In a single isolated atom of hydrogen electron make transition from `4^(th)` excited state to grond state producing maximum possible number of wavelengths if the `2^(nd)` lowest energy of these lines is used to further excite an already excited sample of `Li^(2+)` ion then transition will be

Options

A. `12rarr 15`
B. `9 rarr 12`
C. `6 rarr 9`
D. `3 rarr 6`

Answer

To solve the problem, we need to analyze the transitions of an electron in a hydrogen atom and then relate it to the lithium ion (Li²⁺). Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Energy Levels in Hydrogen
In hydrogen, the energy levels are given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 2: Determine the Initial and Final States for Hydrogen
The electron in hydrogen transitions from the 4th excited state to the ground state. The 4th excited state corresponds to \( n = 5 \) (since the ground state is \( n = 1 \)):
- Ground state: \( n_1 = 1 \)
- 4th excited state: \( n_2 = 5 \)

### Step 3: Calculate the Possible Transitions
The possible transitions from \( n = 5 \) to lower energy levels are:
- \( 5 \to 4 \)
- \( 5 \to 3 \)
- \( 5 \to 2 \)
- \( 5 \to 1 \)

This gives us a total of 4 transitions.

### Step 4: Identify the Wavelengths Produced
Each transition produces a spectral line. The maximum number of wavelengths produced when transitioning from \( n = 5 \) to all lower states is 4.

### Step 5: Determine the Second Lowest Energy Transition
The transitions in order of energy (from highest to lowest) are:
1. \( 5 \to 1 \)
2. \( 5 \to 2 \) (this is the second lowest)
3. \( 5 \to 3 \)
4. \( 5 \to 4 \)

### Step 6: Use the Second Lowest Energy Transition to Excite Li²⁺
The second lowest transition corresponds to the transition from \( n = 5 \) to \( n = 2 \).

### Step 7: Analyze the Lithium Ion (Li²⁺)
For the lithium ion (Li²⁺), the energy levels are given by:
\[ E_n = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
where \( Z = 3 \) for lithium.

### Step 8: Calculate the Energy Levels for Li²⁺
For Li²⁺:
- Ground state: \( n = 1 \)
- First excited state: \( n = 2 \)
- Second excited state: \( n = 3 \)
- Third excited state: \( n = 4 \)

### Step 9: Determine the Transition for Li²⁺
If we use the energy from the \( 5 \to 2 \) transition in hydrogen to excite Li²⁺, we need to find out what transition this energy corresponds to in Li²⁺.

Using the energy levels:
- For \( n = 2 \) in Li²⁺:
\[ E_2 = -\frac{3^2 \cdot 13.6 \, \text{eV}}{2^2} = -\frac{122.4}{4} = -30.6 \, \text{eV} \]

- For \( n = 3 \) in Li²⁺:
\[ E_3 = -\frac{3^2 \cdot 13.6 \, \text{eV}}{3^2} = -\frac{122.4}{9} = -13.6 \, \text{eV} \]

### Final Transition
The transition will be from \( n = 2 \) to \( n = 3 \) in Li²⁺, as this corresponds to the energy absorbed from the \( 5 \to 2 \) transition in hydrogen.

### Conclusion
The transition in the Li²⁺ ion after using the energy from the second lowest transition in hydrogen is:
**Transition from \( n = 2 \) to \( n = 3 \) in Li²⁺.**