Considering Bohr's atomic model for hydrogen atom :
(A) the energy of H atom in ground state is same as energy of He⁺ion in its first excited state.
(B) the energy of H atom in ground state is same as that for Li⁺⁺ ion in its second excited state.
(C) the energy of H atom in its ground state is same as that of He⁺ion for its ground state.
(D) the energy of He⁺ion in its first excited state is same as that for Li⁺⁺ion in its ground state.
Choose the correct answer from the options given below :

Solution

Physics sir · 3 steps

Energy goes as Z²/n², so any ion sitting at n = Z matches hydrogen's ground state.

1. Bohr energy formula. For a one-electron atom or ion, the energy of level n depends on the nuclear charge Z and on n. Two states have the same energy whenever Z²/n² is the same.

   Eₙ = -13.6 Z²/n² eV

2. Name the states. The ground state is n = 1, the first excited state is n = 2 and the second excited state is n = 3. The hydrogen ground state has Z = 1 and n = 1, so E = -13.6 eV.

   Statement	First state (Z, n, E)	Second state (Z, n, E)	Equal?
   (A)	H: 1, 1, -13.6 eV	He⁺: 2, 2, -13.6 × 4/4 = -13.6 eV	Yes
   (B)	H: 1, 1, -13.6 eV	Li²⁺: 3, 3, -13.6 × 9/9 = -13.6 eV	Yes
   (C)	H: 1, 1, -13.6 eV	He⁺: 2, 1, -13.6 × 4 = -54.4 eV	No
   (D)	He⁺: 2, 2, -13.6 eV	Li²⁺: 3, 1, -13.6 × 9 = -122.4 eV	No

3. So. Only (A) and (B) give equal energies, so the answer is option (a).

The idea to keep. For hydrogen-like species, E ∝ Z²/n². Any ion in the state n = Z has the same energy as hydrogen's ground state. Remember that the k-th excited state is n = k + 1.

Answer from the NTA final answer key. Solution by PrepWiser.