Calculate the shortest wavelength of Lyman series of H-atom spectrum.

To calculate the shortest wavelength of the Lyman series in the hydrogen atom spectrum, we need to understand a few key concepts about atomic transitions and the Rydberg formula. The Lyman series consists of electronic transitions from higher energy levels to the first energy level (n=1) of the hydrogen atom. The shortest wavelength corresponds to the transition from the highest energy level to the first energy level, which is an important aspect of the series.

Understanding Energy Levels

In a hydrogen atom, electrons occupy specific energy levels, denoted by the principal quantum number n. The energy levels are given by the formula:

• E_n = -13.6 eV / n²

Here, E_n is the energy of the level, and n is the principal quantum number (n = 1, 2, 3, ...). The energy difference between two levels can be calculated as:

• ΔE = E_final - E_initial

Finding the Shortest Wavelength

The transition that produces the shortest wavelength in the Lyman series is from n = ∞ (a very high energy level) to n = 1. The energy difference for this transition is:

• ΔE = E_1 - E_∞ = -13.6 eV - 0 eV = -13.6 eV

This energy is released as a photon, and the energy of the photon is related to its wavelength (λ) by the equation:

• E = hc / λ

Where:

• E is the photon energy
• h is Planck's constant (6.626 x 10^-34 J·s)
• c is the speed of light (3 x 10^8 m/s)

Calculating Wavelength

First, we need to convert the energy from electronvolts to joules. Since 1 eV = 1.602 x 10^-19 J, we have:

• ΔE = 13.6 eV × 1.602 x 10^-19 J/eV = 2.179 x 10^-18 J

Now, we can rearrange the photon energy equation to solve for wavelength:

• λ = hc / E

Substituting the values into the equation gives:

• λ = (6.626 x 10^-34 J·s)(3 x 10^8 m/s) / (2.179 x 10^-18 J)

Calculating this will yield:

• λ ≈ 1.214 x 10^-7 m or 121.4 nm

Final Result

The shortest wavelength of the Lyman series for the hydrogen atom spectrum is approximately 121.4 nanometers. This wavelength falls within the ultraviolet range, making it significant in spectroscopy and astrophysics, as it can be used to study various astronomical phenomena and the characteristics of hydrogen in different environments.