Angular Momentum of Electron

In the world of physics, angular momentum helps us understand how objects behave when they rotate. It is not only limited to big objects like planets but it also helps us gain insight into the motion of smaller particles such as electrons. Let’s understand angular momentum of an electron with the basic introduction of momentum in this article.

What is Angular Momentum ?

Momentum is a physical quantity that an object might possess by virtue of its motion. Angular momentum is when the motion is rotational. Angular momentum is defined as

“The quantity of rotation of a body, which is the product of its moment of inertia and its angular velocity.”

In simpler terms, angular momentum measures how much motion an object has while it’s spinning around a point. The greater the mass of the object and the faster it spins, the more angular momentum it has.

Momentum of an Electron

Bohr put forward a theory about the rotational energy of an electron inside the atom. In Bohr’s atomic model, electrons are positioned in specific paths around the nucleus, and these paths play a crucial role in defining the electron’s angular momentum. Bohr proposed that the orbital angular momentum of an electron is quantized, meaning it can only be found in certain fixed values.

Bohr also specified that the angular momentum of an electron must be a multiple of h/2π, where h is Planck’s constant. This limitation confines electrons to specific stable paths around the nucleus.

Angular Momentum of an Electron

Understanding the chemical properties of elements and the structure of atoms depends significantly on the electron’s angular momentum. It describes the reasons behind atoms absorbing or releasing light at specific wavelengths and why some elements exhibit specific chemical reactions. This knowledge forms the basis for areas such as chemistry, physics, and also materials science.

De Broglie’s Explanation to the Quantization of Angular Momentum of Electron

Expanding on Bohr’s research, Louis de Broglie later proposed the concept that electrons exhibit characteristics of both particles and waves during motion. This duality was a crucial development in comprehending how electrons behave on a quantum scale.

De Broglie applied his hypothesis to the Bohr model of the atom. he said that if electrons behave like waves, then for an electron to exist in a stable orbit around the nucleus, the circumference of its orbit must accommodate an integer number of wavelengths.

This can be shown by this formula, 2πr=nλ. Here, if we substitute λ by h/p where p = mv, we get

2πr = nh/p

and if we use the formula of angular momentum L = mvr and rearrange the expression we get,

L = nh/2π

In terms of mathematics, the formula for the angular momentum of an electron can be expressed as:

The angular momentum is equal to the product of mass, velocity, and radius, which is also equivalent to half of the reduced Planck constant divided by 2 times pi.

L = mvr = nh/2π

Where

• Angular momentum is represented by L.
• The mass of the electron is denoted by m.
• v represents the speed at which the electron is moving.
• r represents the size of the orbit.
• n is the main quantum number that indicates the orbit in which the electron can be found.
• Planck’s constant is denoted by h.

This equation also serves as a foundational principle for contemporary comprehension of atomic structure.

Orbital Angular Momentum

If we use the right formula we can also obtain the Orbital Angular momentum of an electron. As we have already learned that angular momentum is quantized so to make things simpler here we make use of orbital quantum number ‘l’ instead of ‘n’. The formula is given as :

L = √l(l+1)h

This makes it easier to calculate the orbital angular momentum of any orbit in an atom. Let’s go through some examples and find orbital angular momentum of p and d electrons.

Orbital Angular Momentum of p-electron

In an atom, electrons are present in different shells and each shell is divided into subshells named s,p,d and f, based on the shape of the orbital in which electrons move. The quantum number of p orbital is ‘1’. So using the formula we learned before and putting l = 1 we get,

L = √1(1+1)h = √2h

This is the magnitude of the angular momentum of an electron in p orbital.

Orbital Angular Momentum of d-electron

In case for d orbital, we have quantum number ‘l’ = ‘2’. Now using the same formula for orbital angular momentum, L = √l(l+1)h we get,

L = √2(2+1)h = √6h

As we can clearly observe this value is higher than that of p- electrons which reflects the more complex spatial arrangement and higher angular momentum associated with d-electrons.

Spin Angular Momentum

Spin angular momentum is an intrinsic property. For easy understanding you can say that electrons have two options for spinning: they can either spin in an “up” direction or in a “down” direction. This does not indicate they are actually spinning in a physical manner; it is merely a method for scientists to explain their magnetic alignment. Actual spin angular momentum is given by the formula :

[Tex]S = \sqrt{s(s+1)} \times \hbar [/Tex]

Where [Tex]\hbar[/Tex] is reduced plank’s constant and s is quantum spin number.

Conclusion

In the above article, we understood angular momentum and learned how to calculated the angular momentum of an electron. Angular momentum is always quantized and is given as multiple of h/2π. By studying the angular momentum of electrons in different orbitals, such as p and d, scientists can gain deeper insights into the nature of materials and the fundamental forces at play at the atomic level. This understanding will help us in solving question related to angular momentum.

FAQs on Angular Momentum of Electron

What is the formula for the orbital angular momentum of an electron?

The orbital angular momentum for an electron revolving in an orbit is given by √l(l+1)h2π.

What is the angular momentum of S?

For s-orbital angular momentum is zero and for a p-orbital angular momentum is non-zero.

What is the de Broglie’s equation?

De Broglie Wavelength for an Electron is given by the equation λ = h/mv.

What is the dual nature of electrons?

The dual nature of electrons is particle nature as well as wave nature was established by this equation. The three-dimensional region in space around the nucleus where the probability of finding an electron was killer energy maximum is called an atomic orbital.

What is the value of angular momentum of Bohr orbit?

The angular momentum of an electron in a Bohr orbit of H-atom is 4.2178×10⁻³⁴kgm²/sec.