Question

Multiplication of electron velocity and radius for an orbit in an atom is:
(A) Proportional to mass of electron.
(B) Proportional to square of mass of electron.
(C) Inversely proportional to mass of electron.
(D) Does not depend upon the mass of the electron.

Answer 1

An electron revolves in stationary orbits without radiating any energy for which the angular momentum of an electron is an integral multiple of $h{\text{/}}2\pi$. This is the postulate on which the Bhor’s model of atom is based.

Step by step answer: The angular momentum is the product of mass, radius and velocity of any moving particle.
The Bhor’s postulate is ‘an electron revolves in stationary orbits without radiating any energy for which the angular momentum of an electron is an integral multiple of $h{\text{/}}2\pi$’. Thus, the equation is,
$mvr = n\dfrac{h}{{2\pi }}$
Where $m$ is the mass of electron,
$v$ is the velocity of electron,
$r$ is the radius of the orbit in which the electron is moving,
$h$ is the Planck’s constant,
$n$ is the integral number or the Principal quantum number ($n = 1$ for K shell, $n = 2$ for L shell, $n = 3$ for M shell, $n = 4$ for N shell, etc.)
Rearrange the equation for the product of velocity of an electron and radius of the orbit in which the electron. Thus,
$vr = n\dfrac{h}{{2\pi }} \times \dfrac{1}{m}$
From this equation, we have,
$vr \propto \dfrac{1}{m}$
Thus, the product of velocity of an electron and radius of the orbit in which the electron is moving is inversely proportional to the mass of the electron.
Thus, the multiplication of electron velocity and radius for an orbit in an atom is inversely proportional to mass of electron.

Thus, the correct option is (C) inversely proportional to mass of electron.

Note: The equation shows that the values of energy and angular momentum of an electron are quantized. This anyhow justifies and proves the validity of plank quantum theory.