Considering Bohr's atomic model for hydrogen atom :

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

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

(C) the energy of H atom in its ground state is same as that of $\mathrm{He}^{+}$ion for its ground state.

(D) the energy of $\mathrm{He}^{+}$ion in its first excited state is same as that for $\mathrm{Li}^{++}$ion in its ground state.

Choose the correct answer from the options given below :

<p>In Bohr's atomic model, the energy of an electron in an atom is given by the formula:</p> <p>$ E \propto \frac{Z^2}{n^2} $</p> <p>where $ Z $ is the atomic number and $ n $ is the principal quantum number of the electron's orbit. Let’s consider the implications:</p> <p><p>For hydrogen ($ \mathrm{H} $), $ Z = 1 $.</p></p> <p><p>For helium ion ($ \mathrm{He}^{+} $), $ Z = 2 $.</p></p> <p><p>For lithium ion ($ \mathrm{Li}^{++} $), $ Z = 3 $.</p></p> <p>Now, identifying the relevant states:</p> <p><p><strong>Ground State</strong>: This corresponds to $ n = 1 $.</p></p> <p><p><strong>First Excited State</strong>: This corresponds to $ n = 2 $.</p></p> <p><p><strong>Second Excited State</strong>: This corresponds to $ n = 3 $.</p></p> <p>Let's evaluate the energy comparisons:</p> <p><p><strong>Hydrogen Atom in Ground State</strong>: $ E \propto \frac{1^2}{1^2} = 1 $.</p></p> <p><p><strong>Helium Ion ($ \mathrm{He}^{+} $) in the First Excited State</strong>:</p> <p>$ E \propto \frac{2^2}{2^2} = 1 $</p> <p>This matches the energy of hydrogen in its ground state.</p></p> <p><p><strong>Lithium Ion ($ \mathrm{Li}^{++} $) in the Second Excited State</strong>:</p> <p>$ E \propto \frac{3^2}{3^2} = 1 $</p> <p>This also matches the energy of hydrogen in its ground state.</p></p> <p>From this analysis, we can conclude:</p> <p><p>The energy of a hydrogen atom in its ground state is equal to the energy of a $ \mathrm{He}^{+} $ ion in its first excited state.</p></p> <p><p>The energy of a hydrogen atom in its ground state also equals the energy of a $ \mathrm{Li}^{++} $ ion in its second excited state.</p></p> <p>Hence, statements (A) and (B) are correct.</p>