An electron jumps from `5^(th)` excited level to `2^(nd)` excited level in hydrogen atom. Maximum number of lines obtained in this transition will be

Options

A. 10
B. 12
C. 21
D. 6

Answer

To find the maximum number of spectral lines obtained when an electron jumps from the 5th excited level to the 2nd excited level in a hydrogen atom, we can follow these steps:

### Step 1: Understand the Excited Levels
In a hydrogen atom, the excited states are defined by the principal quantum number \( n \). The 5th excited level corresponds to \( n = 6 \) (since the ground state \( n = 1 \) is the first level). The 2nd excited level corresponds to \( n = 3 \).

### Step 2: Identify the Values of \( n_2 \) and \( n_1 \)
- For the 5th excited state (where the electron jumps from):
\[ n_2 = 6 \]
- For the 2nd excited state (where the electron jumps to):
\[ n_1 = 3 \]

### Step 3: Use the Formula for Maximum Number of Spectral Lines
The maximum number of spectral lines \( N \) that can be obtained from a transition between two energy levels is given by the formula:
\[ N = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} \]

### Step 4: Substitute the Values into the Formula
Now, substituting \( n_2 = 6 \) and \( n_1 = 3 \) into the formula:
\[ N = \frac{(6 - 3)(6 - 3 + 1)}{2} \]
\[ N = \frac{(3)(4)}{2} \]
\[ N = \frac{12}{2} = 6 \]

### Step 5: Conclusion
Thus, the maximum number of spectral lines obtained in this transition is:
\[ \boxed{6} \]

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