Question

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.

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

Hint:

In Bohr's atomic model, the energy levels depend on the atomic number \( Z \) and the principal quantum number \( n \). For ions like He\(^+\) and Li\(^2+\), the energy will differ due to their increased nuclear charge.

Answer 1

The Correct Option is B

The energy of an electron in a specific orbit is given by: \[ E \propto \frac{Z}{n^2} \] For hydrogen atom, \( Z_H = 1 \), for He\(^+\), \( Z_{He^+} = 2 \), and for Li\(^2+\), \( Z_{Li^{2+}} = 3 \). 
1st excited state \( n = 2 \) and 2nd excited state \( n = 3 \). 
From the given statements, only (A) and (B) are correct.

Answer 2

The problem asks us to compare the energies of an electron in different states for hydrogen-like species (H, He+, Li++) based on Bohr's atomic model and identify the correct statements.

Concept Used:

According to Bohr's model, the energy of an electron in the \(n^{th}\) orbit of a hydrogen-like atom with atomic number \(Z\) is given by the formula:

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

Where:

• \(Z\) is the atomic number (number of protons in the nucleus).
• \(n\) is the principal quantum number, representing the energy level or orbit.
• The ground state corresponds to \(n=1\).
• The first excited state corresponds to \(n=2\).
• The second excited state corresponds to \(n=3\).

Step-by-Step Solution:

We will evaluate the energy for each case mentioned in the statements.

Step 1: Evaluate Statement (A)

The statement compares the energy of a H atom in the ground state with a He+ ion in its first excited state.

• For H atom (ground state): \(Z=1\), \(n=1\). \[ E_{\text{H, ground}} = -13.6 \times \frac{1^2}{1^2} = -13.6 \, \text{eV} \text{} \]
• For He+ ion (first excited state): \(Z=2\), \(n=2\). \[ E_{\text{He+, 1st excited}} = -13.6 \times \frac{2^2}{2^2} = -13.6 \times \frac{4}{4} = -13.6 \, \text{eV} \text{} \]

Since both energies are equal (-13.6 eV), Statement (A) is true.

Step 2: Evaluate Statement (B)

This statement compares the energy of a H atom in the ground state with a Li++ ion in its second excited state.

• For H atom (ground state): As calculated before, \(E_{\text{H, ground}} = -13.6 \, \text{eV}\).
• For Li++ ion (second excited state): \(Z=3\), \(n=3\). \[ E_{\text{Li++, 2nd excited}} = -13.6 \times \frac{3^2}{3^2} = -13.6 \times \frac{9}{9} = -13.6 \, \text{eV} \text{} \]

Since both energies are equal (-13.6 eV), Statement (B) is true.

Step 3: Evaluate Statement (C)

This statement compares the energy of a H atom in its ground state with a He+ ion in its ground state.

• For H atom (ground state): \(E_{\text{H, ground}} = -13.6 \, \text{eV}\).
• For He+ ion (ground state): \(Z=2\), \(n=1\). \[ E_{\text{He+, ground}} = -13.6 \times \frac{2^2}{1^2} = -13.6 \times 4 = -54.4 \, \text{eV} \]

Since \( -13.6 \, \text{eV} \neq -54.4 \, \text{eV} \), Statement (C) is false.

Step 4: Evaluate Statement (D)

This statement compares the energy of a He+ ion in its first excited state with a Li++ ion in its ground state.

• For He+ ion (first excited state): As calculated before, \(E_{\text{He+, 1st excited}} = -13.6 \, \text{eV}\).
• For Li++ ion (ground state): \(Z=3\), \(n=1\). \[ E_{\text{Li++, ground}} = -13.6 \times \frac{3^2}{1^2} = -13.6 \times 9 = -122.4 \, \text{eV} \]

Since \( -13.6 \, \text{eV} \neq -122.4 \, \text{eV} \), Statement (D) is false.

Final Computation & Result:

From the step-by-step analysis, we have found that:

• Statement (A) is true.
• Statement (B) is true.
• Statement (C) is false.
• Statement (D) is false.

Therefore, the correct option is the one that includes only statements (A) and (B).

The correct option is (A), (B) only.