Question

The energy levels for \[_{\text{Z}}{{\text{A}}^{( + {\text{Z}} - 1)}}\] can be fiven by:
A.\[{{\text{E}}_{\text{n}}}{\text{ for }}{{\text{A}}^{( + {\text{Z}} - 1)}} = {{\text{Z}}^2} \times {{\text{E}}_{\text{n}}}{\text{ for H}}\]
B.\[{{\text{E}}_{\text{n}}}{\text{ for }}{{\text{A}}^{( + {\text{Z}} - 1)}} = {\text{Z}} \times {{\text{E}}_{\text{n}}}{\text{ for H}}\]
C.\[{{\text{E}}_{\text{n}}}{\text{ for }}{{\text{A}}^{( + {\text{Z}} - 1)}} = \dfrac{1}{{{{\text{Z}}^2}}} \times {{\text{E}}_{\text{n}}}{\text{ for H}}\]
D.\[{{\text{E}}_{\text{n}}}{\text{ for }}{{\text{A}}^{( + {\text{Z}} - 1)}} = \dfrac{1}{{\text{Z}}} \times {{\text{E}}_{\text{n}}}{\text{ for H}}\]

Answer 1

Hint: The above question is based on Bohr Theory for hydrogen and hydrogen like species. The formula for calculation of radius of states in hydrogen level will be used here. Put the value of atomic number and find the relation
Formula used: \[{{\text{E}}_{\text{n}}}{\text{ = 13}}{\text{.6}}\dfrac{{{{\text{Z}}^2}}}{{\text{n}}}\] here E is energy of state, n is the number if stationary state and Z is atomic number.

Complete step by step answer:
According to Bohr model for hydrogen atom, the energy in the stationary states for hydrogen atoms can be calculated as
\[{{\text{E}}_{\text{n}}}{\text{ = 13}}{\text{.6}}\dfrac{{{{\text{Z}}^2}}}{{\text{n}}}\].
We know atomic number for hydrogen is 1 and hence the energy of stationary states will be:\[{{\text{E}}_{\text{n}}}{\text{ = 13}}{\text{.6}}\dfrac{1}{{\text{n}}}\].
This is very obvious from the formula that the energy of any other species will be \[{{\text{Z}}^2}\] times the energy of the hydrogen atom because atomic number of hydrogen is 1. That means:
\[{{\text{E}}_{\text{n}}}{\text{ for }}{{\text{A}}^{( + {\text{Z}} - 1)}} = {{\text{Z}}^2} \times {{\text{E}}_{\text{n}}}{\text{ for H}}\]

Hence the correct option is A.

Additional information:
 Electron revolves around the positively charged nucleus. Each orbital in which atom revolves has fixed energy or quantized energy. These states are known as stationary states. The electron does not lose energy when it moves in the stationary state. The electron will move from the lower energy level to the higher energy level if it has the energy equal to the gap between the stationary states.

Note:
There were some limitations of the Bohr model. Bohr model gave very satisfactory results for the atomic spectra for hydrogen and hydrogen like species but it fails to give information for spectra of higher elements. The Bohr model was proposed in 1915 by Neil Bohr.