. The solution of the Schrödinger equation for quantum systems directly leads to energy quantization.

H^ψ = Eψ (Equation 3)

Here in equation (3) H^ is the Hamiltonian operator, ψ is the wavefunction and E is the energy of the electrons.

Quantization of energy also yields insights in quantum field theory and quantum electrodynamics. 

Real-World Implications

A variety of appliances use the electrons’ movements between quantized energy levels in atoms or substances to generate energy, light, or accurate measurements. The following is a list of some energy quantization-based applications:

•  Photons are released when electrons in a higher energy state transition back to a lower energy state. This mechanism is used in lasers to produce coherent beams of light.
• Electroluminescence, which occurs when electrons in a semiconductor material change energy levels, is the basis for LEDs.
• Photovoltaics use the principle of photoelectric effect to produce electrons from excited photons.
• Information regarding molecular orbitals and electron configurations can be found in absorption and emission spectra. This facilitates the study of complicated compounds and how they interact.
• The energy taken for extracting an electron from an atom is provided by spectrum information, which is vital for grasping chemical reactions and bonds.
• Auger electron spectroscopy (AES) and X-ray fluorescence (XRF) are used to examine the surface characteristics and structure of materials.
• Spectroscopy is used in material science to get high accuracy in experiments.

Conclusions

The idea of atoms’ energy levels is essential to comprehending how matter behaves at the microscopic level. The quantization of energy in quantum physics gives birth to the idea that electrons in atoms possess distinct energy levels. This fundamental concept drives issues like emission and absorption spectra, which act as distinct imprints for every element, in addition to explaining the stability of atoms. 

Advances in a variety of disciplines, such as spectroscopy, astronomy, chemistry, and material science, have been made possible by our increasing awareness of atomic energy levels. This idea is directly applied in technologies like lasers, and LEDs, which show how it impacts daily life. Additionally, the analysis of atomic structure, chemical bonds, and the interactions of light and matter is supported by energy level transitions, which facilitate advances in science and technology.

References

1. Eisberg, R., & Resnick, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.). John Wiley & Sons.
2. Oxtoby, D. W., Gillis, H. P., & Butler, L. J. (2015). Principles of Modern Chemistry (8th ed.). Cengage Learning.
3. Beiser, A. (2003). Concepts of Modern Physics (6th ed.). McGraw-Hill Education.
4. National Institute of Standards and Technology (NIST). (n.d.). Atomic Spectra Database. Retrieved from https://www.nist.gov/pml/atomic-spectra-database
5. Georgia State University. (n.d.). HyperPhysics – Atomic Energy Levels. Retrieved from http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html
6. ChemLibreTexts. (2023). Atomic Structure and Spectroscopy. Retrieved from https://chem.libretexts.org
7. Bohr, N. (1913). On the constitution of atoms and molecules. Philosophical Magazine, 26(1), 1-25.
8. Khan Academy. (n.d.). Quantum Physics: Atomic Energy Levels. Retrieved from https://www.khanacademy.org
9. Massachusetts Institute of Technology (MIT). (n.d.). OpenCourseWare: Quantum Mechanics. Retrieved from https://ocw.mit.edu