Bohr's atomic model, proposed in 1913 by the Danish physicist Niels Bohr, was a groundbreaking theory in the field of atomic physics. It fundamentally changed our understanding of atomic structure by addressing the contradictions in classical physics, particularly regarding the stability of atoms and the nature of electron orbits. By building on the Rutherford model and incorporating the emerging concepts of quantum theory, Bohr developed a model that could explain the discrete spectral lines observed in hydrogen. This model laid the foundation for the development of quantum mechanics and provided critical insights into atomic structure, electron behavior, and the emission of light.
Bohr’s Postulates: The Foundation of the Model
Bohr’s atomic model is based on two crucial postulates, which were instrumental in resolving the limitations of classical mechanics in explaining atomic phenomena. These postulates address the behavior of electrons in atoms, specifically the issue of why electrons do not radiate energy and spiral into the nucleus, which classical electromagnetism would predict.
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Electron Orbits Without Radiation:
According to Bohr's first postulate, electrons move in stable, circular orbits around the nucleus without emitting any radiation. This was a radical departure from classical electromagnetism, which predicted that accelerating charged particles (like electrons) should continuously emit electromagnetic radiation, causing them to lose energy and spiral into the nucleus. Bohr’s solution to this paradox was to propose that electrons in an atom only occupy specific, quantized orbits. These orbits are associated with particular energy levels, and electrons do not emit radiation while moving in these stable orbits. -
Energy Absorption and Emission During Transitions:
Bohr’s second postulate addresses the issue of energy absorption and emission. According to this postulate, electrons can absorb or emit energy only when they transition between these quantized orbits. The energy absorbed or emitted during such a transition corresponds to the difference in energy between the initial and final orbits. The energy of the emitted or absorbed photon is given by the equation:ΔE = hν
Where:
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ΔE is the energy difference between two orbits,
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h is Planck’s constant,
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ν is the frequency of the emitted or absorbed photon.
This equation successfully explained the discrete spectral lines observed in the hydrogen atom's emission spectrum. Bohr’s theory showed that the frequency of the emitted or absorbed radiation is directly related to the energy difference between the two orbits involved in the transition.
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Derivation of the Radius and Energy of Orbits
Bohr also provided a mathematical framework for determining the radius and energy of an electron’s orbit. He used classical physics principles, such as Coulomb’s law of electrostatic attraction, and combined them with the concept of quantized angular momentum. By equating the centripetal force required to keep the electron in orbit to the electrostatic force between the electron and the nucleus, Bohr was able to derive expressions for both the radius and the energy of the electron’s orbit.
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Radius of the Electron’s Orbit:
The radius of the electron’s orbit in the nth energy level is given by the formula:rₙ = (ε₀h²n²) / (πme²Z)
Where:
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ε₀ is the permittivity of free space,
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h is Planck’s constant,
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n is the principal quantum number (indicating the energy level),
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m is the mass of the electron,
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e is the charge of the electron,
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Z is the atomic number of the element.
For hydrogen (Z = 1), this formula simplifies to:
rₙ = 0.529 × n² Å
This equation provides the radius of the electron’s orbit for any given energy level (n). The value 0.529 Å represents the Bohr radius, which is the radius of the electron’s orbit in the ground state (n = 1) of hydrogen.
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Energy of the Electron in the nth Orbit:
The energy of the electron in the nth orbit is given by the formula:Eₙ = -13.6 Z² / n² eV
The negative sign indicates that the electron is bound to the nucleus. As the value of n increases, the energy becomes less negative, indicating that the electron is less tightly bound to the nucleus. For the hydrogen atom (Z = 1), this formula simplifies to:
Eₙ = -13.6 / n² eV
This equation gives the energy of an electron in any orbit, and it explains why the energy levels are discrete. For example, the energy of the electron in the first orbit (n = 1) is -13.6 eV, and the energy increases (becomes less negative) as the electron moves to higher orbits.
Spectral Lines and Energy Transitions
Bohr’s model also provided an explanation for the discrete spectral lines observed in the hydrogen atom’s emission spectrum. When an electron transitions from a higher energy level (n₂) to a lower energy level (n₁), it emits a photon. The energy of the emitted photon corresponds to the energy difference between the two levels, and the frequency of the emitted radiation is given by:
ν = ΔE / h
Where:
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ΔE is the energy difference between the two levels,
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h is Planck’s constant.
The wavelength of the emitted radiation can then be found using the relation:
λ = c / ν
Where:
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λ is the wavelength of the radiation,
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c is the speed of light,
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ν is the frequency of the radiation.
The energy difference between two energy levels in the hydrogen atom is given by:
ΔE = 13.6 Z² (1/n₁² - 1/n₂²) eV
This equation allows us to calculate the energy difference between two levels and, consequently, the wavelength of the emitted radiation. The spectral lines observed in the hydrogen atom correspond to transitions between these quantized energy levels. The most well-known spectral lines are found in the Balmer series, which corresponds to transitions where electrons fall to the second energy level (n = 2). These spectral lines are in the visible region of the electromagnetic spectrum and are responsible for the characteristic colors seen in the hydrogen spectrum.
Rydberg Formula and Hydrogen-Like Species
Bohr’s model also applies to hydrogen-like ions, such as He+, Li²+, and Be³+, which have only one electron. The model predicts that these ions will have deeper energy levels due to the increased nuclear charge, and their electron orbits will be smaller. The energy of the nth orbit for a hydrogen-like ion is given by:
Eₙ = -13.6 Z² / n² eV
The radius of the nth orbit is given by:
rₙ = 0.529 × n² / Z Å
Where Z is the atomic number of the ion. The Rydberg formula, which Bohr derived, can be used to calculate the wavelengths of spectral lines for these ions:
1/λ = R Z² (1/n₁² - 1/n₂²)
Where:
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R is the Rydberg constant (1.097 × 10⁷ m⁻¹),
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Z is the atomic number,
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n₁ and n₂ are integers with n₂ > n₁.
This formula predicts the wavelengths of spectral lines for both hydrogen and hydrogen-like ions. It also explains the different series of spectral lines, including the Lyman, Balmer, Paschen, Brackett, and Pfund series. These series correspond to transitions between different energy levels in the atom, and the wavelengths of the spectral lines depend on the atomic number (Z) of the ion.
Advanced Applications in Atomic Physics
Although Bohr’s atomic model was eventually superseded by quantum mechanical models, it remains crucial for understanding the basics of atomic structure and spectral lines. In competitive exams like JEE Advanced, Bohr’s model is still used to solve problems related to transition wavelengths, energy absorption or emission, and the use of the Rydberg formula for hydrogen-like ions. Additionally, Bohr’s model has applications in the study of ionization energy, binding energy, and the calculation of electron speeds and orbital radii.
Bohr’s model has applications beyond simple atomic systems. It is also used to analyze hydrogen-like ions, multi-photon emission pathways (such as cascade transitions), and the Doppler effect in spectral lines. Despite being a semi-classical theory, Bohr’s atomic model provides a powerful and intuitive framework for understanding the quantum nature of atoms. It remains an essential building block in the study of quantum mechanics and atomic physics.