Question

If the series limit of wavelength of the Lyman series for the hydrogen atom is \(912 \wedge\) then the series limit of wavelength for the Balmer series of the hydrogen atom is (1) \(912 \AA\) (2) \(912 \times 2 \AA\) (3) \(912 \times 4 \AA\) (4) \(912 / 2 \AA\)

Short answer

Option (3) 912 × 4 Å

Step-by-step solution

- Understand the Given Information

The problem provides the series limit of the Lyman series for the hydrogen atom as 912 Å.

- Recall the Series Formula

The Lyman series corresponds to electronic transitions where the electron falls to the n=1 level (ground state). The Balmer series corresponds to electronic transitions where the electron falls to the n=2 level.

- Use the Relationship Between Series Limits

The series limit for the Lyman series is given as 912 Å, which is the transition to the ground state (n=1). For the Balmer series, the transitions are to the n=2 level. The ratio of the series limits is given by the square of the ratio of the levels: \[\frac{1}{\text{Balmer limit}} = \frac{1}{\text{Lyman limit}} \times \frac{1}{4} \] Therefore, the wavelength of the Balmer series limit is 4 times that of the Lyman series limit.

- Calculate the Balmer Limit

Given the Lyman series limit is 912 Å, the Balmer series limit will be: \[\text{Balmer limit} = 912 \times 4 \text{ Å} = 3648 \text{ Å}\]

- Select the Correct Option

From the calculation, the series limit of the wavelength for the Balmer series of the hydrogen atom is 3648 Å. This matches the given option (3).

Key concepts

Lyman series

The Lyman series pertains to electronic transitions in a hydrogen atom that lead to the release of photons. This series involves electrons descending from higher energy levels (n >= 2) to the ground state (n=1).
When these electrons fall to the lowest energy level, they emit ultraviolet radiation, making the Lyman series significant in UV spectroscopy.
Key points to remember about the Lyman series:

• Transitions end at the principal quantum number n=1.
• Emits ultraviolet light.
• Key wavelength limit is 912 ångströms (Å).

Understanding these aspects will aid in comprehending its role in hydrogen's spectral lines.

Balmer series

The Balmer series is associated with electronic transitions where electrons fall to the second energy level (n=2) in a hydrogen atom. Unlike the Lyman series, the Balmer series ejects photons in the visible part of the electromagnetic spectrum.
Key points to remember about the Balmer series:

• Transitions end at the principal quantum number n=2.
• Emits visible light.
• Key wavelength components are visible in the range 3648 Å (for the series limit).

This series is highly significant in understanding the visible spectra of hydrogen and setting the foundation for detecting other elements in stars.

Electronic transitions

Electronic transitions describe movements of electrons between different energy levels within an atom. When an electron jumps from a higher energy level to a lower one, it emits a photon corresponding to the energy difference.
Key points to remember about electronic transitions:

• Energy levels are quantized, meaning electrons can only exist at specific energy values.
• The frequency and wavelength of the emitted photon directly correlate to the energy gap between the levels.
• Each transition's characteristics define the spectral series (Lyman, Balmer, etc.).

Understanding these transitions is fundamental for interpreting atomic spectra and analyzing the emission and absorption lines in various elements.