Question 147: Atoms & Nuclei

Question

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 :

Accepted Answer

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

Answer

(A), (B) only

Answer

(A), (D) only

Answer

(A), (C) only

Answer

(B), (D) only

Atoms & Nuclei

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