Question

How many different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are excited to states with principal quantum number n?

• A. $\frac{n^2}{2}$
• B. $\frac{(n-1{)}^2}{2}$
• C. $\frac{n(n-1)}{2}$
• D. $\frac{\left(n\right(n-1){)}^2}{2}$

Answer 1

The correct option is C $\frac{n(n-1)}{2}$

From the nth state, the atom may go to (n – 1)th state, …., $2^{nd}$ state or 1st state. So there are (n– 1) possible transitions starting from the nth state. The atom that reached (n – 1)th state may make (n – 2) different transitions to reach 1st state. Similarly for the one reaching at (n-2), it will take (n – 3) different transitions. Similarly for other lower states. The total number of possible transitions is
(n – 1) + (n – 2) + (n – 3) + …. 2 + 1
=$\frac{n(n-1)}{2}$

Just simple maths .!!!!