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Question: In a sample of \(H - \) atom electrons make transition from \[{5^{th}}\] excited state, upto ground ...

In a sample of H−H - atom electrons make transition from 5th{5^{th}} excited state, upto ground state, producing all types of possible electrons, then find out the number of lines in the infrared region?

Explanation

Solution

Bohr’s gives the concept of producing atomic spectrum by hydrogen atom. The electron is initially present in the ground state and after providing some energy to the hydrogen atom, its electron jumps from ground state to the excited state associated with higher energy state.

Complete answer:
During the transition of an electron from one energy state to another energy state a light of specific wavelength is produced. This light is observed in a form of straight line in the spectrum of hydrogen atoms. Depending upon the energy level from which the electron jumps to higher energy state there are a number of lines formed which are known by the scientist who discovered them.
When an electron from its ground state (n=1)\left( {n = 1} \right) jumps to the fifth energy state (n=5)\left( {n = 5} \right). The spectrum created in this condition is commonly known as the Lyman series.
The general rule to calculate the total number of emission lines formed by the electron when it jump from level n2{n_2} to level n1{n_1} is expressed as:
E=(n2−n1)(n2−n1+1)2E = \dfrac{{\left( {{n_2} - {n_1}} \right)\left( {{n_2} - {n_1} + 1} \right)}}{2}
In the case of lyman series value of (n2=5)\left( {{n_2} = 5} \right) and (n1=1)\left( {{n_1} = 1} \right). Now put these values in above equation to calculate total number of emission lines formed by the electron during its transition
E=(5−1)(5−1+1)2E = \dfrac{{\left( {5 - 1} \right)\left( {5 - 1 + 1} \right)}}{2}
After solving the above equation, we get
E=202E = \dfrac{{20}}{2}
On solving, we get
E=10E = 10
Out of these 1010 lines, 44 lines comes under the ultraviolet region, 33 comes under the visible region and the remaining 33 comes under the infrared region.
ultraviolet region-5→4,5→3,5→2,5→15 \to 4,5 \to 3,5 \to 2,5 \to 1
visible region- 4→3,4→2,4→14 \to 3,4 \to 2,4 \to 1
infrared region- 3→2,3→13 \to 2,3 \to 1
infrared region- 2→12 \to 1
⇒\Rightarrow Hence, there is formation of 1010 lines when electrons of hydrogen atom jump from 5th{5^{th}}excited state, upto ground state out of which 33 comes under the infrared region.

Note:
Balmer series is produced if transition occurs from higher energy state to 2nd{2^{nd}} energy state. The Paschen series is produced if the transition of electrons occurs from a higher energy state to the 3rd{3^{rd}} energy state. The Paschen series is associated with the infrared region.