What is Momentum?
It is the product of mass and velocity of a particle traveling in a definite medium. It It is the product of mass and velocity of a particle traveling in a definite medium. It is a vector quantity (has both magnitude and direction).It is very well defined and explained in Newton's second law of motion which states that rate of change of momentum is equal to the force acting on that object (if it follows the conditions for law of motion).
Real Life Example of Momentum -
Taking the example of a gun, before firing, both the guns remain in position of rest. This implies that the total momentum of the gun (when firing is not done) is zero. But, when the bullet is fired, it gains some momentum. Also, the total moment of the system is conserved by the recoiling of the gun.
Momentum of Electron
A scientist known as Bohr, has put forward the hypothesis of angular momentum of an electron. If we follow Bohr’s atomic model, we can study the different arrangements of electrons that are located among different orbits near the nucleus.
Also, using Bohr’s atomic model, we can manipulate the angular momentum of the electron that is quantized, revolving around the nucleus.
In his further statement, he included the movement of electrons by stating that the only electron which can revolve around its orbit possesses the angular momentum of its integral multiple of h/2.
Later on, Louis de Broglie explained the postulate of the quantization of the angular momentum of an electron given by Bohr. He stated that an electron in motion shows the behavior of a particle-wave.
Mathematically,
Angular momentum = mvr or \[\frac{nh}{2\pi }\]
Here,v = velocity
n = the orbit where electrons are present
m = mass of the electron
r = the radius of the nth orbit
Angular Momentum of Electron
Louis de Broglie explained the quantization of angular momentum of the electron. According to him, the particle waves travel in a cord and have an analogous nature (behavior).
Under the resonant conditions, particle waves can drive towards the standing waves. If we pull out a stationary string plucked, the excitement of wavelength occurs. As per the findings, it is inferred that the waves that have nodes can survive the angular momentum, and later on, they create standing waves in the string.
Angular Momentum of Electron Formula
Standing waves are formed in a string when a wave completes the total distance of its integral number of wavelengths.
That is why, for an electron in motion in the kth circular orbit, having the radius of rk, the total distance covered by any electron = circumference of the orbit of an electron.
The formula of orbital angular momentum is given by:
2πrk = kλ …. 1
Here,
λ = de Broglie wavelength
According to the de Broglie equation, the expression is:
λ = \[\frac{h}{p}\]
Here,
h is Planck’s constant
p is the momentum of an electron
Therefore, λ = \[\frac{h}{mvk}\] …. 2
Here,
mvk is momentum of electron in the kth orbit
So, by putting the value of λ from equation 2 in equation 1, we get:
\[\frac{kh}{mvk}\] = 2πrk
\[\frac{kh}{2\pi }\] = mvkrk
So, we can infer that Bohr’s second postulate is successfully proven by de Broglie hypothesis.
The above inference states the quantization of the angular momentum of the orbiting electron. We can also find the nature of the wave of an electron, which explains the energy states of the electron along with the quantized electron orbits.
Orbital Angular Momentum of Electron
The orbital angular momentum of an electron can be obtained using the right formula. To make the learning easier; the orbital angular momentum is quantized. We need to apply the orbital angular momentum quantum number l.
Here is the formula for orbital angular momentum:
\[L=\sqrt{l(l+1)h}\]
Mvr Formula
We can calculate the angular momentum of a particle having the mass M with radius and velocity v.
The expression is given as:
L = mvr sin θ
Here,
θ = tangential angle at a certain point with the circumference of the orbit.
If we do the vector product, then L = \[\vec{r}\] х \[\vec{p}\]
The angular momentum magnitude of a ring-shaped orbit, L = Mvr
How to Calculate Orbital Angular Momentum?
We can calculate the orbital angular momentum by using the following orbitals taken as an example:
Example: 1
Find out the orbital angular momentum of these orbitals: 3p, 3d, and 3s.
Ans: The formula to calculate the orbital angular momentum is:
\[L=\sqrt{l(l+1)h}\]
Here, h = \[\frac{h}{2\pi }\]
, l = angular momentum
The values of l for different orbitals are given below.
s-orbital = 0
p-orbital = 1
d-orbital = 2
f-orbital = 3
Here, For 3p, so Angular momentum =
\[\sqrt{l(l+1)h}\]
= \[\sqrt{2h}\]
For orbital 3d, angular momentum =
\[\sqrt{2(2+1)h}\]
= \[\sqrt{6h}\]
For orbital 3s, angular momentum =
\[\sqrt{0h}\]
= 0
Angular Momentum of Electron in Hydrogen Atom
We can calculate the angular momentum by applying de Broglie’s hypothesis upon an electron of a Hydrogen atom.
We know that the associated wavelength of any particle according to Broglie hypothesis is:
λ = hMv.....(1)
From classical mechanics, the orbital angular momentum is:
L = Mvr …. (2)
It is also noted that
2πr = mλ
λ = 2πrm
By inserting the value of in equation 1, we get:
2πrm = hmv....3
From equation 1 we can get:
Mv = Lr
By substituting the Mv in equation 3 we get:
2πrm = h(Lr) = rhL
Now, solve this relation for angular momentum L:
L = rh х m2πr
or, L = mh2π
or, L = m . h2π
Here, h2π is designated as h.
So, the angular momentum of the electron orbiting in the Hydrogen atom is:
L = m.h
Here, m = 1, 2, 3, 4 …
Also, the angular momentum is quantized in units of ħ.
Law of Conservation of Momentum :
Acc. to this law, the total momentum of the system is always conserved. Therefore, the momentum before the movement of objects = momentum after the movement of objects in a system. Also, value momentum depends on mass and velocity of the object in that system.
Following are some important applications of this law -
Air - Filled Balloon : Nowadays, balloons are filled with air for various uses. Here, the balloon is filled with air and this air forms a system in the balloon. When the balloon stays at rest, the momentum of that balloon is zero. But, when the balloon's knot is loosen up, The balloon accelerates with some velocity in the air.
The Example of a Gun - the momentum of a gun changes when firing a gun.
Rockets and Jets - These aircrafts also follow the law of conservation of momentum. In these, burning of fuel rush produces large momentum. Due to this, the aircrafts launches. Also, the total momentum in this case also remains conserved.
Similarly, when playing games like football, basketball etc. When we kick the ball or throw it, certain momentum is gained by the ball. This moment is conserved in the environment.
To understand this topic, students must know about torque.
Torque
Torque is also called rotational force or turning effect. This force acts as the rotational equivalent of linear force in a system having angular rotation. Torque is a vector quantity and represented by 'T' . It is calculated by the formula T = position vector × Force vector.
Angular Momentum by Bohr's Model of Hydrogen Atom
In 1913, Niels Bohr studied hydrogen atoms in depth and proposed the atomic Hydrogen model to the world. He described the hydrogen model as a positively charged nucleus which contains protons and neutrons. Also, he explained the negatively charged electron cloud which surrounds these protons and neutrons.
The angular momentum of an electron can also be calculated with the formula given by bohr. When the velocity (v), number of orbits (n) and mass of electron (m) are known, mvr or nh/2π gives the angular momentum of the electron.
FAQs on Angular Momentum of Electron
1. What is the angular momentum of an electron in an atom?
The angular momentum of an electron is a physical property that describes the electron's rotational motion as it orbits the nucleus. According to Niels Bohr's atomic model, an electron can only revolve in specific orbits where its angular momentum is an integral multiple of h/2π. This property is fundamental to understanding the stability and structure of atoms as described by quantum mechanics.
2. How is the angular momentum of an electron calculated according to Bohr's model?
According to Bohr's second postulate for atomic models, the angular momentum (L) of an electron in a specific orbit is calculated using the formula: L = mvr = nh/2π. In this formula:
- m is the mass of the electron.
- v is the velocity of the electron in its orbit.
- r is the radius of the orbit.
- n is the principal quantum number (or the orbit number, e.g., 1, 2, 3...).
- h is Planck's constant.
3. What does it mean for an electron's angular momentum to be 'quantized'?
To say an electron's angular momentum is quantized means it can only have specific, discrete values rather than any possible value. It cannot exist in a state between these allowed values. This is a core concept of quantum mechanics, explained by Bohr's formula L = n(h/2π), where 'n' can only be an integer (1, 2, 3, ...). This quantization is the reason electrons occupy stable, distinct energy levels or orbits around the nucleus.
4. How did de Broglie's hypothesis explain Bohr's quantization of angular momentum?
Louis de Broglie proposed that electrons have wave-like properties. He explained Bohr's quantization by suggesting that for an electron to exist in a stable orbit, its associated wave must form a standing wave around the nucleus. This condition is only met when the circumference of the orbit (2πr) is an integral multiple of the electron's wavelength (λ), leading to the equation 2πr = nλ. By substituting de Broglie's wavelength formula (λ = h/mv), this directly derives Bohr's postulate: mvr = nh/2π, thus providing a physical justification for quantization.
5. What are the different types of angular momentum an electron possesses?
An electron in an atom possesses two distinct types of angular momentum:
- Orbital Angular Momentum: This arises from the electron's motion or revolution around the nucleus. It is associated with the electron's orbital shape, described by the azimuthal quantum number (l).
- Spin Angular Momentum: This is an intrinsic property of the electron, analogous to a planet spinning on its own axis. This 'spin' is a purely quantum mechanical property and is not a physical rotation.
6. How do you calculate the orbital angular momentum for an electron in different orbitals, such as s, p, and d?
The magnitude of the orbital angular momentum (L) is calculated using the formula L = √[l(l+1)]ħ, where ħ (h-bar) is h/2π and 'l' is the azimuthal quantum number for the specific orbital. The value of 'l' is:
- For an s-orbital, l = 0, so L = 0.
- For a p-orbital, l = 1, so L = √[1(1+1)]ħ = √2ħ.
- For a d-orbital, l = 2, so L = √[2(2+1)]ħ = √6ħ.
7. Why is the angular momentum of an electron considered a vector quantity?
Angular momentum is a vector quantity because it has both magnitude and direction. The magnitude is calculated by the appropriate formula (like mvr or the quantum mechanical version), while its direction is perpendicular to the plane of the electron's rotation. The direction is typically determined using the right-hand rule, where curling the fingers in the direction of rotation points the thumb in the direction of the angular momentum vector.
8. How is the angular momentum of an electron in an atom different from the angular momentum of a planet orbiting the sun?
The key difference lies in quantization. A planet's angular momentum is classical and can change continuously; it can have any value depending on its speed and distance. In contrast, an electron's angular momentum is quantized, meaning it is restricted to discrete, specific values defined by quantum numbers. An electron cannot have an angular momentum value that falls between these allowed levels, which is fundamentally different from the continuous possibilities in a classical system.
