Angular Momentum of Electron

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).

• 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.