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Angular momentum of electrons, according to Bohr’s atomic model, orbiting around the nucleus is quantized. Angular momentum of an electron by Bohr is given by mvr or nh/2π, where v is the velocity, n is the orbit in which the electron is, m is the mass of the electron, and r is the radius of the nth orbit. Here, we will discuss the concept in detail along with De Broglie’s Explanation of the quantization of angular momentum of electrons.
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Key Takeaways: Momentum of electron, angular momentum of electron, De Broglie’s explanation to the quantization of angular momentum, Bohr’s second postulate, Electron, Nucleus, Velocity, Ordit
Momentum of Electron
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The atomic model of Bohr established a number of postulates regarding the arrangement of electrons in various orbits around the nucleus. The angular momentum of electrons orbiting around the nucleus is quantized in Bohr's atomic model. He went on to say that electrons only move in orbits where their angular momentum is an integral multiple of h/2. Louis de Broglie later stated this hypothesis about the quantization of an electron's angular momentum. A moving electron in a circular orbit, according to him, behaves like a particle wave. mvr or nh/2 is Bohr's expression for an electron's angular momentum. Here v stands for the velocity, n stands for the orbit in which the electron is, m stands for the mass of the electron, and r stands for the radius of the nth orbit).
Read more: Magnetic Field due to a Current Element
What is Angular Momentum of Electron?
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Angular momentum is a property of mass in motion around a fixed axis that is conserved in a closed domain. The rotating equivalent of linear momentum is angular momentum (also known as moment of momentum or rotational momentum). Because it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics.
Angular Momentum of Electron
The angular momentum of a point particle in three dimensions is a pseudovector r p, which is the cross product of the particle's position vector r (relative to some origin) and its momentum vector, which in Newtonian physics is p = mv. Because the particle's position is measured from the origin, angular momentum, unlike momentum, is dependent on where the origin is chosen.
L = r x p
L = m x v x r
Read more: Magnetic Field on the Axis of a Circular Current Loop
Quantization of Angular Momentum of Electron
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Rutherford postulated that electrons orbit around an atom's nucleus. One issue with this model is that circling electrons experience centripetal acceleration in classical physics, and accelerating charges lose energy by radiating; a stable electronic orbit is classically impossible. Despite this, Bohr postulated stable electronic orbits, with electronic angular momentum quantified as:
When angular momentum is quantized, the radius of the orbit and its energy are quantized as well. The discrete lines in the spectrum of the hydrogen atom, according to Bohr, were caused by transitions of an electron from one permissible orbit/energy to another. He also thought that, as Einstein predicted, the energy for a transition is obtained or released in the form of a photon, such that:
The Bohr frequency condition is what this is called. This criterion, when combined with Bohr's equation for the permitted energy levels, produces a close match to the observed hydrogen atom spectrum. It only works for atoms with one electron, though.
Read more: Derivation of lorentz transformation
De Broglie’s Explanation to the Quantization for Angular Momentum of Electron
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De Broglie devised a theory to explain why angular momentum might be quantized in the way Bohr imagined it was. De Broglie observed that if you take the electron's wavelength and assume that an integral number of wavelengths must fit in the perimeter of an orbit, you get the same quantized angular momenta as Bohr.
De Broglie's Explanation to the Quantization for Angular Momentum of Electron
The circular orbit's circumference must be an integer number of wavelengths. Therefore, for an electron that is moving in the kth circular orbit of radius rk, the total distance is equal to the circumference of the orbit, 2πrk.
Where,
λ = de Broglie wavelength.
It is known that de Broglie wavelength is given by,
p = electrons momentum
h = Plank’s constant
Therefore,
Where mvk = the momentum of an electron revolving in the kth orbit.
Now, putting the value of λ from equation (2) in equation (1) we get,
2πrk = kh/mvk
mvk rk = kh/2π
Thus it can be concluded that de Broglie's hypothesis successfully proves Bohr’s second postulate which asserts the quantization of angular momentum of the orbiting electron. It can also be expressed that due to the wave nature of the electron, the quantized electron orbits and energy states are.
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Angular Momentum of Electron Formula
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- L = m x v x r
- ΔE = h x v
- m x v x r = (n x h)/(2 x π)
Things to Remember
- The angular momentum of electrons orbiting around the nucleus is quantized in Bohr's atomic model.
- According to him, electrons only move in orbits where their angular momentum is an integral multiple of h/2.
- The amount of momentum that an object possesses is determined by two factors: the amount of material moving and the speed at which it is traveling.
- The rotating equivalent of linear momentum is angular momentum (also known as moment of momentum or rotational momentum).
- L = m x v x r
- When angular momentum is quantized, the radius of the orbit and its energy are quantized as well.
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Sample Questions
Ques: Who proposed the quantization of angular momentum? (2 marks)
Ans: The quantization of angular momentum was proposed by Bohr.
Ques: How is the angular momentum of an electron calculated? (2 marks)
Ans: The angular momentum of an electron can be calculated using Bohr’s relation which is given by mvr or nh/2π.
Ques: The angular momentum of an electron in Bohr’s hydrogen atom whose energy is -3.4 eV, is it? (2 marks)
Ans: The energy of the electron in the nth orbit of the hydrogen atom
En=−13.6/ n2 eV
⇒3.4=−13.6/ n2
⇒n2=4
or,
n=2
Ques: Is angular momentum always conserved quantity? (2 marks)
Ans: Yes, because angular momentum is a significant quantity, it is conserved. The rotating equivalent of linear momentum, according to physicists, is angular momentum. As a result, the total angular momentum in a closed-loop/structure is considered to be Constant.
Ques: Is angular momentum a vector quantity? (2 marks)
Ans: Yes, angular momentum can be expressed as a vector quantity. Angular velocity is a vector quantity as well. The reason for this is that they both have magnitude and direction. The angular momentum works perpendicular to the plane of rotation in this case.
Ques: Angular Momentum of the electron in 4th orbit is? (2 marks)
Ans: The angular momentum (mvr) of the electron in the nth orbit is equal to nh/2π.
For the electron in 4th orbit,
angular momentum = mvr = 4h/2π = 2h/π
Ques: The angular momentum of an electron at a distance of 4.768 ? from the nucleus of a hydrogen atom is? (3 marks)
Ans: Radius of orbit (r) = 0.529 x n2/Z ?
Or, 4.768 ? = 0.529 x n2/1 ?
Or, n2 = 4.768 / 0.529 = 9.013
Therefore,
n = 3
And the angular momentum, mvr = 3h/2π
Ques: What is the relation between angular and linear momentum? (2 marks)
Ans: The linear momentum is defined as the product of mass and velocity while the angular momentum is the product of angular velocity and mass. The unit is measured in a kilogram meter square.
Ques: Derive the expression for the radius of nth electron orbit using Bohr’s postulates of the atomic model. Hence obtain the expression for Bohr’s radius. (5 marks)
Ans: Basic postulates of Bohr’s atomic model are as follows:
(i) Every atom consists of a central core called a nucleus wherein the entire positive charge and mass of the atom are concentrated. In a circular orbit, a suitable number of electrons are revolving around the nucleus. The centripetal force is required for the revolution which is provided by the electrostatic force of attraction between the electron and the nucleus.
(ii) Electron can revolve only in certain discrete non-radiating orbits, called the stationary orbit. The total angular momentum of the revolving electron in an integral multiple of h/2π, where [h is plank constant]
Mvr = nh/2π.
(iii) The radiation of energy takes place only when an electron jumps from one permitted orbit to another. When the electron jumps from the inner to outer orbit, the difference in the total energy of electrons in the two permitted orbits is absorbed and emitted when an electron jumps from the outer to the inner orbit.
Radii of Bohr’s stationary orbits. As per Bohr’s postulates, the angular momentum of electron for any permitted orbit is,
Mvr = nh/2π or, v = nh/ 2πmr … (i)
Again, according to Bohr’s postulates, the centripetal force is equal to the electrostatic force between the electron and nucleus.
r = h2/ 4πmke2 is the exression for the Bohr’s radius.
Ques: (a) Show that the circumference of the electron in the n,h orbital state in a hydrogen atom is n times the de-Broglie wavelength associated with it using Bohr’s second postulate of quantization of orbital angular momentum.
(b) The electron in a hydrogen atom is initially in the third excited state. Find the maximum number of spectral lines which can be emitted when it finally moves to the ground state. (5 marks)
Ans: (a) Based on the de-Broglie hypothesis, this electron is connected with the wave character.
Therefore a circular orbit can be assumed as stationary energy which states only if it contains an integral number of de- Broglie wavelengths, that is we must have
2πr = nλ
Mvr = nh/2π or, 2πr = nh/ mv or, 2πr = nλ
Thus the circumference of the electron in the nth orbital in a hydrogen atom is n times the de-Broglie wavelength connected with it.
(b) For third excited state n = 4
For ground-state n = 1
Thus, possible transitions are :
ni = 4 to nf = 3, 2, 1
ni = 3 to nf = 2, 1
ni = 2 to nf = 1
Therefore, the total number of transitions =6.
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