Question

Three energy levels ${L_1}$, ${L_2}$ and ${L_3}$ of a hydrogen atom correspond to increasing values of energy i.e., ${E_{L_1}}$ < ${E_{L_2}}$< ${E_{L_3}}$ , if the wavelength corresponding to the transitions ${L_3}$ to ${L_2}$ , ${L_2}$ to ${L_1}$ and ${L_3}$ to ${L_1}$ are ${\lambda_3}$, ${\lambda_2}$ and ${\lambda_1}$ respectively then,
A. $\lambda_3 = \lambda_1 + \lambda_2$
B. $\lambda_1 = \dfrac{{\lambda_2\lambda_3}}{{\lambda_2 + \lambda_3}}$
C. $\lambda_1 = \lambda_2 + \lambda_3$
D. $\lambda_1 = \dfrac{{\lambda_2\lambda_3}}{{\lambda_2 - \lambda_3}}$

Answer 1

Hint: According to Bhor's atomic model, the electrons revolve around the nucleus in an orbit that has specific size and energy. The energy of the orbit is proportional to the size of the orbit. The smallest orbit has the lowest energy level. Electrons move from one orbit to another by absorbing or emitting a photon or energy.

Complete step by step answer:Let ${L_1}$ (represents the ground state), ${L_2}$(represents the intermediate state) and ${L_3}$(represent the highest level) represent the levels of hydrogen atom and ${E_{L_1}}$,${E_{L_2}}$ and ${E_{L_3}}$ represent the energy of ${L_1}$, ${L_2}$ and ${L_3}$ levels respectively.
Change in energy is given by,
\[\Delta E = \dfrac{{hc}}{\lambda }\] ……….(1)
$
{L_2} - {L_3} = \dfrac{{hc}}{{\lambda_3}}.......(2) \\
\Rightarrow{L_1} - {L_3} = \dfrac{{hc}}{{\lambda_1}} .....(3) \\
\Rightarrow{L_1} - {L_2} = \dfrac{{hc}}{{\lambda_2}} .....(4) \\
$
Adding equation (2) and (4),
${L_2} - {L_3} + {L_1} - {L_2} = \dfrac{{hc}}{{\lambda_3}} - \dfrac{{hc}}{{\lambda_2}} \\
\Rightarrow{L_2}-{L_3}+{L_1} - {L_2} = hc(\dfrac{{\lambda_2 - \lambda_3}}{{\lambda_2\lambda_3}})$
Equating (5) and (3),
$\Rightarrow\dfrac{{hc}}{{\lambda_1}}=hc(\dfrac{{\lambda_2-\lambda_3}}{{\lambda_2\lambda_3}})$
$\dfrac{1}{{\lambda_1}} = \dfrac{{\lambda_2 - \lambda_3}}{{\lambda_2\lambda_3}}$
Upon reversing the above equation we get,
$\therefore\lambda_1 = \dfrac{{\lambda_2\lambda_3}}{{\lambda_2 - \lambda_3}}$

Therefore the correct answer is option B.

Note:The energy of the emitted or absorbed photon when the electron jumps from one energy level to the other must be equal to the energy difference between those two energy levels.Bohr model of the hydrogen atom was the first atomic model to successfully explain the radiation spectra of atomic hydrogen. Niels Bohr introduced the atomic Hydrogen model in 1913. Bohr Model of the hydrogen atom attempts to plug in certain gaps as suggested by Rutherford’s model. It holds a special place in history as it gave rise to quantum mechanics by introducing quantum theory.