Understanding Hydrogen Spectral Series and Quantum Numbers

The spectrum of the hydrogen atom is composed of several series of spectral lines, each corresponding to electron transitions between different energy levels. These energy levels are quantized and are described by principal quantum numbers, denoted by \(n\).

When an electron transitions from a higher energy level (initial state, \(n_i\)) to a lower energy level (final state, \(n_f\)), it emits a photon of specific energy, corresponding to a line in the spectrum. Each series is defined by the final energy level \(n_f\) to which the electron transitions.

Spectral Series and Final Quantum Numbers (\(n_f\))

Let's identify the final quantum number (\(n_f\)) for each of the listed spectral series:

• Lyman Series: Electron transitions end at the \(n_f = 1\) energy level (the ground state). Transitions occur from \(n_i = 2, 3, 4, ...\) to \(n_f = 1\).
• Balmer Series: Electron transitions end at the \(n_f = 2\) energy level. Transitions occur from \(n_i = 3, 4, 5, ...\) to \(n_f = 2\).
• Paschen Series: Electron transitions end at the \(n_f = 3\) energy level. Transitions occur from \(n_i = 4, 5, 6, ...\) to \(n_f = 3\).
• Brackett Series: Electron transitions end at the \(n_f = 4\) energy level. Transitions occur from \(n_i = 5, 6, 7, ...\) to \(n_f = 4\).
• Pfund Series: Electron transitions end at the \(n_f = 5\) energy level. Transitions occur from \(n_i = 6, 7, 8, ...\) to \(n_f = 5\).

We can summarize this information in a table:

Arranging in Increasing Order of Quantum Number (\(n_f\))

The question asks to arrange the series in increasing order of the quantum number when coming from an excited energy state. This refers to the increasing order of the final quantum number (\(n_f\)).

Looking at the table or the list above, the final quantum numbers are 1, 2, 3, 4, and 5 for Lyman, Balmer, Paschen, Brackett, and Pfund series, respectively.

Arranging these series based on the increasing values of \(n_f\):

1. \(n_f = 1\): Lyman Series (A)
2. \(n_f = 2\): Balmer Series (B)
3. \(n_f = 3\): Paschen Series (C)
4. \(n_f = 4\): Brackett Series (D)
5. \(n_f = 5\): Pfund Series (E)

Therefore, the increasing order of the series based on the final quantum number is Lyman, Balmer, Paschen, Brackett, Pfund, which corresponds to A, B, C, D, E.

Conclusion

The arrangement of the given spectral series in increasing order of the final quantum number (\(n_f\)) is Lyman, Balmer, Paschen, Brackett, and Pfund.

Revision Table: Hydrogen Spectral Series

Additional Information: Atomic Energy Levels and Transitions

In the Bohr model and quantum mechanical description of the hydrogen atom, electrons occupy discrete energy levels. Each level is associated with a principal quantum number \(n\), where \(n = 1, 2, 3, ...\).

• The lowest energy level is \(n=1\), called the ground state.
• Levels with \(n > 1\) are called excited states.
• An electron in an excited state can transition to a lower energy state by emitting a photon. The energy of the emitted photon is equal to the difference in energy between the initial (\(E_i\)) and final (\(E_f\)) states: \(\Delta E = E_i - E_f\).
• The frequency \(\nu\) and wavelength \(\lambda\) of the emitted photon are related to the energy difference by the equation: \(\Delta E = h\nu = \frac{hc}{\lambda}\), where \(h\) is Planck's constant and \(c\) is the speed of light.
• The energy of a level \(n\) in the hydrogen atom is given by the formula \(E_n = -\frac{13.6}{n^2}\) eV.
• The spectral lines observed correspond to these specific energy differences or transitions. Each series represents transitions ending at a particular final energy level \(n_f\).