Question

The angular momentum of an electron in Bohr's hydrogen atom whose energy is$ - 3.4eV$, is

A.)$\dfrac{{5h}}{{2\pi }}$
B.)$\dfrac{h}{{2\pi }}$
C.)$\dfrac{h}{\pi }$
D.)$\dfrac{{2h}}{{3\pi }}$

Answer 1

Hint – Start by describing the Bohr’s atomic model. Then use the equation $E = - \dfrac{{13.6}}{{{n^2}}}$ to find the value of $n$and$L = \dfrac{{nh}}{{2\pi }}$, to find the value of L. Use this method to reach the solution of this problem.

Complete step by step answer:
This model also implies that electrons in a fixed orbit have a fixed energy and angular momentum. The maximum energy of an electron in an atom is $13.6eV$

We know, for a hydrogen atom,

$E = - \dfrac{{13.6}}{{{n^2}}}$

Here,

$E = $Energy of an electron in the nth orbit,
$n = $Number of the orbit in which the electron is in.

So for the given condition, as

$E = - 3.4eV$, So

$ - 3.4 = - \dfrac{{13.6}}{{{n^2}}}$

$ \Rightarrow {n^2} = \dfrac{{13.6}}{{3.4}}$
$ \Rightarrow {n^2} = 4$
$ \Rightarrow n = 2$(Equation 1)

Angular momentum – The product of the moment of inertia of a body and its angular velocity is called the angular momentum of a body. To understand it in simpler terms, consider it as a quantity that measures the rotation of a body.

According to Bohr’s model of hydrogen atom,

$L = \dfrac{{nh}}{{2\pi }}$(Equation 2)
Here,

$L = $Angular momentum of the electron,
$\pi = $3.14,

Substituting the value of n from equation 1 in equation 2 we get,

$ \Rightarrow L = \dfrac{{2h}}{{2\pi }}$
$ \Rightarrow L = \dfrac{h}{\pi }$


Additional Information: Neil Bohr proposed the atomic model of Hydrogen in 1913. He proposed the atomic model of Hydrogen in 1913. He proposed an atom is a positively charged nucleus (composed of protons and neutrons) and is surrounded by an electron cloud that is negatively charged. The electrons orbit the nucleus in fixed circular atomic shells. Electrostatic force (the forces of interaction between two charged bodies, i.e. in this case protons and electrons) is the force that holds the atom together.

Note – The Bohr’s atomic model has some errors, but it is a very important model because it contains most of the accepted features of atomic theory.