Atoms

12.9 It is given that the energy of the electron beam used to bombard gaseous hydrogen at room temperature is 12.5 eV. Also, the energy of the gaseous hydrogen in its ground state at room temperature is – 13.6 eV.

When gaseous hydrogen is bombarded with an electron beam, the energy of the gaseous hydrogen becomes -13.6 + 12.5 eV i.e. -1.1 eV

Orbital energy is related to orbit level(n) as:

E = $\frac{-13.6}{n^2}$ eV

For n = 3, E = $\frac{-13.6}{9}$ eV = - 1.5 eV

This energy is approximately equal to the energy of the gaseous hydrogen. It can be concluded that the electron has jumped from n = 1 to n = 3 level.

During the de-excitation, the electron can jump from n=3 to n=1 directly, which forms a line of the Lyman series of hydrogen spectrum.

We have the relation for wave number for Lyman series as:

$\frac{1}{?}$ = $R_y\left(\frac{1}{1^2}-\frac{1}{n^2}\right)$ where

$R_y$ = Rydberg constant = 1.097 $*{10}^7{m}^{-1}$

$?=$ Wavelength of radiation emitted by the transition of electron

For n = 3, we get

$\frac{1}{?}$ = 1.097 $*{10}^7*\left(\frac{1}{1^2}-\frac{1}{3^2}\right)$

$?=1.215*{10}^{-7}m$ = 102.6 nm

If the electron jumps from n = 2 to n = 1, then the wavelength of the radiation is given as:

$\frac{1}{?}$ = 1.097 $*{10}^7*\left(\frac{1}{1^2}-\frac{1}{2^2}\right)$

$?=1.215*{10}^{-7}m$ = 121.5 nm

If the transition takes place from n = 3 to n = 2, then the wavelength is given as:

$\frac{1}{?}$ = 1.097 $*{10}^7*\left(\frac{1}{2^2}-\frac{1}{3^2}\right)$

$?=6.563*{10}^{-7}m$ = 656.3 nm

This radiation corresponds to the Balmer series of the hydrogen spectrum.

Hence, in Lyman series, two wavelengths i.e. 102.6 nm and 121.5 nm are emitted. And in the Balmer series, one wavelength i.e. 656.3 nm is emitted.

12.7 Let $v_1$ be the speed of the electron of the hydrogen atom in the ground state level, $n_1=1.$ For charge e of an electron, $v_1$ is given by the relation

$v_1=\frac{e^2}{n_{1}4?{?}_0\left(\frac{h}{2?}\right)}$ = $\frac{e^2}{2n_1{?}_{0}h}$

$where$ , e = 1.6 $*{10}^{-19}$ C 

${?}_0=$ Permittivity of free space = 8.85 $*{10}^{-12}$ $N^{-1}{C}^2{m}^{-2}$

$h$ = Planck's constant = 6.626 Js $*{10}^{-34}$

$Hence$ $v_1=\frac{e^2}{2n_1{?}_{0}h}$ $\frac{{(1.6*{10}^{-19})}^2}{2*1*8.85*{10}^{-12}*6.626*{10}^{-34}}$ = 2.182 $*{10}^6$ m/s

For level 2, $n_2=2$ ,

$v_2=\frac{e^2}{2n_2{?}_{0}h}$ = $\frac{{(1.6*{10}^{-19})}^2}{2*2*8.85*{10}^{-12}*6.626*{10}^{-34}}$ = 1.091 $*{10}^6$ m/s

For level 3, $n_3=3$ ,

$v_3=\frac{e^2}{2n_3{?}_{0}h}$ = $\frac{{(1.6*{10}^{-19})}^2}{2*3*8.85*{10}^{-12}*6.626*{10}^{-34}}$ = 0.727 $*{10}^6$ m/s

Let be the orbital period of the electron when it is in level . Orbital period is given by the expression

$T_1$ = $\frac{2?r_1}{v_1}$ where $r_1$ = radius of the orbit = $\frac{n_1^2{h}^2{?}_0}{?m{e}^2}$ , where

$m=$ mass of an electron = 9.1 $*{10}^{-31}$ kg

e = 1.6 $*{10}^{-19}$ C

${?}_0=$ Permittivity of free space = 8.85 $*{10}^{-12}$ $N^{-1}{C}^2{m}^{-2}$

$h$ = Planck's constant = 6.626 $*{10}^{-34}$ Js

$T_1$ = $\frac{2?}{v_1}*\frac{n_1^2{h}^2{?}_0}{?m{e}^2}$ = $\frac{2*n_1^2*h^2*{?}_0}{v_1*m*e^2}$ = $\frac{2*1*{(6.626*{10}^{-34})}^2*8.85*{10}^{-12}}{2.182*{10}^6*9.1*{10}^{-31}*{(1.6*{10}^{-19})}^2}$ = $\frac{7.77*{10}^{-78}}{5.083*{10}^{-62}}$

=1.53 $*{10}^{-16}s$

For level $n_2$ =2, $T_2$ = $\frac{2*n_2^2*h^2*{?}_0}{v_2*m*e^2}$ = $\frac{2*4*{(6.626*{10}^{-34})}^2*8.85*{10}^{-12}}{1.091*{10}^6*9.1*{10}^{-31}*{(1.6*{10}^{-19})}^2}$ = $\frac{3.108*{10}^{-77}}{2.542*{10}^{-62}}$ =

1.22 $*{10}^{-15}$ s

For level $n_3$ = 3, $T_3$ = $\frac{2*n_3^2*h^2*{?}_0}{v_3*m*e^2}$ = $\frac{2*9*{(6.626*{10}^{-34})}^2*8.85*{10}^{-12}}{0.727*{10}^6*9.1*{10}^{-31}*{(1.6*{10}^{-19})}^2}$ = $\frac{6.993*{10}^{-77}}{1.694*{10}^{-62}}$ = 4.13 $*{10}^{-15}$ s