A singly ionized helium atom in an excited state (n = 4) emits a photon of energy 2.6 eV. Given that the ground state energy of hydrogen atom is 13.6 eV, the energy `(E_t)` and quantum number (n) of the resulting state are respectively,

Options

A. `E_t = -13.6eV , n =1`
B. `E_t =- 6.0 eV , n =3`
C. `E_t = -6.0 eV , n=2`
D. `E_t = - 13.6 eV , n=2`

Explanation

To solve the problem, we need to find the energy \(E_t\) and the quantum number \(n\) of the resulting state of a singly ionized helium atom after it emits a photon of energy 2.6 eV from an excited state (n = 4).

### Step-by-Step Solution:

1. **Identify the Initial Energy State**:
The energy of a hydrogen-like atom in the nth state is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]
For singly ionized helium (He\(^+\)), the atomic number \(Z = 2\). Therefore, for \(n = 4\):
\[ E_4 = -\frac{13.6 \cdot 2^2}{4^2} = -\frac{13.6 \cdot 4}{16} = -3.4 \, \text{eV} \]

2. **Calculate the Energy of the Final State**:
When the atom emits a photon of energy 2.6 eV, it loses this energy. Thus, the energy of the resulting state \(E_t\) can be calculated as:
\[ E_t = E_4 + 2.6 \, \text{eV} = -3.4 \, \text{eV} + 2.6 \, \text{eV} = -0.8 \, \text{eV} \]

3. **Determine the Quantum Number of the Resulting State**:
We need to find the quantum number \(n\) such that:
\[ E_n = -\frac{13.6 \cdot Z^2}{n^2} \]
Setting \(E_n = -0.8 \, \text{eV}\):
\[ -0.8 = -\frac{13.6 \cdot 2^2}{n^2} \]
Simplifying this:
\[ 0.8 = \frac{54.4}{n^2} \]
Rearranging gives:
\[ n^2 = \frac{54.4}{0.8} = 68 \]
Taking the square root:
\[ n = \sqrt{68} \approx 8.25 \]
Since \(n\) must be a whole number, we check \(n = 8\).

4. **Final Energy Calculation**:
To confirm, we calculate the energy for \(n = 8\):
\[ E_8 = -\frac{13.6 \cdot 2^2}{8^2} = -\frac{54.4}{64} = -0.85 \, \text{eV} \]
This is close to -0.8 eV, confirming that the quantum number is indeed \(n = 8\).

### Final Answer:
The energy \(E_t\) of the resulting state is approximately \(-0.8 \, \text{eV}\) and the quantum number \(n\) is \(8\).

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