Question

The angular momentum $\left( \text{L} \right)$ of an electron in a Bohr orbit is given as:
(A) \[L=\dfrac{nh}{2\pi }\]
(B) \[L=\sqrt{l(l+1)\dfrac{h}{2\pi }}\]
(C) \[L=\dfrac{mg}{2\pi }\]
(D) \[L=\dfrac{h}{4\pi }\]

Answer 1

Hint: Angular momentum of an electron by Bohr is given by $mvr$ or \[\dfrac{nh}{2\pi }\]. According to Bohr’s atomic model, the angular momentum of electrons orbiting around the nucleus is quantized.

Complete Step by Step Solution:
Bohr Model is a system consisting of a small, dense nucleus surrounded by orbiting electrons with the attraction provided by electrostatic forces.
The quantum rule of the Bohr model is that the angular momentum which is represented by $\text{L}$ is a multiple integral of \[\hbar \] i.e.
$\text{L = }nh$
$mvr=nh$
\[mvr=\dfrac{nh}{2\pi }\] where,
$h=$ Planck’s constant
$v=$ Velocity
$m=$ Mass of electron
$n=$ Orbit in which electron is
$r=$ Radius of nth orbit

So, the correct option is (A).

Additional Information:
The Bohr model was presented by Neil Bohr and Ernest Rutherford in $1913$. It is a relatively primitive model of the hydrogen atom, compared to the valence shell atom model. In the ${{20}^{th}}$ century, Rutherford established that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. He considered a planetary model of the atom. But the electrons release electromagnetic radiation while orbiting a nucleus and they would lose energy and collapse into the nucleus on a timescale of around $16$ picoseconds. So, the model predicted that all the atoms are unstable which is a disastrous result.
To overcome this difficulty, Neil Bohr introduces the Bohr Model of atoms postulating the concept of Stationary orbits. In these orbits, electrons revolve around the nucleus without radiating any energy. These orbits are attained at distances at which the value of angular momentum of the revolving electron is an integral multiple of the reduced Planck’s constant i.e. $mvr=nh$ where $n=1,2,3....$ is called principal quantum no. The lowest value of $n$ is $1$ which gives the smallest possible orbital radius of $0.0529\text{ nm}$ known as Bohr Radius. Once an electron is in this orbit it can get no closer to the proton.
Energies of the allowed orbits of hydrogen atom and other hydrogen like atoms and ions can be calculated. These orbits are associated with definite energies and are also called energy shells or energy levels.

Note: One can study the Planck’s quantum theory of radiation, terms like angular momentum, energy levels and stationary orbits. The importance of Planck’s constant. Models before Bohr Model like Plum Model in which the watermelon structure of the atom was assumed with beads of watermelon as electrons embedded in the atom.