X different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are exited to states with principal quantum number n=6 ? The value of X is ______________.

Answer

To determine the number of different wavelengths (X) that can be observed in the spectrum from a hydrogen sample when the atoms are excited to states with a principal quantum number \( n = 6 \), we can follow these steps:

### Step 1: Understand the Energy Levels
In a hydrogen atom, the energy levels are quantized and are defined by the principal quantum number \( n \). 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 Possible Transitions
When an electron in a hydrogen atom is excited to a higher energy level (in this case, \( n = 6 \)), it can transition to any lower energy level (from \( n = 5 \) down to \( n = 1 \)). The possible transitions are:
- From \( n = 6 \) to \( n = 5 \)
- From \( n = 6 \) to \( n = 4 \)
- From \( n = 6 \) to \( n = 3 \)
- From \( n = 6 \) to \( n = 2 \)
- From \( n = 6 \) to \( n = 1 \)
- From \( n = 5 \) to \( n = 4 \)
- From \( n = 5 \) to \( n = 3 \)
- From \( n = 5 \) to \( n = 2 \)
- From \( n = 5 \) to \( n = 1 \)
- From \( n = 4 \) to \( n = 3 \)
- From \( n = 4 \) to \( n = 2 \)
- From \( n = 4 \) to \( n = 1 \)
- From \( n = 3 \) to \( n = 2 \)
- From \( n = 3 \) to \( n = 1 \)
- From \( n = 2 \) to \( n = 1 \)

### Step 3: Count the Transitions
To count the number of unique transitions, we can use the formula for the number of ways to choose 2 energy levels from \( n \) levels, which is given by:
\[ \text{Number of transitions} = \frac{n(n-1)}{2} \]
In this case, \( n = 6 \):
\[ \text{Number of transitions} = \frac{6(6-1)}{2} = \frac{6 \times 5}{2} = 15 \]

### Conclusion
Thus, the number of different wavelengths \( X \) that may be observed in the spectrum from a hydrogen sample when the atoms are excited to states with principal quantum number \( n = 6 \) is:
\[ \boxed{15} \]