Question

Assume that a hydrogen atom's electron has been excited to the \(n=6\) level. How many different wavelengths of light can be emitted as this excited atom loses energy?

Short answer

The electron in the hydrogen atom is excited to the \(n=6\) level and can fall to any level below the current level (n=1, 2, 3, 4, or 5). Each transition corresponds to a different energy and, consequently, a different wavelength of emitted light. Counting all the unique transitions, we find that the hydrogen atom can emit 5 different wavelengths of light as it loses energy.

Step-by-step solution

Identify the energy levels involved in the transitions

The electron in the hydrogen atom is excited to the n=6 level, and it can fall to any level below the current level (n=1, 2, 3, 4, or 5). Each transition corresponds to a different energy, and consequently, a different wavelength of emitted light.

Calculate the number of possible transitions (wavelengths)

When the electron falls from the n=6 level to a lower level, we can calculate the total number of possible transitions by adding up the possible transitions from each lower level to n=6. Transitions to n=1: 1. n=6 → n=1 Transitions to n=2: 2. n=6 → n=2 Transitions to n=3: 3. n=6 → n=3 Transitions to n=4: 4. n=6 → n=4 Transitions to n=5: 5. n=6 → n=5

Count all the unique transitions

We can now count all the unique transitions that we listed above, which will give us the number of different wavelengths of light emitted as the excited hydrogen atom loses energy. Number of unique transitions = 5 Therefore, the hydrogen atom with an electron excited to the n=6 level can emit 5 different wavelengths of light as it loses energy.