Question

Find the wavelength of radiation required to excite an electron in the ground level of $L{i^{ + 2}}$(z=3) to the third energy level.

Answer 1

To solve this question firstly we have to use the energy formula for hydrogen like atoms where we use the rydberg`s constant term and initial state of electron and final state of the electron. After manipulating the equation in terms of wavelength we can finally get the wavelength of required energy.

Complete step-by-step solution: Now we will write the energy equation:
$Energy = 13.6{Z^2}(\dfrac{1}{{{n_1}^2}} - \dfrac{1}{{{n_2}^2}}) \\\ \\\$
As we all know that rydberg's constant is value which is constant over the earth.
So we have, $\dfrac{1}{{{R_H}}} = 912\dot A$
$\therefore Energy = \dfrac{{hc}}{\lambda }$where h=planck's constant ,c is velocity of light and $\lambda$ is wavelength.
So after again manipulating the equation:
$\dfrac{{hc}}{\lambda } = 13.6{Z^2}(\dfrac{1}{{{n_1}^2}} - \dfrac{1}{{{n_2}^2}}) \\\ \dfrac{1}{\lambda } = {R_H} \times {Z^2}(\dfrac{1}{{{n_1}^2}} - \dfrac{1}{{{n_2}^2}}) \\\$ $\therefore \dfrac{{13.6}}{{hc}} = {R_H}$
So,$\dfrac{1}{\lambda } = {R_H} \times {Z^2}(\dfrac{1}{{{n_1}^2}} - \dfrac{1}{{{n_2}^2}})$
$\therefore$Initial state is first shell and final state is third shell.
Hence, ${n_1} = 1{\text{ }}and{\text{ }}{n_2} = 3$
Now putting the value of ${n_1}$and ${n_2}$in equation $\dfrac{1}{\lambda } = {R_H} \times {Z^2}(\dfrac{1}{{{n_1}^2}} - \dfrac{1}{{{n_2}^2}})$
So we get,
$\dfrac{1}{\lambda } = {R_H} \times 9(1 - \dfrac{1}{9})$ $\therefore$z =3 for lithium
On solving above equation:
$\dfrac{1}{\lambda } = 8{R_H}$
$\Rightarrow \lambda = \dfrac{1}{8}{R_H}$
$\Rightarrow \lambda = 114\mathop A\limits^o$

Hence, calculated value of wavelength will be 114$\mathop A\limits^o$.

Note: Interpretation of hydrogen spectrum: only electrons in the hydrogen atom reside under ordinary condition on the first Orbit when energy is supplied the electron moves to the higher energy shells depending on the amount of energy observed. when this electron returns to any of the lower energy shells it emits energy. Lyman series is formed when the electron returns to the lowest energy state while Balmer series is formed when the electron returns to the second energy state similarly Paschen, Brackett and Pfund series are formed when electrons return to the third fourth and fifth energy shells from higher energy shells respectively.