Question

Ultraviolet light of wavelength \[{\lambda _1}\] ​ and \[{\lambda _2}\] ​ when allowed to fall on hydrogen atoms in their ground state is found to liberate electron with kinetic energy 1.8 eV and 4.0 eV respectively. Find the value of \[\dfrac{{{\lambda _2}}}{{{\lambda _1}}}\] ​​.
A. $\dfrac{8}{7} \\ $
B. $\dfrac{7}{8} \\ $​
C. $\dfrac{{20}}{9} \\ $
D. $\dfrac{9}{{20}}$

Answer 1

Hint:The postulates of Bohr say energy can only be emitted or absorbed when an electron changes from one non-radiating orbit to another. Here energy is falling so this energy is absorbed by electrons.

Formula used:
Energy is given by,
$E = hf = \dfrac{{hc}}{\lambda }$
Where, $h$ is the planck’s constant and $f$ is the frequency of the radiation.

Complete step by step solution:
The Neil Bohr Model has three Bohr's Postulates, here the third postulate is explained in more detail below: only when an electron jumps from one non-radiating orbit to another does energy either emit or absorb. When an electron jumps from the inner to the outer orbit, the energy difference between the two stationary orbits is absorbed, and when an electron jumps from the outer to the inner orbit, it is expelled.

If $E_1$ and $E_2$ are equal to the total energy (T.E.) of an e- in its inner and outer stationary orbits, respectively, the frequency of radiation released during a jump from the outer to the inner orbit is given by:
\[\;E{\text{ }} = \;{\text{ }}hf\;{\text{ }} = {\text{ }}{E_2} - {\text{ }}{E_1} \ldots .\left( 3 \right)\]

We are aware that the majority of hydrogen atoms exist in the ground state, and that when this atom is exposed to energy from an electron collision or heat, the electrons may need to be raised to a higher energy level, such as from n = 1 to n = 2, 3, etc. The difference between their energies may be determined using equation (3).

If we apply $E = hf = \dfrac{{hc}}{\lambda }$.
So, by applying this relation we get,
$\dfrac{{{E_1}}}{{{E_2}}} = \dfrac{{\dfrac{{hc}}{{{\lambda _1}}}}}{{\dfrac{{hc}}{{{\lambda _2}}}}} = \dfrac{{{\lambda _2}}}{{{\lambda _1}}} \\ $
From it we can get
$\dfrac{{{\lambda _2}}}{{{\lambda _1}}} = \dfrac{{{E_1}}}{{{E_2}}} \\
\Rightarrow \dfrac{{{\lambda _2}}}{{{\lambda _1}}}= \dfrac{{1.8}}{4} \\
\therefore \dfrac{{{\lambda _2}}}{{{\lambda _1}}}= \dfrac{9}{{20}}$

Therefore, option D is the correct answer.

Notes There are three Bohr's Postulates in the Neil Bohr Model, and the third one is detailed in greater detail below: The only time energy emits or absorbs is when an electron transitions from one non-radiating orbit to another. When an electron jumps from the inner to the outer orbit, the energy difference between the two stationary orbits is absorbed, and when an electron jumps from the outer to the inner orbit, it is ejected.