Question

What is the velocity of electrons present in the first Bohr orbit of a hydrogen atom?
A. $2.18 \times {10^5}{\text{m}}{{\text{s}}^{ - 1}}$
B. \[2.18 \times {10^6}{\text{m}}{{\text{s}}^{ - 1}}\]
C. \[2.18 \times {10^{ - 18}}{\text{m}}{{\text{s}}^{ - 1}}\]
D. \[2.18 \times {10^{ - 9}}{\text{m}}{{\text{s}}^{ - 1}}\]

Answer 1

Hint: The Bohr model of atoms is the basis of solving this question. This could derive radius and energy of an orbit. A wave number was also calculated by him. The velocity of an electron changes with each element. Atomic number is a factor influencing the velocity of electrons in a certain orbit.

Given data:
The orbit number, \[{\text{n}} = 1\]

Complete step by step solution:
An atom is made up of mainly three particles-electrons, protons, and neutrons. Since protons and electrons are equal in number and have opposite charge, they attract each other.
The main postulates of Bohr model of atom are:
Electrons revolve around the nucleus with definite velocities in circular orbits. Energy of an electron in a certain orbit remains constant.
Angular momentum of an electron is quantized.
Energy of an electron changes when it moves from one orbit to another.
Radius of nth orbit, energy and velocity of electron in nth orbit was also calculated using Bohr model of atom.
From Bohr postulates, the equation for calculating velocity of electrons in nth orbit was derived. It is given below:
\[{\text{v}} = \dfrac{{\text{Z}}}{{\text{n}}} \times 2.88 \times {10^6}{\text{m}}{{\text{s}}^{ - 1}}\], where \[{\text{Z}} \to \] atomic number
\[{\text{v}} \to \] velocity of electron in nth orbit
\[{\text{n}} \to \] orbit number
For a hydrogen atom, atomic number \[{\text{Z}} = 1\], the orbit number, \[{\text{n}} = 1\].
Substituting all these values we get,
$
  {\text{v}} = \dfrac{1}{1} \times 2.18 \times {10^6}{\text{m}}{{\text{s}}^{ - 1}} \\
  {\text{v}} = 2.18 \times {10^6}{\text{m}}{{\text{s}}^{ - 1}} \\
$
Hence the option B is correct.

Additional information:
The motion of electron is restricted to those orbits where its angular momentum is an integral multiple of \[\dfrac{{\text{h}}}{{2\Pi }}\], where \[{\text{h}}\] is known as Planck’s constant.
i.e., \[{\text{mvr}} = \dfrac{{{\text{nh}}}}{{2\Pi }}\], where \[{\text{m}},{\text{v}},{\text{r}}\] are the mass of electron, velocity of electron and radius of orbit.

Note: Bohr model of atom couldn’t explain how the radiation of energy occurred when an electron jumps from one orbit to another. Bohr Theory predicted that only a series of spectral lines exist for hydrogen. It also couldn’t explain the Zeeman Effect and Stark Effect.