Hydrogen atom in a sample are excited to `n = 5` state and it is found that photons of all possible wavelength are present in the emission spectra. The minimum number of hydrogen atom in the sample would be

Options

A. 5
B. 6
C. 10
D. infinite

Explanation

To solve the problem, we need to determine the minimum number of hydrogen atoms required to produce all possible wavelengths in the emission spectrum when the hydrogen atoms are excited to the n = 5 state.

### Step-by-Step Solution:

1. **Understanding the Energy Levels**:
The hydrogen atom has quantized energy levels denoted by the principal quantum number \( n \). For the hydrogen atom, the energy levels are given by \( n = 1, 2, 3, 4, 5, \ldots \).

2. **Possible Transitions from n = 5**:
When an electron in a hydrogen atom is excited to the \( n = 5 \) state, it can transition to lower energy levels (n = 4, 3, 2, 1). Each transition emits a photon of a specific wavelength.

3. **Calculating the Number of Spectral Lines**:
The number of possible transitions (or spectral lines) from a given energy level \( n \) to lower levels can be calculated using the formula:
\[ \text{Number of lines} = \frac{n(n-1)}{2} \]
For \( n = 5 \):
\[ \text{Number of lines} = \frac{5 \times 4}{2} = 10 \]
This means there are 10 possible transitions from \( n = 5 \) to lower levels.

4. **Identifying Required Atoms for Each Transition**:
- Some transitions can be achieved by a single hydrogen atom. For example:
- Transition from \( n = 5 \) to \( n = 4 \) (one atom)
- Transition from \( n = 4 \) to \( n = 3 \) (one atom)
- Transition from \( n = 3 \) to \( n = 1 \) (one atom)
- Other transitions may require separate atoms. For example:
- Transition from \( n = 5 \) to \( n = 3 \) (one atom)
- Transition from \( n = 3 \) to \( n = 1 \) (one atom)
- The remaining transitions will also require additional atoms.

5. **Counting the Minimum Number of Atoms**:
- By analyzing the transitions:
- For the transitions \( 5 \to 4 \), \( 4 \to 3 \), and \( 3 \to 1 \): 1 atom can handle these.
- For the transition \( 5 \to 3 \) and \( 3 \to 1 \): 1 atom can handle these.
- The other transitions (like \( 5 \to 2 \), \( 4 \to 2 \), etc.) will require separate atoms.

After careful consideration, we find that:
- 1 atom can handle transitions \( 5 \to 4 \), \( 4 \to 3 \), and \( 3 \to 1 \).
- 1 atom can handle \( 5 \to 3 \) and \( 3 \to 1 \).
- 4 more atoms are needed for the remaining transitions.

Thus, the total minimum number of hydrogen atoms required is:
\[ 1 (for 5 \to 4, 4 \to 3, 3 \to 1) + 1 (for 5 \to 3 and 3 \to 1) + 4 (for remaining transitions) = 6 \]

### Final Answer:
The minimum number of hydrogen atoms in the sample would be **6**.