`He^(+)` is in `n^(th)` state. It emits two successive photons of wavelength `103.7nm` and `30.7nm`, to come to ground state the value of `n` is:

Options

A. 4
B. 5
C. 6
D. 7

Answer

To solve the problem, we follow these steps:

### Step 1: Calculate the energy of the emitted photons
The energy of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant (\( 4.14 \times 10^{-15} \) eV·s),
- \( c \) is the speed of light (\( 3 \times 10^8 \) m/s),
- \( \lambda \) is the wavelength of the photon in meters.

First, we convert the wavelengths from nanometers to meters:
- \( \lambda_1 = 103.7 \, \text{nm} = 103.7 \times 10^{-9} \, \text{m} \)
- \( \lambda_2 = 30.7 \, \text{nm} = 30.7 \times 10^{-9} \, \text{m} \)

Now we calculate the energies:
1. For \( \lambda_1 = 103.7 \, \text{nm} \):
\[ E_1 = \frac{1240}{103.7} \approx 11.95 \, \text{eV} \]
2. For \( \lambda_2 = 30.7 \, \text{nm} \):
\[ E_2 = \frac{1240}{30.7} \approx 40.41 \, \text{eV} \]

### Step 2: Calculate the total energy emitted
The total energy emitted when the helium ion transitions to the ground state is the sum of the energies of the two photons:
\[ E_{\text{total}} = E_1 + E_2 = 11.95 + 40.41 \approx 52.36 \, \text{eV} \]

### Step 3: Use the energy level formula to find \( n \)
For a hydrogen-like atom, the energy levels are given by:
\[ E = -\frac{Z^2 \cdot 13.6}{n^2} \quad \text{(in eV)} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

For helium ion \( He^+ \), \( Z = 2 \). The energy difference between the initial and final states is:
\[ E_{\text{total}} = 13.6 \cdot Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Here, \( n_1 = 1 \) (ground state) and \( n_2 = n \) (initial state). Thus, we can write:
\[ 52.36 = 13.6 \cdot 4 \left( 1 - \frac{1}{n^2} \right) \]
Simplifying gives:
\[ 52.36 = 54.4 \left( 1 - \frac{1}{n^2} \right) \]
\[ \frac{52.36}{54.4} = 1 - \frac{1}{n^2} \]
Calculating the left side:
\[ 0.963 = 1 - \frac{1}{n^2} \]
Rearranging gives:
\[ \frac{1}{n^2} = 1 - 0.963 = 0.037 \]
Taking the reciprocal:
\[ n^2 \approx \frac{1}{0.037} \approx 27.03 \]
Taking the square root:
\[ n \approx 5.2 \]
Since \( n \) must be an integer, we round to the nearest whole number:
\[ n = 5 \]

### Final Answer
Thus, the value of \( n \) is \( 5 \).

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