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Which is correct for any H-like species?
(A) \[({E_2} - {E_1}) > ({E_3} - {E_2}) > ({E_4} - {E_3})\]
(B) \[({E_2} - {E_1}) < ({E_3} - {E_2}) < ({E_4} - {E_3})\]
(C) \[({E_2} - {E_1}) = ({E_3} - {E_2}) = ({E_4} - {E_3})\]
(D) \[({E_2} - {E_1}) = \dfrac{1}{4}({E_3} - {E_2}) = \dfrac{1}{9}({E_4} - {E_3})\]

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Last updated date: 20th Jun 2024
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Answer
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Hint: Quantum numbers are characteristic quantities that are used to describe the various properties of an electron in an atom like the position, energy or spin of the electrons. There are four quantum numbers namely, principal quantum number, azimuthal quantum number, magnetic quantum number and spin quantum number.

Complete step by step solution:
Principal quantum number is denoted by ‘n’ and it represents the electron shell of an atom. For an atom, its energy is inversely proportional to the square of its principal quantum number.
\[{\text{E}} \propto - \dfrac{{\text{1}}}{{{{\text{n}}^{\text{2}}}}}\]
Where E is the energy of an electron and n is a principal quantum number. Now we calculate the values of \[({E_2} - {E_1})\],\[({E_3} - {E_2})\]and \[({E_4} - {E_3})\].
For \[({E_2} - {E_1})\], \[{n_1} = 1,{n_2} = 2\]
\[
  {{\text{E}}_2} - {E_1} \propto - \left( {\dfrac{{\text{1}}}{{{2^2}}} - \dfrac{1}{{{1^2}}}} \right) \\
  {{\text{E}}_2} - {E_1} \propto 0.75 \\
  \]
For \[{\text{(}}{{\text{E}}_{\text{3}}}{\text{ - }}{{\text{E}}_{\text{2}}}{\text{)}}\],\[{{\text{n}}_3}{\text{ = 3,}}{{\text{n}}_{\text{2}}}{\text{ = 2}}\]
\[
  {{\text{E}}_3} - {E_2} \propto - \left( {\dfrac{{\text{1}}}{{{3^2}}} - \dfrac{1}{{{2^2}}}} \right) \\
  {{\text{E}}_3} - {E_2} \propto 0.14 \\
  \]
For \[{\text{(}}{{\text{E}}_{\text{4}}}{\text{ - }}{{\text{E}}_{\text{3}}}{\text{)}}\], \[{{\text{n}}_3}{\text{ = 3,}}{{\text{n}}_4}{\text{ = 4}}\]
\[
  {{\text{E}}_4} - {E_3} \propto - \left( {\dfrac{{\text{1}}}{{{4^2}}} - \dfrac{1}{{{3^2}}}} \right) \\
  {{\text{E}}_4} - {E_3} \propto 0.049 \\
  \]
So, the correct relationship for H-like species is, \[({E_2} - {E_1}) > ({E_3} - {E_2}) > ({E_4} - {E_3})\].

Hence the correct option is (A).

Note: Similarly, Azimuthal quantum number is denoted by ‘l’ and it represents the number of angular nodes. Magnetic quantum number is denoted by ‘m’ and explains the angular momentum. Spin quantum number is denoted by ‘s’ and explains the direction of spin.