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Bohr Model Of The Hydrogen Atom

Bohr Model Of The Hydrogen Atom

Edited By Vishal kumar | Updated on Jul 02, 2025 05:50 PM IST

The Bohr model of the hydrogen atom represents a groundbreaking advancement in atomic theory, providing a clear and quantifiable explanation of atomic structure. Proposed by Niels Bohr in 1913, this model introduces the concept that electrons orbit the nucleus in discrete energy levels or "shells," rather than in continuous orbits. This revolutionary idea helped explain the discrete lines observed in atomic spectra, known as spectral lines. In real life, the Bohr model's principles underpin technologies such as lasers, which rely on electron transitions between energy levels, and various spectroscopy techniques used in chemical analysis and astronomy. By understanding the Bohr model, we gain insights into the fundamental workings of atoms, influencing both scientific research and practical applications. In this article, we will explore the Bohr model of the hydrogen atom, its key concepts, and its implications for modern science and technology.

This Story also Contains
  1. Bohr's Model of Hydrogen Atom
  2. Radius of Orbit and Velocity of the Electron
  3. Solved Examples Based on the Bohr Model of the Hydrogen Atom
  4. Summary
Bohr Model Of The Hydrogen Atom
Bohr Model Of The Hydrogen Atom

Bohr's Model of Hydrogen Atom

Bohr proposed a model for the hydrogen atom which is also applicable to some lighter atoms in which a single electron revolves around a stationary nucleus of positive charge Ze (called a hydrogen-like atom)
Bohr's model is based on the following postulates

(1). Bohr’s first postulate was that an electron in an atom could revolve in certain stable orbits without the emission of radiant energy, contrary to the predictions of electromagnetic theory. According to this postulate, each atom has certain definite stable states in which it can exist, and each possible state has definite total energy. These are called the stationary states of the atom

For electrons revolving in a stable orbit, the necessary centripetal force is provided by the coulomb's force

mvn2rn=kze2rn2
(2). Bohr’s second postulate defines these stable orbits. This postulate states that the electron revolves around the nucleus only in those orbits for which the angular momentum is some integral multiple of h2π where h is the Planck’s constant (= 6.6 × 10–34 J s). Thus the angular momentum (L) of the orbiting electron is quantised. That is L=mvnrn=nh2π;n=1,2,3
(3). Bohr’s third postulate incorporated into atomic theory the early quantum concepts that had been developed by Planck and Einstein. It states that an electron might make a transition from one of its specified non-radiating orbits to another of lower energy. When it does so, a photon is emitted having energy equal to the energy difference between the initial and final states. The frequency of the emitted photon is then given by $h\nu = E_i - E_f$

Ei is the energy of the initial state and Ef is the energy of the final state. Also, Ei > Ef.

rn-radius of the nth orbit

vn- speed of an electron in the nth orbit

Radius of Orbit and Velocity of the Electron

In the Bohr model of the hydrogen atom, the radius of the electron's orbit and its velocity are crucial parameters that help define the atom's structure. According to Bohr's theory, electrons orbit the nucleus in quantized orbits, and each orbit corresponds to a specific energy level.

The Radius of the Orbit

The radius of an electron's orbit in the Bohr model of the hydrogen atom is a key concept in understanding atomic structure. According to Bohr's theory, electrons orbit the nucleus in quantized paths or orbits with specific radii. The radius of the electron's orbit depends on the principal quantum number n. For an electron around a stationary nucleus, the electrostatic force of attraction provides the necessary centripetal force.

ie. 14πε0(Ze)er2=mv2r
also mvr=nh2π
From equations (i) and (ii) radius of the r orbit
rn=n2h24π2kZme2=n2h2ε0πmZe2=0.53n2Z (k=14πε0)rnn2Zrn=0.53n2Z


Speed of Electron

In the Bohr model of the hydrogen atom, the speed of the electron in its orbit is a crucial aspect of understanding atomic structure. According to Bohr's theory, From the above relations, the speed of electrons in nth orbit can be calculated as

vn=2πkZe2nh=Ze22ε0nh=(c137)Zn=2.2×106Znm/sec
where (c= speed of light =3×108 m/s)

Solved Examples Based on the Bohr Model of the Hydrogen Atom

Example 1: According to Bohr’s theory, the time average magnetic field at the centre (i.e. nucleus) of a hydrogen atom due to the motion of electrons in the nth orbit is proportional to : (n=principal quantum number)

1) n−4

2) n−5

3) n−3

4) n−2

Solution:

Magnetic field due to current at the centre of the circle =μ0I2r

I=qt=e(2πrv)=ev2πrB=μ(ev2πr)2r=μ0ev4πr2B=(μee4π)(c137)zn(r0n2z)2=μ0ec4π×137r02×z3n5Bn5

Hence, the answer is the option (2).

Example 2: An electron from various excited states of a hydrogen atom emits radiation to come to the ground state. Let λn, and λg be the de Broglie wavelength of the electron in the nth state and the ground state respectively. Let λn be the wavelength of the emitted photon in the transition from the nth state to the ground state. For large n (A, B are constants)

1) Λn2λ
2) ΛnA+Bλn2
3) ΛnA+Bλn
4) Λn2A+Bλn2

Solution:

The velocity of the electron in the nth orbital

v=(e22ϵ0h)zn wherein vαzne22ϵ0h=c137Vn=(e22ϵ0h)1nEn=12mVn2λn=hmVn=hme22ϵ0nh=2ϵ0nh2me2(λn>>>λg)

hcΛn=EnE1=12m(v12vn2)=12m((h2mλg)(hmλn))hcΛn=h22m(1λg21λn2)=h22mλg2(1λg2λn2)Λn0hc=2mλg2h211λ22λn2λgλn<<1 so using Binomial expansion Λn=2mcλg2h(1+λg2λn2)Λn=A+Bλn2

Hence, the answer is the option (2).

Example 3: The ratio of radii of the first three orbits in a hydrogen atom is:

1) 1: 4: 9

2) 1: 2: 3

3) 1: 2: 4

4) 9: 4: 1

Solution:

Atomic number, Z is equal to 1
Hence the radius of nth orbit, rn=0.529n2A0.
For the first three orbits, n values are 1,2 and 3.
Therefore:
The ratio of radii of first three orbits =r1:r2:r3=n12:n22:n32=12:22:32=1:4:9

Hence, the answer is the option (1).

Example 4: The time period of revolution of an electron in its ground state orbit in a hydrogen atom is 1.6×1016s. The frequency of revolution of the electron in its first excited state (in s-1) is:-

1) 5.6×1012
2) 1.6×1014
3) 7.8×1014
4) 6.2×1015

Solution:

The velocity of the nth orbit

Vnzn
the radius of the nth orbit
rnαn2zT=2πrVTn2×nz×z=n3z2T1ffz2n3f1=11.6×1016s1
and
f1f2=(n2n1)3=23=8f2=f18=7.8×1014 s1

Hence, the answer is the option (3).

Example 5: In Bohr's model of a hydrogen-like atom, the force between the nucleus and the electrons is modified as F=e24πϵ0(1r2+βr3) Where β is a constant. For the atom, the radius of the nth orbit in terms of the Bohr radius

(a0=ϵ0h2mπe2)

1) rn=a0nβ
2) rn=a0n2+β
3) rn=a0n2β
4) rn=a0n+β

Solution:

F=mv2r=e24πε0(1r3+βr3)v2=e24πε0m(1r+βr2)

From Bohr's postulate
mvr=nh2πv=nh2πmr
comparing both we get
n2h24π2m2r2=e24πε0m(1r+βr2)

So,
rn=a0n2β

Hence, the answer is the option (3).

Summary

The Bohr model of the hydrogen atom introduced significant insights into atomic structure by proposing that electrons orbit the nucleus in discrete, quantized orbits. Bohr's theory quantifies the radius of these orbits and the speed of electrons, with the radius increasing with the principal quantum number and the speed decreasing in higher orbits. This model explains the stability of electron orbits and the emission of photons during transitions between these orbits. The principles derived from Bohr's model are foundational in understanding atomic spectra and underpin technologies like lasers and spectrometers, demonstrating their relevance to both scientific and practical applications.

Frequently Asked Questions (FAQs)

1. What is the Bohr model of the hydrogen atom?
The Bohr model is a simple representation of the hydrogen atom proposed by Niels Bohr in 1913. It depicts the atom as a small, positively charged nucleus surrounded by an electron moving in circular orbits. The model introduced the concept of quantized energy levels and explained the hydrogen spectrum, but it has limitations and is not applicable to more complex atoms.
2. Why does the Bohr model only work well for hydrogen atoms?
The Bohr model works well for hydrogen atoms because they have only one electron. This simplifies the electron-electron interactions that occur in more complex atoms. For multi-electron atoms, the model fails to account for these interactions and cannot accurately predict their behavior or spectra.
3. How does the Bohr model explain the stability of atoms?
The Bohr model explains atomic stability by proposing that electrons can only exist in specific, stable energy levels or orbits. Electrons do not continuously emit energy as classical physics would predict, but instead only emit or absorb energy when transitioning between these discrete levels. This concept of quantized energy states prevents electrons from spiraling into the nucleus.
4. What is meant by "ground state" in the Bohr model?
The ground state in the Bohr model refers to the lowest energy level an electron can occupy in an atom. For hydrogen, this is the first energy level (n=1). An electron in the ground state is in its most stable configuration and requires energy input to move to a higher energy level.
5. How does the Bohr model explain atomic spectra?
The Bohr model explains atomic spectra by proposing that electrons can only transition between specific energy levels. When an electron moves from a higher to a lower energy level, it emits a photon with energy equal to the difference between the two levels. This results in the characteristic spectral lines observed for each element, as each transition corresponds to a specific wavelength of light.
6. What is the relationship between orbital radius and energy levels in the Bohr model?
In the Bohr model, the orbital radius is directly related to the energy level. As the energy level (n) increases, the orbital radius increases proportionally to n². This means that electrons in higher energy levels orbit farther from the nucleus. The relationship is given by the formula: r = n²a₀, where a₀ is the Bohr radius (the smallest possible orbital radius).
7. What is the role of angular momentum quantization in the Bohr model?
Angular momentum quantization is a key postulate of the Bohr model. It states that the angular momentum of an electron in an orbit must be an integer multiple of ℏ (h/2π). This quantization condition determines the allowed orbital radii and energy levels. It was introduced by Bohr to explain the stability of atoms and the discrete nature of atomic spectra.
8. What is the significance of the Bohr radius?
The Bohr radius (a₀) is the smallest possible orbital radius in the Bohr model, corresponding to the ground state of the hydrogen atom. It serves as a fundamental unit of length in atomic physics and is approximately 5.29 × 10⁻¹¹ meters. The Bohr radius is significant because it provides a characteristic scale for atomic systems and appears in many atomic physics calculations.
9. What is the physical meaning of the principal quantum number in the Bohr model?
In the Bohr model, the principal quantum number (n) determines the energy level and size of the electron's orbit. It takes positive integer values (1, 2, 3, ...) and is related to the total energy of the electron, the orbital radius, and the angular momentum. Larger values of n correspond to higher energy levels, larger orbits, and greater angular momentum.
10. How does the Bohr model relate to de Broglie's matter wave theory?
The Bohr model can be connected to de Broglie's matter wave theory by considering the electron's wavelength. De Broglie proposed that particles can exhibit wave-like properties, with a wavelength inversely proportional to their momentum. In the Bohr model, the allowed electron orbits correspond to standing waves, where the circumference of the orbit is an integer multiple of the electron's de Broglie wavelength.
11. How does the Bohr model explain the ionization energy of hydrogen?
The Bohr model explains the ionization energy of hydrogen as the energy required to move an electron from the ground state (n=1) to infinity (n=∞). This energy corresponds to the work needed to overcome the electrostatic attraction between the electron and the proton. The model accurately predicts the ionization energy of hydrogen, which is about 13.6 eV.
12. How does the Bohr model explain the concept of electron shells?
The Bohr model introduces the concept of electron shells, which are groups of electrons with similar energies. Each shell corresponds to a principal quantum number (n) and can hold a maximum number of electrons given by 2n². This concept helps explain the periodic table structure and chemical properties of elements, although it's oversimplified compared to modern quantum mechanical models.
13. What is the Bohr frequency condition?
The Bohr frequency condition states that the frequency of light emitted or absorbed during an electron transition is proportional to the energy difference between the initial and final states. Mathematically, it's expressed as: ΔE = hf, where ΔE is the energy difference, h is Planck's constant, and f is the frequency of the emitted or absorbed photon. This condition is crucial for explaining the discrete nature of atomic spectra.
14. How does the Bohr model explain the emission of photons?
According to the Bohr model, photons are emitted when an electron transitions from a higher energy level to a lower one. The energy of the emitted photon is equal to the difference in energy between the two levels. This process explains the discrete nature of atomic spectra, as only specific energy transitions are allowed, resulting in specific wavelengths of light being emitted.
15. What is the relationship between the Bohr model and the quantum mechanical model of the atom?
The Bohr model is a precursor to the quantum mechanical model of the atom. While it introduced key concepts like quantized energy levels and electron transitions, it has been superseded by quantum mechanics. The quantum model retains the idea of discrete energy levels but describes electrons as probability clouds rather than particles in definite orbits. The Bohr model can be seen as a simplified, semi-classical approximation of the more accurate quantum mechanical description.
16. How does the Bohr model explain the concept of excited states?
In the Bohr model, excited states occur when an electron occupies an energy level higher than its ground state. This happens when an atom absorbs energy, typically in the form of a photon. Excited states are unstable, and electrons tend to return to lower energy levels by emitting photons. The model explains how different excited states correspond to different spectral lines in atomic emission spectra.
17. What is the significance of the Bohr-Sommerfeld model?
The Bohr-Sommerfeld model is an extension of the Bohr model that introduces elliptical orbits and additional quantum numbers. It attempts to explain the fine structure of spectral lines by considering relativistic effects and the quantization of angular momentum in three dimensions. While more complex than the original Bohr model, it still falls short of the full quantum mechanical description but represents an important step in the development of atomic theory.
18. What is the significance of the Rydberg constant in the Bohr model?
The Rydberg constant is a fundamental physical constant that appears in the Bohr model and the Rydberg formula for atomic spectra. It relates the energy levels of an atom to the wavelengths of light emitted during electron transitions. The Bohr model provides a theoretical derivation of this constant, expressing it in terms of fundamental constants like the electron mass, charge, and Planck's constant.
19. How does the Bohr model explain the concept of binding energy?
In the Bohr model, binding energy is the energy required to remove an electron from its orbit and bring it to rest at infinity. It's equal to the negative of the electron's total energy in a given state. The model shows that electrons in lower energy levels have higher binding energies, explaining why ground state electrons are more tightly bound to the nucleus than those in excited states.
20. What is the significance of the n² term in the Bohr model's energy equation?
The n² term in the Bohr model's energy equation (En = -13.6 eV / n²) reflects the quantization of energy levels. It shows that energy levels are not evenly spaced but become closer together as n increases. This explains why spectral lines converge at higher frequencies and why electrons in higher energy levels are more easily excited or ionized.
21. How does the Bohr model explain the concept of quantum numbers?
The Bohr model introduces the principal quantum number (n) as a way to describe electron energy levels. This laid the foundation for the more comprehensive set of quantum numbers used in modern quantum mechanics. While the Bohr model only uses one quantum number, it paved the way for understanding how discrete quantum numbers could describe electron states in atoms.
22. How does the Bohr model explain the concept of spectral line intensity?
While the basic Bohr model doesn't directly address spectral line intensity, it provides a foundation for understanding this concept. The intensity of spectral lines is related to the probability of electron transitions between energy levels. More probable transitions result in more intense spectral lines. This idea, further developed in quantum mechanics, has its roots in the Bohr model's description of electron transitions.
23. What is the relationship between the Bohr model and the concept of wave-particle duality?
The Bohr model indirectly supports the concept of wave-particle duality. By explaining atomic spectra using both particle-like electrons in orbits and wave-like photons emitted during transitions, it bridges classical and quantum ideas. The model's success in explaining spectral lines using quantized energy levels hinted at the dual nature of matter and energy, a concept fully developed in later quantum theories.
24. How does the Bohr model relate to the periodic table of elements?
While the Bohr model was primarily developed for hydrogen, its concept of electron shells provided a basis for understanding the periodic table. The idea that electrons fill shells with increasing energy levels helps explain the periodicity of chemical properties. Although oversimplified, this concept from the Bohr model is still useful for introducing students to the structure of the periodic table.
25. What is the significance of the Balmer series in relation to the Bohr model?
The Balmer series is a set of spectral lines in the visible region of hydrogen's emission spectrum. The Bohr model successfully explained these lines by showing that they result from electron transitions to the second energy level (n=2) from higher levels. This was a major triumph for the model and provided strong support for its validity.
26. How does the Bohr model incorporate Planck's quantum theory?
The Bohr model incorporates Planck's quantum theory by proposing that electrons can only exist in discrete energy levels, rather than at any arbitrary energy. The energy of these levels is quantized according to Planck's constant (h), which relates the energy of a photon to its frequency. This quantization is fundamental to explaining the discrete nature of atomic spectra.
27. How does the Bohr model explain the Rydberg formula?
The Bohr model provides a theoretical basis for the empirical Rydberg formula, which describes the wavelengths of spectral lines in hydrogen. By calculating the energy levels of the hydrogen atom, Bohr showed that the differences between these levels correspond exactly to the observed spectral lines. This derivation of the Rydberg formula from first principles was a major success of the Bohr model.
28. What are the main limitations of the Bohr model?
The main limitations of the Bohr model include: 1) It only works well for hydrogen-like atoms, 2) It doesn't explain the fine structure of spectral lines, 3) It violates the Heisenberg uncertainty principle, 4) It doesn't account for electron spin, and 5) It fails to explain molecular bonding. Despite these limitations, it remains a useful introductory model for understanding basic atomic structure.
29. How does the Bohr model explain the Lyman series?
The Lyman series is a set of spectral lines in the ultraviolet region of hydrogen's emission spectrum. The Bohr model explains these lines as resulting from electron transitions to the ground state (n=1) from higher energy levels. Each line in the series corresponds to a specific transition, with the energy differences between levels determining the wavelengths of the emitted photons.
30. How does the Bohr model explain the stability of electron orbits?
The Bohr model explains the stability of electron orbits by postulating that electrons can only exist in certain allowed energy states. In these states, the electron's centripetal force exactly balances the electrostatic attraction to the nucleus. The model asserts that electrons in these orbits do not radiate energy continuously, contrary to classical electromagnetic theory, thus maintaining stable orbits.
31. How does the Bohr model explain the concept of quantum jumps?
The Bohr model introduces the concept of quantum jumps, which are instantaneous transitions of electrons between energy levels. When an electron moves from a higher to a lower energy level, it emits a photon; when it moves to a higher level, it absorbs a photon. These discrete jumps explain the quantized nature of atomic spectra and represent a departure from classical physics, where energy changes were thought to be continuous.
32. What is the role of Coulomb's law in the Bohr model?
Coulomb's law plays a crucial role in the Bohr model by describing the electrostatic force between the positively charged nucleus and the negatively charged electron. This force provides the centripetal force necessary to keep the electron in its circular orbit. The balance between the Coulomb force and the centripetal force determines the allowed orbital radii and energy levels in the model.
33. How does the Bohr model relate to the concept of quantized angular momentum?
The Bohr model introduces the concept of quantized angular momentum, postulating that the angular momentum of an electron in an orbit must be an integer multiple of ℏ. This quantization condition is crucial for determining the allowed orbital radii and energy levels. It represents a fundamental departure from classical physics and is a key aspect of the model's quantum nature.
34. What is the relationship between the Bohr model and the hydrogen emission spectrum?
The Bohr model provides a theoretical explanation for the hydrogen emission spectrum. It shows that the discrete spectral lines result from electrons transitioning between specific energy levels. Each spectral series (Lyman, Balmer, Paschen, etc.) corresponds to transitions ending at a particular energy level. The model's ability to explain these spectra was a major triumph and provided strong support for its validity.
35. How does the Bohr model explain the concept of electron shells and subshells?
While the Bohr model primarily deals with principal energy levels (shells), it laid the groundwork for understanding electron shells and subshells. Each principal quantum number (n) in the Bohr model corresponds to a shell. The model suggests that each shell can hold a maximum of 2n² electrons, which aligns with the more detailed descriptions of subshells in later atomic models.
36. What is the significance of the Bohr magneton in relation to the Bohr model?
The Bohr magneton is a physical constant of magnetic moment named after Niels Bohr. While not directly part of the original Bohr model, it arises from considering the magnetic moment of an electron in a Bohr orbit. It serves as a natural unit for expressing the magnetic moments of electrons in atoms and is important in understanding atomic magnetism and the Zeeman effect.
37. How does the Bohr model explain the concept of atomic size?
The Bohr model provides a simple way to conceptualize atomic size. In this model, the size of an atom is determined by the outermost occupied electron orbit. As the principal quantum number (n) increases, the orbital radius increases as n², explaining why atoms in excited states are larger than those in the ground state. This concept helps in understanding trends in atomic radii across the periodic table.
38. What is the role of the Planck-Einstein relation in the Bohr model?
The Planck-Einstein relation (E = hf) is fundamental to the Bohr model. It connects the energy of a photon to its frequency, allowing the model to explain the discrete nature of atomic spectra. When an electron transitions between energy levels, the energy difference is emitted or absorbed as a photon with a specific frequency, directly linking the atomic energy levels to observable spectral lines.
39. What is the significance of the Bohr model in the development of quantum mechanics?
The Bohr model was a crucial stepping stone in the development of quantum mechanics. It introduced key concepts like quantized energy levels, electron transitions, and the wave-particle duality of light. While ultimately superseded by more accurate quantum mechanical models, it provided a conceptual framework that guided further research and helped bridge the gap between classical and quantum physics.
40. What is the significance of the Bohr model in explaining atomic emission and absorption spectra?
The Bohr model

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