The ionization enegry of the electron in the hydrogen atom in its ground state is `13.6 ev`. The atoms are excited to higher energy levels to emit radiations of `6` wavelengths. Maximum wavelength of emitted radiation corresponds to the transition between

Options

A. `n = 3` to `n = 2` states
B. `n = 3` to `n = 1` states
C. `n = 2` to `n = 1` states
D. `n = 4` to `n = 3` states

Explanation

To solve the problem, we need to determine the maximum wavelength of emitted radiation corresponding to a transition in the hydrogen atom when it is excited to higher energy levels.

### Step-by-Step Solution:

1. **Understanding Ionization Energy**:
The ionization energy of the electron in the hydrogen atom in its ground state is given as 13.6 eV. This energy corresponds to the transition from the ground state (n=1) to the point where the electron is completely removed (n=∞).

2. **Identifying the Number of Wavelengths**:
The problem states that the atom is excited to higher energy levels to emit radiation of 6 wavelengths. The number of spectral lines (wavelengths) emitted when an electron transitions between energy levels can be calculated using the formula:
\[ \text{Number of transitions} = \frac{N(N-1)}{2} \]
where \(N\) is the number of energy levels involved.

3. **Setting Up the Equation**:
We know that the number of wavelengths emitted is 6. Thus, we set up the equation:
\[ \frac{N(N-1)}{2} = 6 \]
Multiplying both sides by 2 gives:
\[ N(N-1) = 12 \]

4. **Solving the Quadratic Equation**:
Rearranging gives us:
\[ N^2 - N - 12 = 0 \]
Now we can factor this quadratic equation:
\[ (N - 4)(N + 3) = 0 \]
This gives us two possible solutions:
\[ N = 4 \quad \text{or} \quad N = -3 \]
Since \(N\) must be a positive integer, we take \(N = 4\).

5. **Identifying the Maximum Wavelength Transition**:
The maximum wavelength corresponds to the transition from the highest energy level (n=4) to the lowest energy level (n=3). This is because the energy difference between these two levels will be the smallest, resulting in the longest wavelength.

6. **Conclusion**:
Therefore, the maximum wavelength of emitted radiation corresponds to the transition between the energy levels \(n = 4\) and \(n = 3\).

### Final Answer:
The maximum wavelength of emitted radiation corresponds to the transition between \(n = 4\) and \(n = 3\).