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 ______________.

Solution

Physics sir · 4 steps

Every pair of levels from n = 1 to n = 6 gives one line, so just count the pairs.

1. One line per jump. An electron in level n = 6 can fall to any lower level, and from there fall again. Each downward jump between two different levels emits a photon of one particular wavelength.
2. Count the pairs. There are 6 levels, n = 1 to n = 6. A sample has many atoms, so every possible pair of levels shows up. The number of pairs is ⁶C₂.

   N = n(n-1)/2 = (6 × 5)/2 = 15

3. Check by listing. From n = 6 there are 5 jumps, from n = 5 there are 4, then 3, 2 and 1. That gives 5 + 4 + 3 + 2 + 1 = 15.
4. So. The spectrum shows 15 different wavelengths, so X = 15.

The common slip: 5. Counting only the jumps that start from n = 6 gives 5, not 15. The atoms also make further jumps from the lower levels on the way down.

The idea to keep. For a sample of atoms excited to level n, the number of spectral lines is n(n-1)/2. A single atom can give at most n - 1 lines.

Answer from the NTA final answer key. Solution by PrepWiser.