Embibe Experts Solutions for Chapter: Atomic Physics, Exercise 4: Exercise-4

A single electron orbits a stationary nucleus of charge Ze where Z is a constant and $e$ is the electronic charge. It requires $47.2eV$ to excite the electron from the $2 nd$ Bohr orbit to $3 rd$ Bohr orbit. Find

(i) the value of Z,

(ii) energy required to excite the electron from the third to the fourth orbit.

(iii) the wavelength of radiation required to remove the electron from the first orbit to infinity

(iv) the kinetic energy, potential energy and angular momentum in the first Bohr orbit.

(v) the radius of the first Bohr orbit.

A gas of identical hydrogen like atoms has some atoms in the lowest (ground) energy level $A$ & some atoms in a particular upper (excited) energy level $B$ & there are no atoms in any other energy level. The atoms of the gas make transition to a higher energy level by the absorbing monochromatic light of photon energy $2.7eV$. Subsequently, the atoms emit radiation of only six different photon energies. Some of the emitted photons have energy $2.7eV$. Some have energy more and some have less than $2.7eV$.

(i) Find the principal quantum number of the initially excited level B.

(ii) Find the ionisation energy for the gas atoms.

(iii) Find the maximum and the minimum energies of the emitted photons.

A classical model for the hydrogen atom consists of a single electron of mass $m_e$ in circular motion of radius r around the nucleus (proton). Since the electron is accelerated, the atom continuously radiates electromagnetic
waves. The total power P radiated by the atom is given by $P=P_0/r^4$ where $P_0=\frac{e^6}{96{\pi }^3{\epsilon }_0^3{c}^3{m}_e^2}$ (c $=$ velocity of ight)

(i) Find the total energy of the atom.

(ii) Calculate an expression for the radius $r\left(t\right)$ as a function of time. Assume that at $t=0$, the radius is $r_0={10}^{-10} \mathrm{m}$,

(iii) Hence or otherwise find the time $t_0$ when the atom collapses in a classical model of the hydrogen atom. Take: $\left[\frac{2}{\sqrt{3}}\frac{e^2}{4\pi {\epsilon }_0}\frac{1}{{ m}_e{c}^2}=r_e\approx 3\times {10}^{-15} \mathrm{m}\right]$

An imaginary particle has charge equal to that of an electron and mass hundred times the mass of the electron. It moves in a circular orbit around a nucleus of charge $+4 e$. Take the mass of nucleus to be very large. Applying Bohr's model.

(i) Find radius of $n^{th }$ Bohr's orbit.

(ii) Energy of photon emitted when the particle makes a transition from fourth orbit to second orbit.