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.
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.
Show Hint
- (B), (C) only
- (A), (B) only
- (A), (D) only
- (A), (C) only
The Correct Option is B
Approach Solution - 1
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.
Approach Solution -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.
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