Bohr Radius

Bohr radius has a value of 5.29177210903(80)1011 m. It is almost equal to the distance between the nucleus and the electron in a hydrogen atom, so it is known as the Bohr radius (a0).

Bohr Radius

:eight_pointed_black_star: What Is Bohr Radius?

:small_red_triangle_down: Bohr Radius In Different Units

ao in Bohr radius
SI units 5.29×10−11 m
Imperial or US units 2.08×10−9 in
Natural units 2.68×10−4 /eV

The Bohr radius (a0) is approximately equal to the distance between a hydrogen atom’s nucleus and its electron in its ground state. As part of the Bohr model of an atom, it bears his name. 5.29177210903(80)1011 m is the calculated value.

Niels Bohr proposed the Bohr model of atomic structure in 1913, in which electrons are attracted to a central nucleus by electrostatic forces. Using the original derivation, electrons were shown to have orbital angular momentum in integer multiples of the decreased Planck constant, which matched the observed emission spectra and predicted a fixed radius for each level.

:eight_pointed_black_star: Bohr Radius Of Hydrogen

The shortest conceivable orbit of an electron around the nucleus has an orbital radius almost equal to the Bohr radius in hydrogen, the most basic element. As a result, the Bohr radius is somewhat less than it would otherwise be. There is a 0.05 percent difference between them.)

The electron probability cloud following the Schrödinger equation, introduced in 1926, replaced the Bohr model of the atom. Spin and quantum vacuum effects complicate things further, resulting in fine and hyperfine structure. Due to its easy link to basic constants, the Bohr radius formula remains an important tool in computations in atomic physics. As a result, it was adopted as the atomic unit of length.

In Schrödinger’s quantum mechanical theory of the hydrogen atom, the Bohr radius is the most likely value of the radial coordinate of the electron location, and hence the most probable distance of the electron from the nucleus.

The generalization of this conclusion may be applied to other systems, such as positronium and muonium, which are electrons circling positrons and anti-muons, utilizing the lower mass and the probable change in charge of the system.


Simply substituting the electron’s mass with the lower mass for the system is usually all that is needed to modify the Bohr model’s radii, energies, and so on (as well as adjusting the charge when appropriate). As an illustration, the radius of positronium is only an approximation of the real thing.



At a0, the hydrogen atom’s ground state wavefunction is at its highest, but it gradually decreases to an effective limiting radius of around 3a0. The apparent 2 disparity between the 5 computed values of 2a0 and the empirical value of 3a0 raises the question of probable causes. The following are two plausible physical reasons.

  1. One would anticipate R to be an overestimate of the real hydrogen atom’s limiting radius since electromagnetic interactions are not included in the calculation of R, and these extra interactions would enhance the binding strength between the nuclear and electronic components.

  2. The first approximation to General Relativity is used in the following calculation of R. c. The above calculation of R uses a Newtonian gravitational model. For a generic relativistic computation, it is realistic to predict a correction factor of 2 or more.

As anticipated by the discrete fractal paradigm, the gravitational coupling constant was used in the GBR computation, which has substantial implications for atom behavior.

For the discrete fractal paradigm, there is substantial empirical evidence, including numerous retrodictive successes and promising tentative results regarding its major predictions about the nature of enigmatic dark matt.

Which dominates the mass of the observable universe, as well as several very encouraging tentative results. At the Discrete Scale Relativity level, two recent publications show that gravity is the fundamental dynamical interaction within constrained systems.

When G-1 is used as the appropriate gravitational coupling constant, hadrons can be described as Kerr-Newman black holes, as shown in one research. According to the other paper6, adopting G-1 for Atomic Scale systems explains the long-standing enigma around the physical significance of the fine structure constant 6.

Unit electromagnetic and gravitational interactions in bonded atomic systems are proven to be the ratio of their respective strengths.


This suggests that gravitational interactions are 137.036 times stronger than electromagnetic interactions in such systems. Observed dynamics are regulated nearly exclusively by electromagnetic interactions, as G0 applies to external interactions between unbound charged Atomic Scale systems.

:eight_pointed_black_star: About Niels Bohr

:small_red_triangle_down: Fast Facts

Name Niels Bohr
Full name Niels Henrik David Bohr
Born 7 October 1885
Died 18 November 1962 (aged 77)
Known for Physics contributions
Spouse(s) Margrethe Nørlund ​(m. 1912)​
Children 6
Resting place Assistens Cemetery

For his contributions to the knowledge of atomic structure and quantum theory, Niels Henrik David Bohr earned the Nobel Prize in Physics. Furthermore, Bohr was a philosopher and proponent of scientific study.

Since electrons do not always stay in one stable orbit around the nucleus as previously thought, Bohr devised the Bohr model of the atom to account for this potential for quantum leaping between distinct energy levels.

This model’s basic ideas remain true, even though other models have taken their place. Complementarity is the idea that things may be analyzed in terms of their conflicting characteristics, such as a wave or a stream of particles. Bohr’s scientific and philosophical work was dominated by the concept of complementarity.

In 1920, Niels Bohr launched the Niels Bohr Institute at the University of Copenhagen, which he started as the Institute of Theoretical Physics. Hans Kramers, Oskar Klein, George de Hevesy, and Werner Heisenberg were among the scientists Bohr mentored and cooperated with.

Hafnium, the Latin word for Copenhagen, where the new zirconium-like element was found, was named after him. Elements named after him include bohrium and boron. He rescued immigrants fleeing Naazi persecution in the 1930s.

His famous encounter with Wernher von Heisenberg, the German nuclear weapons project’s chief after the Germans conquered Denmark, took place after the occupation. Bohr escaped to Sweden in September 1943 after learning he was due to be imprisoned by the Germans.


The British Tube Alloy’s nuclear weapons program recruited him, and he was part of the British mission to the Manhattan Project from then on out. International collaboration on nuclear energy was advocated by Bohr following World War II While at CERN and the Danish Atomic Energy Commission, he was instrumental in establishing Ris and becoming its first chairman in 1957; he also worked on CERN’s development.

:eight_pointed_black_star: How Bohr Model Was Invented?

How Bohr Model Was Invented?

In September 1911, Bohr traveled to England, where much of the theoretical work on the atomic and molecular structure was being done. He met J. J. Thomson of Trinity College, Cambridge. He went to James Jeans and Joseph Larmor’s electromagnetic lectures and researched cathode rays, but he didn’t impress Thomson.

  • In 1911, Ernest Rutherford, a New Zealander, disputed Thomson’s 1904 plum pudding model with a small central nucleus Rutherford model of the atom. To pursue post-doctoral study at Victoria University of Manchester, Bohr met George de Hevesy and Charles Galton Darwin (whom Bohr referred to as “the grandson of the real Darwin”).

  • Bohr married in July 1912 and honeymooned in England and Scotland. Upon his return, he lectured on thermodynamics at the University of Copenhagen. Martin Knudsen proposed Bohr as a docent, which was authorized in July 1913.

  • His three pieces were published in Philosophical Magazine in July, September, and November of that year. It was based on Rutherford’s nuclear structure and Max Planck’s quantum theory.

  • Bohr’s approach to planetary atom models was novel. To stabilize an atom, electrons travel in quantized “stationary states” orbits around the nucleus. However, it wasn’t until his 1921 paper that he showed the number of electrons in the outer orbits of each atom’s atom determined the chemical properties of each element.

  • He proposed that an electron may drop from a high-energy orbit to a low-energy orbit, emitting a discrete quantum of energy. This constituted the basis for the old quantum theory. The Pickering series defied Balmer’s formula.

When Alfred Fowler pressed Bohr on this, he said they were created by ionized helium, helium atoms with one electron. The Bohr model worked for such ions. Thomson, Rayleigh, and Hendrik Lorentz were among those who disapproved of the trilogy, while Rutherford and his contemporaries like Einstein, Fermi, Max Born, and Arnold Sommerfeld hailed it as a breakthrough.

Experiments confirmed the trilogy’s capacity to explain occurrences that other models couldn’t explain. The Bohr model of the atom has been surpassed, although it is still the most widely used in high school physics and chemistry textbooks.

Bohr despised tutoring medical students Then he returned to Manchester, where Rutherford had hired him to replace Darwin as a reade. Bohr was avowed. He enjoyed a vacation in Tyrol with his brother Harald and aunt Hanna Adler. He also visited Göttingen and Munich’s Ludwig Maximilian University, where he met Sommerfeld and gave seminars on the trilogy.


The outbreak of WWI delayed their return to Denmark and Bohr’s following travel to England, where he landed in October 1914. They stayed until July 1916, when he was appointed to the newly founded Chair of Theoretical Physics at the University of Copenhagen. However, he had to teach physics to medical students after losing his docents. King Christian X was delighted to meet such a great football player.

:eight_pointed_black_star: What Are The Postulates Of Bohr’s Atomic Model?

What are the postulates of Bohr's atomic model?

The experimental Rydberg formula may be used to determine the locations of absorption and emission lines in the spectrum of atomic hydrogen, as shown in the figure. The spectrum of atomic hydrogen cannot be explained using classical physics.

Atomic hydrogen’s radiation spectra were first accurately explained by the Bohr model of its atomic structure, which was developed in the early twentieth century. Before it, the Rutherford nuclear model of the atom had been developed.

According to Rutherford’s concept, an atom is made up of a positively charged point-like nucleus that holds almost all of the atom’s mass, as well as negatively charged electrons that are dispersed throughout the atom’s rest of its mass.

The following are the postulates that Bohr used to construct his model of the hydrogen atom:

  • Electrons rotate around the nucleus in a set circular route referred to as “orbits,” “shells,” or “energy level” as they go around the nucleus.

  • “Stationary orbit” is the word used to describe these orbits.

  • A certain amount of fixed energy will be present in every circular orbit, and these circular orbits were referred to as orbital shells. For as long as the electrons continue to rotate about the nucleus in their set orbital shells, there will be no emission of energy from them.

  • Numbers such as n=1, n=2, n=3, and so on are used to represent the different energy levels. Quantum numbers are what these are referred to as. From the lowest energy level (nucleus side n=1) to the greatest energy level (excitation side n=1), a variety of quantum numbers can be used.

  • The multiple energy levels or orbits are represented in two ways: as numbers such as 1, 2, 3, 4… or as shells such as K, L, M, N. The ground state of an electron is the lowest energy level that it can achieve.

  • When electrons leap from one energy level to another, there is a corresponding shift in energy. The electrons in atom travel from a lower to a higher energy level by collecting the necessary energy. When an electron loses energy, on the other hand, it goes from a higher to a lower energy level.

Atomic hydrogen and hydrogen-like ions with low atomic numbers may be explained using Bohr’s concept of a hydrogen atom, which explains their emission and absorption spectra.


It was the first model to propose the notion of a quantum number to explain atomic states and to assume quantization of electron orbits in the atom, as well as the first model to postulate the existence of a quantum number. The invention of Bohr’s model, which deals with many-electron atoms, is a significant step in the development of quantum mechanics.

:eight_pointed_black_star: The Niels Bohr Atomic Model: Features And Limitations

Features and Limitations

Thomson’s atomic model and Rutherford’s atomic model both failed to provide satisfactory answers to any issues about the energy of an atom or its stability. Niels Bohr proposed an atomic structure model in 1913, describing an atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the positively charged nucleus.

  • Similar to how planets orbit the sun in our solar system, with attraction provided by electrostatic forces. This model is commonly referred to as Bohr’s atomic model.

  • It was essentially an upgraded version of Rutherford’s atomic model that overcame the constraints of the original model. On the majority of topics, he agrees with him, such as the notion of a nucleus and electrons circling the nucleus. The following are the most notable characteristics of Niels Bohr’s atomic model:

  • Electrons rotate around the nucleus in stable orbits, emitting no radiant energy and remaining in their stable orbits. Each orbit contains a specific amount of energy, which is referred to as an energy shell or energy level.

  • The shells of an orbit or energy level are denoted by the letters K, L, M, and N. When the electron is at its lowest energy level, it is referred to as being in the ground state of the particle.

When an electron transitions from one orbit or energy level to another, it either emits or absorbs energy. The transition from a higher to a lower energy level results in the emission of energy, whereas the transition from a lower to a higher energy level results in the absorption of energy (and vice versa).


Plank’s equation states that the amount of energy absorbed or released is equal to the difference between the energies of the two energy levels (E1, E2), with the amount of energy absorbed or emitted equally to the difference between the energies of the two energy levels (E1, E2).

:eight_pointed_black_star: The Atomic Model Theory Of Bohr limitations

It is an infringement of the Heisenberg Uncertainty Principle. According to the Bohr atomic model theory, electrons have both a known radius and orbit, i.e., a known location and momentum at the same time, which is contrary to Heisenberg’s hypothesis.

Even while the Bohr atomic model theory achieved accurate predictions for smaller-sized atoms such as hydrogen, when larger-sized atoms are taken into consideration, the resulting spectrum predictions are poor.

Although it attempted to explain the Zeeman effect when the spectral line is divided into many components in the presence of a magnetic field, it was unsuccessful.


However, it was unable to account for the Stark effect, which occurs when the spectral line is broken into fine lines in the presence of an electric field.

Frequently Asked Questions - FAQs

People asked many questions about the Bohr model. We discussed a few of them below:

:one: What is the definition of Bohr’s radius class 12 physics?

The radius of Bohr’s Model: The radius on which electrons flow around the nucleus in the orbit defined by Bohr’s model is referred to as Bohr’s radius.

:two: What was the process by which Bohr discovered his model?

He came up with the notion after analyzing the way flaming, hot hydrogen emits light in a laboratory. When an incandescent light bulb is turned on, it emits light at all of the distinct wavelengths. Because of the heated filament, all of the distinct wavelengths of light are emitted from the light bulb as the filament becomes hotter.

:three: What is the first Bohr orbit, exactly?

When there are two electrons in the initial Bohr orbit, the orbit is filled, which explains why helium is inert. The second orbit accommodates eight electrons, and when it is filled, the atom is neon, which is once again inert.

:four: What was it that Niels Bohr discovered?

A hypothesis of the hydrogen atom was presented by Niels Bohr in 1913, based on the quantum theory, which states that some physical quantities can only have discrete values. Electronic particles circulate about a nucleus, but only in defined orbits.

:five: What exactly was the Bohr experiment?

The Bohr model depicts the atom as a tiny, positively charged nucleus that is surrounded by electrons in an orbiting state. As the first to establish that electrons move in different orbits around the nucleus and that the number of electrons in the outer orbit affects the characteristics of an element, Niels Bohr is credited with discovering the concept of orbital symmetry.

:six: What languages did Niels Bohr know and understand?

Niels performed admirably in the majority of school courses, but he struggled with his native language, Danish. While he enjoyed chatting, he had a strong aversion to the act of writing essays.

:seven: Who was the person who discovered the proton?

This year marks the 100th anniversary of Ernest Rutherford’s publication of his discoveries establishing the existence of the proton. Over several decades, the proton was regarded as an elementary particle.

:eight: What is the value of Z in the Bohr radius?

The atomic number Z is the same as one. As a result, the radius of the nth orbit is rn = 0.529n2. The n values for the first three orbits are 1, 2, and 3.

:nine: What does Bohr’s model explain?

The Bohr model proposes that electrons circle the nucleus at fixed energy levels as they move around the nucleus. Higher energy levels exist in orbits that are farther away from the nucleus. Light is produced when electrons fall to a lower energy level and emit energy in the form of light.

:keycap_ten: Is the Bohr model accurate?

This model was presented by Niels Bohr in 1915; it is not exact, but it has many nearly true aspects, and it is suitable for most of our discussion since it is accurate enough.

:closed_book: Conclusion

In 1913, Niels Bohr published the Atomic Hydrogen Model, which is still in use today. According to his description, it is made up of a positively charged nucleus made up of protons and neutrons, which is surrounded by a negatively charged electron cloud. In the model, electrons are arranged in atomic shells around the nucleus. Electrostatic forces between the positively charged nucleus and its negatively charged surrounds hold the atom together.

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