Question

The wavelength of the limiting line of the Lyman series is $ 911{A^0} $ . The atomic number of the element which emits minimum wavelength of $ 0.7{A^0} $ X-rays will be:
(A) $ 31 $
(B) $ 33 $
(C) $ 35 $
(D) $ 37 $

Answer 1

Hint :Here, the limiting wavelength of Lyman series is given. Lyman series is the part of the spectral line formed by the hydrogen atom. The wavelengths of the spectral series are calculated by Rydberg formula. The Lyman series is when electron transition takes place from higher energy states $ ({n_h} = 2,3,4,5,6,...) $ to $ {n_l} = 1 $ energy state. The Rydberg formula is given by:
 $ \dfrac{1}{\lambda } = R{Z^2}\left( {\dfrac{1}{{n_l^2}} - \dfrac{1}{{n_h^2}}} \right) $ ; $ \lambda $ is wavelength, $ Z $ is the atomic number of the element, $ R $ is the Rydberg constant and value is $ 1.09737 \times {10^7}{m^{ - 1}} $ ,
 $ {n_l} $ is the lower energy level, $ {n_h} $ is the higher energy level.

Complete Step By Step Answer:
According to the given information we have the wavelength of the limiting line of the Lyman series as $ 911{A^0} $ . We know that for Lyman series the lower energy level is given as $ l = 1 $ so $ {n_1} = 1 $ , and limiting line means electron is ejecting from orbit $ h = 2 $ and $ {n_2} = \infty $
Thus, the Rydberg formula is used as:
 $ \dfrac{1}{\lambda } = R{Z^2}\left( {\dfrac{1}{{n_l^2}} - \dfrac{1}{{n_h^2}}} \right) $
 $ \Rightarrow \dfrac{1}{\lambda } = R{Z^2}\left( {\dfrac{1}{{{1^2}}} - \dfrac{1}{{{\infty ^2}}}} \right) $
We have been asked about the same condition for the different atomic number
Let us consider the wavelengths given for second condition in the question as:
 $ {\lambda _1} = 911{A^0} $ and $ {\lambda _2} = 0.7{A^0} $
Now we have the equation from the above equation that
 $ \Rightarrow \dfrac{{{\lambda _1}}}{{{\lambda _2}}} = {Z^2} $ …. (Comparing two wavelengths)
 $ \Rightarrow \dfrac{{911{A^0}}}{{0.7{A^0}}} = {Z^2} $
 $ \Rightarrow \boxed{Z = 37} $
Thus, the atomic number is calculated as $ 37 $ .
The correct answer is option D.

Note :
We must understand the concept used here as the Lyman series is given and its limiting line wavelength is also given and it must be understood that the limiting line means electrons are ejecting from orbit. Using Rydberg formula we have calculated the required atomic no.