Question

The shortest wavelength of the line in hydrogen atomic spectrum of Lyman series
when \[{{R}_{H}}\]= \[109678\,c{{m}^{-1}}\] is:
(a) 1002.7 \[\overset{\circ }{\mathop{A}}\,\]
(b) 1215.6 \[\overset{\circ }{\mathop{A}}\,\]
(c) 1127.30 \[\overset{\circ }{\mathop{A}}\,\]
(d) 911.7 \[\overset{\circ }{\mathop{A}}\,\]
(e) 1234.7 \[\overset{\circ }{\mathop{A}}\,\]

Answer 1

Hint: Johannes Robert Rydberg then later in 1889, found several series of spectra that would fit a more general relationship which was similar to Balmer’s empirical formula, came to be known as the Rydberg formula. It is given by:
\[\dfrac{1}{\lambda }=R\left[ \dfrac{1}{{{n}_{f}}^{2}}-\dfrac{1}{{{n}_{i}}^{2}} \right]\,\,\,{{n}_{i}}>{{n}_{f}}\]
 where, \[{{n}_{1}}\] and \[{{n}_{2}}\] = 1, 2, 3, 4, 5, ….
 \[R\] (Rydberg constant) =\[1.097\times {{10}^{7}}{{m}^{-1}}\]
For the hydrogen atom, \[{{n}_{1}}\] = 1 corresponds to the Lyman series.

Complete step by step solution: We have been given in the question \[{{R}_{H}}\]= \[109678\,c{{m}^{-1}}\]
According to Rydberg formula, for Hydrogen we can write:
\[\dfrac{1}{\lambda }={{R}_{H}}\left[ \dfrac{1}{{{n}_{f}}^{2}}-\dfrac{1}{{{n}_{i}}^{2}} \right]\,\,\]
where, \[\lambda \] = wavelength of the emitted photon
\[{{R}_{H}}\] = Rydberg constant
\[{{n}_{f}}\] = final energy level of the transition
\[{{n}_{i}}\] = initial energy level of the transition
In our case, you have the \[{{n}_{i}}=\infty \to {{n}_{f}}=1\] transition, which is part of the Lyman series.

After substituting the values, we get:
\[\dfrac{1}{\lambda }=109678\times \left[ \dfrac{1}{{{1}^{2}}}-\dfrac{1}{{{\infty }^{2}}} \right]\,\]
\[\dfrac{1}{\lambda }=109678\times \left[ 1-0 \right]\,\]
\[\lambda =\dfrac{1}{109678}\]
\[\begin{align}
  & \lambda =9.117\times {{10}^{-6}}\, \\
 & \,\,\,\,\,=\,911.7\overset{\circ }{\mathop{A}}\, \\
\end{align}\]
Therefore, the correct option is (d).


Note: There are other series in the hydrogen atom also.
Balmer series with \[{{n}_{i}}\] = 2, present in the ultraviolet lines.
Paschen series with \[{{n}_{i}}\] = 3, present in the infrared region of the spectrum.
Brackett and Pfund series corresponding to \[{{n}_{i}}\] = 4 and \[{{n}_{i}}\] = 5 respectively, present in the infrared region.