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Chapter 7: Problem 12

There are an infinite number of allowed electronic transitions in the hydrogen atom. Why don’t we see more lines in the hydrogen emission spectrum?

Short Answer

Expert verified
In conclusion, the hydrogen emission spectrum has a limited number of visible lines because of two main reasons: 1) the energy differences between higher energy levels decrease with increasing principal quantum number, n, which results in emitted photons with wavelengths outside the visible light spectrum; and 2) the human eye can only detect a certain range of photon energies, corresponding to wavelengths between 400nm and 700nm. Therefore, even though there are infinite allowed electronic transitions in the hydrogen atom, we don't see more lines in the hydrogen emission spectrum as most transitions emit photons that are not visible to us.

Step by step solution

01

Understand the hydrogen atomic structure and electronic transitions

In the hydrogen atom, the electron orbits the nucleus in different energy levels. These energy levels are quantized, meaning the electron can only exist in specific energy levels. When an electron in a hydrogen atom transitions from a higher energy level to a lower energy level, it emits a photon with an energy equal to the energy difference between the two levels.
02

Learn about hydrogen emission spectrum

The hydrogen emission spectrum is created when a high-energy electron collides with a hydrogen atom, causing the electron in the hydrogen atom to jump to a higher energy level. As the electron falls back to its original energy level, it releases energy in the form of photons, which create the lines in the hydrogen emission spectrum. These lines correspond to the different energy transitions.
03

Energy limits and visible light range

Although there are infinite allowed electronic transitions, the energy differences between higher energy levels become very small as the principal quantum number, n, increases. Also, note that the human eye can only detect a limited range of photon energies, corresponding to the visible light spectrum (wavelengths between 400nm and 700nm).
04

Limitations due to decreasing energy differences and visibility

As we go to very high energy levels (large values of n), the energy differences between consecutive levels become very small, and the emitted photons have wavelengths that fall outside the visible spectrum. This means that most of these transitions will not be visible to the human eye, and thus, we don't see more lines in the hydrogen emission spectrum even though there are infinite allowed electronic transitions.
05

Concluding remarks

In conclusion, the hydrogen emission spectrum contains a limited number of visible lines due to the decreasing energy differences between higher energy levels in the hydrogen atom and the limited range of wavelengths that can be detected by the human eye. Although there are infinite allowed electronic transitions in the hydrogen atom, most of these transitions emit photons that fall outside the visible light spectrum and hence are not visible to us.

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