Question

What is the shortest possible wavelength of the Lyman series in hydrogen?

Short answer

Answer: The shortest possible wavelength of the Lyman series in hydrogen is approximately 91.1 nm.

Step-by-step solution

Recall the formula for the hydrogen spectral lines

To calculate the wavelength of the spectral lines in hydrogen, we will use the Rydberg formula for hydrogen: $$\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$ In this formula, λ represents the wavelength, 𝑅_H is the Rydberg constant for hydrogen (approximately 1.097 x 10^7 m⁻¹), n₁ and n₂ are the principal quantum numbers of the initial and final energy levels, respectively.

Determine n₁ and n₂ for the Lyman series

For the Lyman series, electron transitions occur between the first energy level (n = 1) and any higher energy level (n > 1). To find the shortest wavelength, we need the transition with the highest energy difference, which corresponds to an electron transitioning from n -> infinity (n₂ = ∞) to the first energy level (n₁ = 1).

Plug the values into the Rydberg formula

Now that we have n₁ = 1 and n₂ = ∞, we can plug these values into the Rydberg formula: $$\frac{1}{\lambda} = R_H \left(\frac{1}{1^2} - \frac{1}{\infty^2}\right)$$

Simplify the equation

Now, simplifying the equation, we get: $$\frac{1}{\lambda} = R_H (1 - 0)$$ $$\frac{1}{\lambda} = R_H$$

Find the shortest possible wavelength

Now, we can solve for λ by taking the reciprocal of both sides of the equation: $$\lambda = \frac{1}{R_H}$$ Plugging in the value of 𝑅_H (1.097 x 10^7 m⁻¹): $$\lambda = \frac{1}{1.097 \times 10^7 \mathrm{m^{-1}}}$$ Thus, the shortest possible wavelength of the Lyman series in hydrogen is: $$\lambda ≈ 9.11 \times 10^{-8} \mathrm{m}$$ or approximately 91.1 nm.

Key concepts

Rydberg formula

The Rydberg formula is an essential tool in atomic physics, particularly when analyzing the spectral lines of hydrogen. It allows us to calculate the wavelengths of photons emitted or absorbed when an electron transitions between energy levels in a hydrogen atom. The formula is given by:\[\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\]where:

• \(\lambda\) is the wavelength of the emitted or absorbed light.
• \(R_H\), the Rydberg constant, is specific to hydrogen and approximately equals 1.097 x 10^7 m⁻¹.
• \(n_1\) and \(n_2\) are the principal quantum numbers representing the electron’s initial and final energy levels.

The formula essentially relates these elements to predict the light's wavelength, helping us understand atomic transitions.

hydrogen spectral lines

Hydrogen spectral lines are a set of discrete wavelengths that are emitted when the electron in a hydrogen atom transitions between different energy levels. These lines appear when an atom undergoes energy changes, releasing light. The lines are grouped into different series, with the Lyman, Balmer, and other series representing transitions to different base energy levels. Each series is characterized by specific wavelengths that appear in the electromagnetic spectrum:

• The Lyman series involves transitions to the lowest energy level (n = 1) and is found in the ultraviolet region.
• The Balmer series involves transitions to the second-lowest energy level (n = 2) and falls within the visible spectrum.
• The Paschen and other series involve transitions to even higher energy levels and occupy the infrared region.

These spectral lines offer insights into the atomic structure and are used in various applications, such as astrophysics and energy level calculations.

principal quantum numbers

Principal quantum numbers, denoted as \(n\), are fundamentally important in the study of atomic physics. They represent the energy levels or shells that electrons occupy in an atom, determining their distance from the nucleus and the energy associated with their state. The principal quantum number is always a positive integer (\(n = 1, 2, 3, \ldots\)), where:

• \(n = 1\) is the closest shell to the nucleus (the ground state), and hence, it has the lowest energy state.
• Electrons can transition between these levels by absorbing or emitting energy, which results in the formation of spectral lines.
• The larger the \(n\), the higher the energy and the further the electron is from the nucleus.

Understanding principal quantum numbers is crucial for explaining the likelihood of electron transitions, ionization energy, and predicting the behavior of atoms in different environments.

Rydberg constant

The Rydberg constant, \(R_H\), plays a pivotal role in the calculation of hydrogen's spectral lines through the Rydberg formula. It is a fundamental constant that reflects the limit of the highest wave number (the reciprocal of wavelength) of any photon that can be emitted from a hydrogen atom. Its value is approximately 1.097 x 10^7 m⁻¹.
The significance of the Rydberg constant lies in its ability to provide a precise and consistent measure for comparing spectral lines across different hydrogen spectra.

• It directly indicates the scale of energy transitions possible within the hydrogen atom.
• The constant underpins much of quantum physics and frequently appears in formulas related to atomic transitions.
• Understanding the Rydberg constant is crucial for students and researchers working with the hydrogen atom, as it significantly impacts our understanding of atomic spectra.

Being one of the most tested constants, \(R_H\) continues to shed light on quantum mechanics and the nature of light.