JEE Main 2025 Apr 2 Shift 1, Physics Q15: Bohr Model of the Hydrogen Atom

JEE Main2025Apr 2, Shift 1Physics

Considering Bohr's atomic model for hydrogen atom :

(A) the energy of H atom in ground state is same as energy of $H e^{+}$ ion in its first excited state.

(B) the energy of H atom in ground state is same as that for $L i^{++}$ ion in its second excited state.

(D) the energy of $H e^{+}$ ion in its first excited state is same as that for $L i^{++}$ ion in its ground state

Choose the correct answer from the options given below :

A
(B), (D) only
B
(A), (B) only
C
(A), (D) only
D
(A), (C) only

Solution

For a hydrogen-like ion, the Bohr energy is $E_{n} = - 13.6 \frac{Z ^{2}}{n ^{2}} eV$ , so two states share the same energy when $\frac{Z ^{2}}{n ^{2}}$ matches. Equivalently $\frac{Z}{n}$ must be equal.

$Z_{H} = 1$ , $Z_{H e^{+}} = 2$ , $Z_{L i^{2 +}} = 3$ .

$1^{st}$ excited state $\Rightarrow n = 2$ ; $2^{nd}$ excited state $\Rightarrow n = 3$ .

(A) H ground ( $Z / n = 1/1 = 1$ ) vs $H e^{+}$ first excited ( $Z / n = 2/2 = 1$ ). Equal. ✓

(B) H ground ( $1/1 = 1$ ) vs $L i^{2 +}$ second excited ( $3/3 = 1$ ). Equal. ✓

(C) H ground ( $1/1 = 1$ ) vs $H e^{+}$ ground ( $2/1 = 2$ ). Not equal. ✗

(D) $H e^{+}$ first excited ( $2/2 = 1$ ) vs $L i^{2 +}$ ground ( $3/1 = 3$ ). Not equal. ✗

$∴$ Only (A) and (B) are correct. Option (2).

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JEE Main 2025 Apr 2 Shift 1, Physics Q15: Bohr Model of the Hydrogen Atom

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