Question

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 _________.

Hint:

Memorizing the formula \(N = \frac{n(n-1)}{2}\) is the fastest way to solve this common type of problem. It's derived from the concept of combinations, as we are choosing any two energy levels out of n levels for a transition to occur.

Answer 1

Correct Answer: 15

To determine the number of different wavelengths observed in the hydrogen spectrum when the atoms are excited to a principal quantum number \( n = 6 \), we need to calculate the possible transitions. The formula to find the number of spectral lines observed is given by:

Number of spectral lines = \( \frac{n(n-1)}{2} \) 

For \( n = 6 \):

\( \frac{6(6-1)}{2} = \frac{6 \times 5}{2} = 15 \)

Thus, the number of different wavelengths (i.e., spectral lines) that can be observed is 15.

Verification:

• The calculated value is 15.
• The range given is 15,15, which includes the calculated value.

Therefore, the value of \( X \) is 15.