Consider Bohr's theory for hydrogen atom. The magnitude of angular momentum, orbit radius and velocity of the electron in ${\mathrm{n}}^{\text{th }}$ energy state in a hydrogen atom are $\mathrm{l},\mathrm{r}$ and $\mathrm{v}$ respectively. Find out the value of ${}^{\text{'}}\mathrm{x}^{\text{'}},$ if product of $\mathrm{v},\mathrm{r}$ and $\mathrm{l}\left(\mathrm{vrl}\right)$ is directly proportional to ${\mathrm{n}}^{\mathrm{x}}.$

A hydrogen sample is prepared in a particular excited state. Photons of energy $2.55 \mathrm{eV}$ get absorbed into the sample to take some of the electrons to a further excited state $\mathrm{B}.$ Find orbit numbers of the states $\mathrm{A}$ and $\mathrm{B}.$ Given the allowed energies of hydrogen atom:

${\mathrm{E}}_1=-13.6 \mathrm{eV}, {\mathrm{E}}_2=-3.4 \mathrm{eV}, {\mathrm{E}}_3=-1.5 \mathrm{eV}, {\mathrm{E}}_4=-0.85 \mathrm{eV}, {\mathrm{E}}_5=-0.54 \mathrm{eV}$

A single electron atom has nuclear charge $+\mathrm{Ze}$ where $\mathrm{Z}$ is atomic number and $\mathrm{e}$ is electronic charge. It requires $42.7 \mathrm{eV}$ to excite the electron from the second Bohr's orbit to third Bohr's orbit. Find :

(a) the atomic number of element.

(b) the energy required for transition of electron from third to fourth orbit.

(c) the wavelength required to remove electron from first Bohr's orbit to infinity.

(d) the kinetic energy of electron in first Bohr's orbit.