Atomic hydrogen is excited from the grounds state to the `n^(th)` state. The number of lines in the emission spectrum will be:

Options

A. `(n(n+1))/(2)`
B. `(n(n-1))/(2)`
C. `((n-1)^(2))/(2)`
D. `((n+1)^(2))/(2)`

Answer

To find the number of lines in the emission spectrum when atomic hydrogen is excited from the ground state to the \( n^{th} \) state, we can follow these steps:

### Step 1: Understand the Concept of Emission Spectrum
When an electron in a hydrogen atom is excited to a higher energy level (from the ground state to the \( n^{th} \) state), it can return to the ground state by emitting photons. Each transition from a higher energy level to a lower one corresponds to a spectral line in the emission spectrum.

**Hint:** Remember that each transition corresponds to a unique line in the spectrum.

### Step 2: Identify the Transitions
When the electron is in the \( n^{th} \) state, it can transition to any of the lower energy levels (from \( n-1 \) down to 1). The possible transitions are:
- From \( n \) to \( n-1 \)
- From \( n \) to \( n-2 \)
- ...
- From \( n \) to \( 1 \)

**Hint:** Count how many levels the electron can transition to from the \( n^{th} \) state.

### Step 3: Calculate the Number of Unique Transitions
To find the total number of unique transitions, we can use the formula for the number of lines in the emission spectrum:
\[ \text{Number of lines} = \frac{n(n-1)}{2} \]
Where \( n \) is the final excited state.

**Hint:** This formula counts all possible pairs of transitions from the \( n^{th} \) state to lower states.

### Step 4: Apply the Formula
For the \( n^{th} \) state:
- If \( n = 4 \):
\[ \text{Number of lines} = \frac{4(4-1)}{2} = \frac{4 \times 3}{2} = \frac{12}{2} = 6 \]

**Hint:** Substitute the value of \( n \) into the formula carefully to avoid mistakes.

### Step 5: Conclusion
Thus, the total number of lines in the emission spectrum when atomic hydrogen is excited from the ground state to the \( n^{th} \) state is 6.

**Final Answer:** The number of lines in the emission spectrum will be \( 6 \).