Glossary

4.2 Energy levels and spectral lines

4 min read•july 31, 2024

Energy levels and spectral lines are key to understanding the hydrogen atom's behavior. They reveal how electrons move between specific energy states, emitting or absorbing light in the process.

These concepts explain the unique spectral patterns of hydrogen, from the ultraviolet Lyman series to the visible Balmer series. They're crucial for grasping atomic structure and the interaction between matter and light.

Energy Levels in the Bohr Model

Quantized Energy States

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  • describes hydrogen atom with positively charged nucleus and negatively charged electron orbiting in discrete energy levels
  • Energy levels quantized allowing electrons to occupy only specific, allowed energy states
  • Energy of electron in nth energy level given by formula En=−13.6 eV/n2E_n = -13.6 \text{ eV} / n^2, where n represents principal quantum number
  • Ground state (n=1) represents lowest energy level of hydrogen atom, with E1=−13.6 eVE_1 = -13.6 \text{ eV}
  • Excited states occur when electrons occupy higher energy levels (n > 1), with energy values becoming less negative as n increases
    • Example: For n=2, E2=−13.6 eV/22=−3.4 eVE_2 = -13.6 \text{ eV} / 2^2 = -3.4 \text{ eV}
    • Example: For n=3, E3=−13.6 eV/32=−1.51 eVE_3 = -13.6 \text{ eV} / 3^2 = -1.51 \text{ eV}

Ionization Energy

  • Ionization energy of hydrogen measures 13.6 eV
  • Represents energy required to remove electron from ground state to infinity (n = ∞)
  • Calculated as difference between ground state energy and energy at infinity
    • Eionization=E∞−E1=0−(−13.6 eV)=13.6 eVE_{\text{ionization}} = E_{\infty} - E_1 = 0 - (-13.6 \text{ eV}) = 13.6 \text{ eV}
  • Ionization energy corresponds to shortest in Lyman series
    • Wavelength calculated using λ=hc/Eionization\lambda = hc / E_{\text{ionization}}, where h is and c is speed of light

Spectral Lines of Hydrogen

Lyman and Balmer Series

  • Spectral series in hydrogen result from electron transitions between different energy levels, emitting or absorbing photons of specific energies
  • Lyman series originates from transitions to ground state (n = 1) from higher energy levels
    • Produces ultraviolet spectral lines
    • Example: Lyman-alpha line (n=2 to n=1) with wavelength of 121.6 nm
    • Example: Lyman-beta line (n=3 to n=1) with wavelength of 102.6 nm
  • Balmer series involves transitions to second energy level (n = 2) from higher levels
    • Produces visible light spectral lines, with some lines extending into near-ultraviolet region
    • Example: H-alpha line (n=3 to n=2) with wavelength of 656.3 nm (red)
    • Example: H-beta line (n=4 to n=2) with wavelength of 486.1 nm (blue-green)

Paschen Series and Series Characteristics

  • Paschen series results from transitions to third energy level (n = 3) from higher levels
    • Produces infrared spectral lines
    • Example: First Paschen line (n=4 to n=3) with wavelength of 1875 nm
    • Example: Second Paschen line (n=5 to n=3) with wavelength of 1282 nm
  • Each series has characteristic set of wavelengths corresponding to specific energy level transitions involved
  • Spectral lines become more closely spaced and less intense as they approach series limit
    • Corresponds to transitions from increasingly higher energy levels
    • Example: Balmer series limit at 364.6 nm represents transitions from n=∞ to n=2

Energy Levels and Photon Wavelengths

Photon Energy and Frequency

  • Energy difference between two atomic energy levels determines energy of emitted or absorbed photon during
  • Energy of photon related to its frequency by equation E=hfE = hf, where h is Planck's constant and f is frequency
  • Wavelength of photon inversely proportional to its frequency, given by λ=c/f\lambda = c/f, where c is speed of light
  • Combining these relationships yields E=hc/λE = hc/\lambda, directly linking energy of transition to wavelength of emitted or absorbed photon
    • Example: Calculate energy of photon with wavelength 656.3 nm (H-alpha line) E=(6.63×10−34 J⋅s)(3×108 m/s)/(656.3×10−9 m)=3.03×10−19 JE = (6.63 \times 10^{-34} \text{ J⋅s})(3 \times 10^8 \text{ m/s}) / (656.3 \times 10^{-9} \text{ m}) = 3.03 \times 10^{-19} \text{ J}

Emission and Absorption Spectra

  • Longer wavelengths correspond to smaller energy differences between levels, while shorter wavelengths indicate larger energy differences
  • Emission spectra result from electrons transitioning from higher to lower energy levels, releasing photons
    • Example: Hydrogen discharge tube producing visible Balmer series lines
  • Absorption spectra occur when atoms absorb photons, causing electrons to transition from lower to higher energy levels
    • Example: Dark lines in solar spectrum due to absorption by hydrogen in solar atmosphere

Wavelengths of Spectral Lines

Rydberg Formula Application

  • Rydberg formula calculates wavelengths of spectral lines in hydrogen 1/λ=R(1/n12−1/n22)1/\lambda = R(1/n_1^2 - 1/n_2^2)
  • R is , approximately equal to 1.097 × 10^7 m^-1 for hydrogen
  • n1 represents lower energy level and n2 higher energy level involved in transition
  • Formula adapted for different spectral series by fixing n1 and varying n2
    • Lyman series n1 = 1 and n2 = 2, 3, 4, ...
    • Balmer series n1 = 2 and n2 = 3, 4, 5, ...
    • Paschen series n1 = 3 and n2 = 4, 5, 6, ...

Applications and Limitations

  • Rydberg formula predicts wavelengths of unknown spectral lines or identifies observed spectral lines in hydrogen
    • Example: Calculate wavelength of Balmer-alpha line (n1=2, n2=3) 1/λ=(1.097×107 m−1)(1/22−1/32)=1.524×106 m−11/\lambda = (1.097 \times 10^7 \text{ m}^{-1})(1/2^2 - 1/3^2) = 1.524 \times 10^6 \text{ m}^{-1} λ=1/(1.524×106 m−1)=6.563×10−7 m=656.3 nm\lambda = 1 / (1.524 \times 10^6 \text{ m}^{-1}) = 6.563 \times 10^{-7} \text{ m} = 656.3 \text{ nm}
  • Formula derived from Bohr model provides accurate results for hydrogen-like atoms
  • Becomes less accurate for more complex atomic structures with multiple electrons
    • Example: Helium spectrum more complex due to electron-electron interactions not accounted for in Bohr model

Key Terms to Review (18)

Absorption spectrum: An absorption spectrum is a graphical representation that shows the specific wavelengths of light absorbed by a substance, typically in the form of dark lines or bands against a continuous spectrum. This spectrum reveals information about the energy levels within an atom or molecule, as the absorbed wavelengths correspond to specific transitions between these energy levels. Understanding the absorption spectrum is crucial for analyzing the composition and properties of various materials, as well as for studying processes like stellar atmospheres and chemical reactions.
Bohr Model: The Bohr Model is a theoretical framework developed by Niels Bohr in 1913 to explain the behavior of electrons in atoms, specifically hydrogen. It introduces the concept of quantized energy levels, where electrons orbit the nucleus at fixed distances corresponding to specific energy states, leading to distinct spectral lines when electrons transition between these levels. This model links energy levels to the atomic spectra observed in elements and provides a foundation for understanding quantum numbers and atomic orbitals.
Electron transition: Electron transition refers to the process where an electron in an atom moves between different energy levels. This movement can occur when an electron absorbs or emits energy, typically in the form of a photon, leading to changes in the atom's energy state. Electron transitions are essential for understanding how atoms interact with light, resulting in spectral lines that represent the unique fingerprints of elements.
Electronvolt: An electronvolt is a unit of energy that is defined as the amount of kinetic energy gained by an electron when it is accelerated through an electric potential difference of one volt. This unit is especially useful in fields like atomic and nuclear physics, where the energies involved are often on the order of millions of electronvolts, making it easier to discuss energy levels and transitions in atoms or the binding energy within atomic nuclei.
Emission spectrum: An emission spectrum is the spectrum of light emitted by a substance when its atoms or molecules transition from higher energy states to lower energy states. This spectrum appears as a series of bright lines or bands on a dark background, each corresponding to a specific wavelength of light, and is unique to each element or compound, making it a valuable tool for identifying substances.
Excited State: An excited state refers to the condition of an atom or molecule when it has absorbed energy, causing one or more of its electrons to jump from a lower energy level to a higher one. This temporary state is crucial for understanding how atoms interact with light and other forms of energy, leading to the emission of spectral lines and influencing the arrangement of electrons in atomic orbitals and configurations in the periodic table.
Hertz: Hertz is the unit of frequency that measures the number of cycles per second of a periodic wave or signal. It plays a crucial role in understanding phenomena like electromagnetic radiation, sound waves, and quantum mechanics, as it helps quantify how often oscillations occur in a given timeframe, linking energy levels to the wavelengths of light emitted or absorbed by atoms.
Max Planck: Max Planck was a German physicist who is regarded as the father of quantum theory, which revolutionized our understanding of atomic and subatomic processes. His introduction of the concept of quantized energy levels laid the groundwork for various applications in physics, including quantum tunneling, energy levels in atoms, and the historical development of quantum mechanics as a field.
Niels Bohr: Niels Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, particularly through his development of the Bohr model of the atom. His work established the idea that electrons exist in specific energy levels and can transition between these levels by absorbing or emitting photons, connecting various principles of quantum mechanics with observable phenomena.
Pauli Exclusion Principle: The Pauli Exclusion Principle states that no two fermions, such as electrons, can occupy the same quantum state simultaneously within a quantum system. This principle is crucial in understanding the structure of atoms and the behavior of multi-electron systems, influencing energy levels, atomic configurations, and the physical properties of materials.
Photon emission: Photon emission is the process by which an excited atom or molecule releases energy in the form of a photon, resulting in a change in its energy state. This phenomenon is fundamental to understanding how atoms interact with light and is crucial for the formation of spectral lines, the effects of external magnetic fields on energy levels, and the rules governing transitions between energy states.
Planck's constant: Planck's constant is a fundamental physical constant that relates the energy of a photon to its frequency, with a value of approximately $$6.626 imes 10^{-34} ext{ J s}$$. It plays a crucial role in quantum mechanics, connecting the macroscopic and microscopic worlds by introducing the concept of quantization of energy levels, which is vital in understanding phenomena like the time-dependent and time-independent Schrödinger equations and spectral lines.
Quantum mechanics: Quantum mechanics is the branch of physics that deals with the behavior of matter and energy at very small scales, typically at the level of atoms and subatomic particles. It introduces concepts like wave functions, which describe the probability distributions of particles, and reveals phenomena that deviate from classical physics, such as tunneling and superposition. Understanding quantum mechanics is essential for exploring the fundamental nature of energy levels, spectral lines, and interactions in atomic systems.
Quantum state: A quantum state is a mathematical representation of a physical system in quantum mechanics, typically described by a wave function. It encapsulates all the information about the system, including its position, momentum, and energy, and is essential for predicting the probabilities of various outcomes when measurements are made.
Rydberg Constant: The Rydberg constant is a physical constant that represents the maximum wavelength of light emitted from an electron transitioning between energy levels in a hydrogen atom. This constant is vital for understanding the spectral lines produced by hydrogen, as it allows for the calculation of wavelengths associated with transitions between quantized energy levels.
Spectroscopy: Spectroscopy is the study of the interaction between matter and electromagnetic radiation, used to analyze the composition, structure, and properties of substances. This technique enables scientists to observe energy levels in atoms and molecules, leading to the identification of specific spectral lines that correspond to various transitions. By investigating these spectral lines, important insights into phenomena such as atomic structure and magnetic interactions can be uncovered, allowing for deeper understanding in both atomic and nuclear physics.
Wave-particle duality: Wave-particle duality is the concept in quantum mechanics that every particle or quantum entity can exhibit both wave-like and particle-like properties, depending on the experimental conditions. This principle highlights the fundamental nature of matter and energy, linking it to phenomena such as interference patterns and discrete energy levels, while also challenging classical views of physics.
Wavelength: Wavelength is the distance between successive crests or troughs of a wave, commonly used in the context of electromagnetic waves and sound waves. It is a fundamental characteristic of waves that determines their energy and behavior, particularly in relation to how they interact with matter. Understanding wavelength is essential for interpreting energy levels and spectral lines, as it directly relates to the emission or absorption of energy by atoms and molecules.
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