Question on Bolr atom model

Question :

The ionization energy of a hydrogen like Bohr atom is 4 rydbergs.

(a) What is the wavelength of radiation emitted when the electron jumps from the first excited state to the ground state.

(b) Whet is the radius of the first orbit of this atom ? Given that Bohr radius of hydrogen atom = 5×10-11m and 1 rydberg = 2.2×10-18J.

Solution :

In terms of rydberg constant R , the energy of electron in the n'th orbit of hydrogen like atom is

${E}_{n}=-\frac{R{Z}^{2}}{{n}^{2}}$

where R=2.2×10-18J. The ionization energy of the atom is

${\mathrm{\Delta E}=E}_{\infty }-{E}_{1}=-R{Z}^{2}\left(\frac{1}{\infty }-\frac{1}{{1}^{2}}\right)=R{Z}^{2}$

Given that ΔE=4R. Therefore , 4R=RZ2 or, Z=2

(a) The energy of the radiation emitted when the electron jumps from the first excited state (n=2) to the ground state (n=1) is

${E=E}_{2}-{E}_{1}=-R{Z}^{2}\left(\frac{1}{{2}^{2}}-\frac{1}{{1}^{2}}\right)=\frac{3R{Z}^{2}}{4}=3R=3×2.2×{10}^{-18}=6.6×{10}^{-18}J$ (since Z=2)

Therefore wavelength of the radiation is given by

$\lambda =\frac{\mathrm{hc}}{E}=\frac{6.63×{10}^{-34}×3×{10}^{8}}{6.6×{10}^{-18}}=301Å$

(b) Radius of the first bohr orbit of the given atom is $=\frac{{r}_{1}}{Z}=2.5×{10}^{-11}m$