Problem 31 - Fundamentals of Physics

Question

What is the ratio of the shortest wavelength of the Balmer series to the shortest wavelength of the Lyman series?

Step-by-Step Solution

Answer

The ratio of shortest wavelengths is 4.

Step 1: Understanding the Problem

The Balmer and Lyman series are two sets of spectral lines that are due to electron transitions in a hydrogen atom. The shortest wavelength in each series corresponds to the transitions from the highest possible energy level to the ground level of that series. Our task is to find the ratio of these shortest wavelengths.

Step 2: Identify Shortest Wavelength for Balmer Series

The Balmer series corresponds to transitions where the final electron energy level (n') is n = 2. The shortest wavelength occurs for a transition from n = ∞ to n = 2.

Step 3: Identify Shortest Wavelength for Lyman Series

The Lyman series corresponds to transitions where the final electron energy level (n') is n = 1. The shortest wavelength occurs for a transition from n = ∞ to n = 1.

Step 4: Use Rydberg Formula

The Rydberg formula for the wavelength of light emitted during a transition is \( \frac{1}{\lambda} = R_H \left( \frac{1}{n'^2} - \frac{1}{n^2} \right) \), where \( R_H \) is the Rydberg constant, \( n' \) is the final energy level, and \( n \) is the initial energy level.

Step 5: Calculate Shortest Wavelengths

For Balmer: \( \frac{1}{\lambda_{Balmer}} = R_H \left( \frac{1}{2^2} - 0 \right) = \frac{R_H}{4} \).For Lyman: \( \frac{1}{\lambda_{Lyman}} = R_H \left( \frac{1}{1^2} - 0 \right) = R_H \).

Step 6: Find Ratio of Wavelengths

The ratio of \( \lambda_{Balmer} \) to \( \lambda_{Lyman} \) is \( \frac{\lambda_{Balmer}}{\lambda_{Lyman}} = \frac{1/R_H}{1/(R_H/4)} = 4 \).

Key Concepts

Balmer Series

The Balmer series is a famous set of spectral lines named after Johann Balmer, who discovered it in 1885. It belongs to the hydrogen spectral series and is visible in the optical part of the light spectrum. The Balmer series arises when electrons in a hydrogen atom transition from higher energy levels to the second energy level (n = 2). This particular series is notable because it's the only one visible to the human eye without instruments, showcasing some of the most famous visible spectral lines, like the H-alpha line in red.

Within the Balmer series, the shortest wavelength occurs when an electron falls from an infinitely high level (n = ∞) down to n = 2. The resulting spectral lines provide crucial insights into the behavior of hydrogen and are often used in astronomical observations to identify hydrogen gas clouds.

• Transition ends at n = 2
• Visible spectral lines include H-alpha, H-beta, etc.
• Detectable with basic optical equipment

Lyman Series

Similar to the Balmer series, the Lyman series is another set of spectral lines in the hydrogen atom, but with a key difference. It involves electron transitions to the ground state (n = 1) of the hydrogen atom. This series lies in the ultraviolet region of the electromagnetic spectrum, which is why it's not seen by the naked eye but is detectable with specialized equipment.

The Lyman series is critical for understanding high-energy transitions in hydrogen. The shortest wavelength in this series happens when an electron transitions from an extremely high level (n = ∞) to n = 1, resulting in a high-energy ultraviolet spectral line. This behavior is particularly important in fields like astrophysics, where analyzing these spectral lines helps to understand the chemical composition of stars and other celestial bodies.

• Transition ends at n = 1
• Found in the ultraviolet spectrum
• Requires specialized equipment to be detected

Rydberg Formula

The Rydberg formula is a mathematical expression used to predict the wavelengths of spectral lines in many chemical elements. Named after Johannes Rydberg, this formula plays a central role in understanding atomic spectra, particularly hydrogen.

The formula is expressed as:\[\frac{1}{\lambda} = R_H \left( \frac{1}{n'^2} - \frac{1}{n^2} \right)\]Where:

• \( \lambda \) is the wavelength
• \( R_H \) is the Rydberg constant, approximately equal to 1.097 \( \times \) 10⁷ m^-1
• \( n' \) and \( n \) are the final and initial energy levels

The Rydberg formula is instrumental in calculating the emission spectra for hydrogen. Used in tandem with quantum mechanics, it highlights the quantized nature of energy levels within an atom. When applied, it can precisely determine the wavelength of light emitted during electron transitions, underpinning our understanding of both the Balmer and Lyman series.

Spectral Lines

Spectral lines, essential tools in atomic spectroscopy, represent the various wavelengths of light emitted or absorbed by an element. These lines appear as distinct bright or dark lines on a spectral chart. Each element emits a unique set of spectral lines, akin to a fingerprint, allowing scientists to identify the composition of substances far or near, like distant stars.

In the context of hydrogen, spectral lines arise when electrons transition between different energy levels. The characteristics of these lines, such as their position and intensity, are determined by the energy difference between the initial and final states of the electron.

Understanding spectral lines is crucial for many scientific fields, including physics, chemistry, and astronomy. Analyzing these lines helps unveil properties of atoms and molecules and forms the foundation for identifying elements in stars and galaxies.

• Unique to each element
• Appear as bright or dark lines
• Key for identifying element composition