3.2. Hydrogen Atom Spectral Series

Hydrogen, being the simplest one-electron atom, provides a clear example of discrete line spectra. The formula for the wavenumber (which is 1 divided by wavelength) of any line in hydrogen’s spectrum can be written in plain words as:

"The wavenumber equals the Rydberg constant for hydrogen, multiplied by [one divided by the square of the final energy level number minus one divided by the square of the initial energy level number]."

We write that as:

Wavenumber = R_H times (1 ÷ (n_f squared) – 1 ÷ (n_i squared)),

where R_H is the Rydberg constant for hydrogen (about 1.0968 × 10^7 per meter), n_f is the lower principal quantum number (for example 1, 2, 3, …), and n_i is the higher principal quantum number (n_i > n_f).

In this section, we learn about how the hydrogen atom's spectral lines can be calculated using a specific formula. The wavenumber, which is basically a measure of how many wave peaks fit in a given length, can be determined through the Rydberg constant. The formula involves two energy levels: the initial level (n_i) where the electron starts and the final level (n_f) where the electron ends up after emitting or absorbing energy. The differences in energy levels are what create the unique spectral lines for hydrogen. This means that when an electron transitions between these levels, it emits or absorbs light at specific wavelengths, corresponding to those energy differences.

Think of a ladder where each rung represents a specific energy level for the electron. When an electron moves from a higher rung (n_i) to a lower one (n_f), it drops down and emits a flash of light—the lower the rung, the more colorful and bright the light it produces. The Rydberg constant acts like a calculator that helps us figure out exactly how much energy each rung jump produces, just like measuring how far the flash can be seen.

Depending on the value of the final energy level n_f, you get different series of lines:

1. Lyman Series (n_f = 1)

   • These lines lie in the ultraviolet.
   • For example:
     • Transition from n_i = 2 down to n_f = 1 produces a wavelength of about 121.6 nanometers.
     • Transition from n_i = 3 down to n_f = 1 produces about 102.6 nm.
     • Transition from n_i = 4 down to n_f = 1 produces about 97.3 nm, and so on.

2. Balmer Series (n_f = 2)

   • These lines lie in the visible region (and near-ultraviolet).
   • Examples:
     • n_i = 3 → n_f = 2 gives a wavelength of 656.3 nm (red line, known as H-alpha).
     • n_i = 4 → n_f = 2 gives 486.1 nm (blue-green line, H-beta).
     • n_i = 5 → n_f = 2 gives 434.0 nm (blue-violet line, H-gamma).
     • n_i = 6 → n_f = 2 gives 410.2 nm (violet line, H-delta).
     • As n_i increases further, the lines get closer together and approach a limit at 364.6 nm (just into the ultraviolet).

3. Paschen Series (n_f = 3)

   • These lines lie in the infrared.
   • For instance:
     • n_i = 4 → n_f = 3 gives about 1,875 nm.
     • n_i = 5 → n_f = 3 gives about 1,282 nm.
     • n_i = 6 → n_f = 3 gives about 1,094 nm, and so on.

4. Brackett Series (n_f = 4)

   • Also in the infrared (longer wavelengths).
   • For example: n_i = 5 → n_f = 4 gives 4,051 nm; n_i = 6 → n_f = 4 gives 2,625 nm; etc.

5. Pfund Series (n_f = 5)

   • Farther out in the infrared (like 7,460 nm for n_i = 6 → n_f = 5).

6. Humphreys Series (n_f = 6)

   • Even farther into the infrared.

The spectral lines of hydrogen are categorized into different named series based on the principal quantum number to which the electron transitions. Each series corresponds to a specific range of wavelengths. The Lyman Series corresponds to transitions that end at n_f = 1, which are in the ultraviolet region. The Balmer Series is visible light that corresponds to transitions ending at n_f = 2. Each transition releases a specific wavelength of light, leading to the distinct spectral lines we observe. The Paschen, Brackett, Pfund, and Humphreys series, corresponding to higher final energy levels, produce infrared light. These series help us understand the energy level structure of the hydrogen atom.

Imagine a colorful waterfall dropping from different heights (energy levels). Each time the water (electron) jumps from a high level to a lower one, it creates a splash of water (light). The higher the drop, the more colorful the splash—some may visualize as bright colors (visible light in the Balmer series) while others might be less visible (infrared or ultraviolet). By labeling each splash with the height from which it fell (the energy levels), we can categorize them into series, helping us understand the overall 'splashing' behavior of the waterfall.

In each series, there is a “series limit” obtained by letting n_i approach infinity. Then one divided by (n_i squared) goes to zero, and the wavenumber approaches R_H divided by (n_f squared). The corresponding wavelength limit equals (n_f squared) divided by R_H. For example, Balmer series (with n_f = 2) has a series limit at 364.6 nm, which is where 1 ÷ wavelength equals R_H times (1 ÷ 4).

As the initial energy level (n_i) increases without limit (approaching infinity), the energy of the corresponding emitted photon approaches the energy needed to completely remove the electron from the atom. This results in a limit where the added terms become negligible (approaching zero), allowing us to determine a 'series limit'. For the Balmer Series, this means that the emissions will get closer together until they reach a point where they can no longer be emitted as distinct lines, known as the series limit. This relationship can be mathematically expressed, showing the connection between energy levels and their resulting spectral emissions.

Think of a spring being stretched; the more you pull it (higher the energy level), the closer you get to a point where it snaps (the electron presence at infinity). The series limit is like that point of potential—the spring will stretch until it can no longer hold tension, much like how energy levels stretch tighter until they no longer create distinct emissions in the spectral series.

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