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 :

Atoms and Nuclei PYQ — JEE Main 2025 · 2 Apr · Shift 1 · Q63 with Solution

View written solutionFree

Correct answer: A

Use Bohr's energy formula for hydrogen-like species

For a hydrogen-like atom/ion,

E_n = -13.6\,\frac{Z^2}{n^2}\ \text{eV}

where:

$Z$ = atomic number
$n$ = principal quantum number

For the given species:

Hydrogen: $Z=1$
$\mathrm{He}^+$ : $Z=2$
$\mathrm{Li}^{++}$ : $Z=3$

Find the required energies

(i) Hydrogen in ground state

Ground state means $n=1$ .

E(\mathrm{H}, n=1) = -13.6\frac{1^2}{1^2} = -13.6\ \text{eV}

(ii) $\mathrm{He}^+$ in first excited state

First excited state means $n=2$ .

E(\mathrm{He}^+, n=2) = -13.6\frac{2^2}{2^2} = -13.6\ \text{eV}

So statement (A) is true.

(iii) $\mathrm{Li}^{++}$ in second excited state

Second excited state means $n=3$ .

E(\mathrm{Li}^{++}, n=3) = -13.6\frac{3^2}{3^2} = -13.6\ \text{eV}

So statement (B) is true.

(iv) $\mathrm{He}^+$ in ground state

Ground state means $n=1$ .

E(\mathrm{He}^+, n=1) = -13.6\frac{2^2}{1^2} = -54.4\ \text{eV}

This is not equal to $-13.6\,\text{eV}$ . So statement (C) is false.

(v) Compare $\mathrm{He}^+$ first excited state and $\mathrm{Li}^{++}$ ground state

We already have:

E(\mathrm{He}^+, n=2) = -13.6\,\text{eV}

For $\mathrm{Li}^{++}$ ground state, $n=1$ :

E(\mathrm{Li}^{++}, n=1) = -13.6\frac{3^2}{1^2} = -122.4\ \text{eV}

These are not equal. So statement (D) is false.

Evaluate the options

(A) true
(B) true
(C) false
(D) false

Hence the correct choice is:

\boxed{\text{Option A: (A), (B) only}}

Comparison with stored correct answer

Stored correct answer = A

My derived answer = A

So, the derived answer agrees with the stored correct answer.

More from Atoms and Nuclei

View written solutionFree

Correct answer: A

Use Bohr's energy formula for hydrogen-like species

For a hydrogen-like atom/ion,

E_n = -13.6\,\frac{Z^2}{n^2}\ \text{eV}

where:

$Z$ = atomic number
$n$ = principal quantum number

For the given species:

Hydrogen: $Z=1$
$\mathrm{He}^+$ : $Z=2$
$\mathrm{Li}^{++}$ : $Z=3$

Find the required energies

(i) Hydrogen in ground state

Ground state means $n=1$ .

E(\mathrm{H}, n=1) = -13.6\frac{1^2}{1^2} = -13.6\ \text{eV}

(ii) $\mathrm{He}^+$ in first excited state

First excited state means $n=2$ .

E(\mathrm{He}^+, n=2) = -13.6\frac{2^2}{2^2} = -13.6\ \text{eV}

So statement (A) is true.

(iii) $\mathrm{Li}^{++}$ in second excited state

Second excited state means $n=3$ .

E(\mathrm{Li}^{++}, n=3) = -13.6\frac{3^2}{3^2} = -13.6\ \text{eV}

So statement (B) is true.

(iv) $\mathrm{He}^+$ in ground state

Ground state means $n=1$ .

E(\mathrm{He}^+, n=1) = -13.6\frac{2^2}{1^2} = -54.4\ \text{eV}

This is not equal to $-13.6\,\text{eV}$ . So statement (C) is false.

(v) Compare $\mathrm{He}^+$ first excited state and $\mathrm{Li}^{++}$ ground state

We already have:

E(\mathrm{He}^+, n=2) = -13.6\,\text{eV}

For $\mathrm{Li}^{++}$ ground state, $n=1$ :

E(\mathrm{Li}^{++}, n=1) = -13.6\frac{3^2}{1^2} = -122.4\ \text{eV}

These are not equal. So statement (D) is false.

Evaluate the options

(A) true
(B) true
(C) false
(D) false

Hence the correct choice is:

\boxed{\text{Option A: (A), (B) only}}

Comparison with stored correct answer

Stored correct answer = A

My derived answer = A

So, the derived answer agrees with the stored correct answer.

Atoms and Nuclei question

Atoms and Nuclei question

(i) Hydrogen in ground state

(ii) $\mathrm{He}^+$ in first excited state

(iii) $\mathrm{Li}^{++}$ in second excited state

(iv) $\mathrm{He}^+$ in ground state

(v) Compare $\mathrm{He}^+$ first excited state and $\mathrm{Li}^{++}$ ground state

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Atoms and Nuclei question

Atoms and Nuclei question

(i) Hydrogen in ground state

(ii) $\mathrm{He}^+$ in first excited state

(iii) $\mathrm{Li}^{++}$ in second excited state

(iv) $\mathrm{He}^+$ in ground state

(v) Compare $\mathrm{He}^+$ first excited state and $\mathrm{Li}^{++}$ ground state

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(i) Hydrogen in ground state

(ii) He+\mathrm{He}^+He+ in first excited state

(iii) Li++\mathrm{Li}^{++}Li++ in second excited state

(iv) He+\mathrm{He}^+He+ in ground state

(v) Compare He+\mathrm{He}^+He+ first excited state and Li++\mathrm{Li}^{++}Li++ ground state

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(i) Hydrogen in ground state

(ii) He+\mathrm{He}^+He+ in first excited state

(iii) Li++\mathrm{Li}^{++}Li++ in second excited state

(iv) He+\mathrm{He}^+He+ in ground state

(v) Compare He+\mathrm{He}^+He+ first excited state and Li++\mathrm{Li}^{++}Li++ ground state

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(ii) $\mathrm{He}^+$ in first excited state

(iii) $\mathrm{Li}^{++}$ in second excited state

(iv) $\mathrm{He}^+$ in ground state

(v) Compare $\mathrm{He}^+$ first excited state and $\mathrm{Li}^{++}$ ground state

(ii) $\mathrm{He}^+$ in first excited state

(iii) $\mathrm{Li}^{++}$ in second excited state

(iv) $\mathrm{He}^+$ in ground state

(v) Compare $\mathrm{He}^+$ first excited state and $\mathrm{Li}^{++}$ ground state