Question

Show that in the Balmer series that the frequencies of successive lines tend toward a limiting value of \(\mathrm{cR} / 4\), where \(\mathrm{R}\) is the Rydberg constant and \(\mathrm{c}\) the speed of light.

Short answer

In the Balmer series, the frequencies of successive lines tend toward a limiting value of \(\frac{cR}{4}\), where R is the Rydberg constant and c is the speed of light. This is found by using the Rydberg formula to obtain the frequency formula for the Balmer series, then finding the difference between successive frequencies and determining the limit as the energy level approaches infinity. The limit of the frequency differences is 0, meaning the frequencies converge to the first emission line value, which is \(\frac{cR}{4}\).

Step-by-step solution

Balmer Series Formula

The Balmer series is represented by the Rydberg formula, which gives the wavenumber of any spectral line in the hydrogen atom: \(\bar{\nu} = R\left(\frac{1}{m^2} - \frac{1}{n^2}\right)\), where \(\bar{\nu}\) is the wavenumber, R is the Rydberg constant, \(m\) is the lower energy level (2 in the case of the Balmer series), and \(n\) is the higher energy level (\(n > m\)). The emission frequency (\(f\)) and the wavenumber (\(\bar{\nu}\)) are related by the speed of light c: \(f = c\bar{\nu}\)

Balmer Series Frequency

Substitute the Balmer series formula into the frequency formula: \(f = cR\left(\frac{1}{m^2} - \frac{1}{n^2}\right)\) For the Balmer series, \(m = 2\), so we have: \(f = cR\left(\frac{1}{4} - \frac{1}{n^2}\right)\)

Find the Difference between Successive Frequencies

Now we find the difference between the frequencies of successive energy levels in the Balmer series. Let's call these differences \(\Delta f_{n}\): \(\Delta f_{n} = f_{n+1} - f_{n}\) Substitute the Balmer series frequency formula for both\(f_{n+1}\) and \(f_{n}\): \(\Delta f_{n} = cR\left(\frac{1}{4} - \frac{1}{(n+1)^2}\right) - cR\left(\frac{1}{4} - \frac{1}{n^2}\right)\) Simplify the expression: \(\Delta f_{n} = cR\left(\frac{1}{n^2} - \frac{1}{(n+1)^2}\right)\)

Determine the Limit of Successive Frequencies

Find the limit of \(\Delta f_{n}\) as \(n \rightarrow \infty\): \(\lim_{n\to\infty}\Delta f_{n} = \lim_{n\to\infty} cR\left(\frac{1}{n^2} - \frac{1}{(n+1)^2}\right)\) As \(n \rightarrow \infty\), both \(n^2\) and \((n+1)^2\) becomes very large, making their fractions very small. Therefore, the limit goes to: \(\lim_{n\to\infty}\Delta f_{n} = cR \left( 0 \right) = 0\) This limit indicates that the successive frequency differences approach zero, meaning the frequency values converge to a particular value as n approaches infinity.

Find the Limiting Value of Successive Frequencies

To find the limiting value of frequencies towards which the successive frequencies converge, we compute the sum of the frequency differences: \(\lim_{n\to\infty} f_{n} = f_{1} + \lim_{n\to\infty} \sum_{k=1}^{n} \Delta f_{k}\) Since the limit of the frequency differences is 0, the frequency values converge to that of the first emission line in the Balmer series: \(f_{1} = cR\left(\frac{1}{4} - \frac{1}{3^2}\right) = cR\left(\frac{1}{4} - \frac{1}{9}\right) = \frac{cR}{4}\) Thus, in the Balmer series, the frequencies of successive lines tend toward a limiting value of \(\frac{cR}{4}\).

Key concepts

Rydberg constant

The Rydberg constant is a fundamental physical constant associated with atomic spectra. It plays a critical role in calculating the wavelengths of spectral lines emitted by hydrogen atoms.
The Rydberg constant, denoted by "R," has a value of approximately 10,973,731.56816 m^-1. This value is derived from experimental observations of hydrogen's spectral lines.
The Rydberg constant appears in the Rydberg formula, which is used to predict the wavelength of light emitted by electrons transitioning to lower energy states in a hydrogen atom.

• It is essential for understanding atomic spectra and quantum mechanics.
• Helps determine the energy levels of electrons in a hydrogen atom.

Spectroscopists use the Rydberg constant together with other formulas to explore the structure of different atoms and molecules.

spectral lines

Spectral lines are the unique wavelengths of light that are emitted or absorbed by an element. They arise from electronic transitions within an atom, where electrons jump between energy levels.
Each element has a distinct set of spectral lines, functioning like a fingerprint for identification in spectroscopy.
In the hydrogen atom, the Balmer series represents the spectral lines seen when the electron transitions from higher energy levels down to the second energy level (n=2).

• Visible spectral lines correspond to electron transitions that release energy in the form of light.
• Balmer series specifically involves transitions from levels n > 2 to n = 2.

Observing and analyzing spectral lines allow scientists to determine the composition and properties of stars and other distant astronomical objects.

hydrogen atom

The hydrogen atom, consisting of one proton and one electron, is the simplest and most abundant atom in the universe.
Despite its simplicity, the hydrogen atom plays a significant role in understanding atomic structure and quantum mechanics.
In the hydrogen atom, the electron orbits the nucleus in defined energy levels, each quantized and associated with specific energies.

• When electrons in a hydrogen atom transition between these energy levels, they emit or absorb photons.
• The spectral lines produced by these transitions are key to studying the atom's electronic structure.

The study of hydrogen's spectral lines laid the groundwork for the development of quantum mechanics, providing insights into the discrete nature of energy levels.

wavenumber

Wavenumber is a measure of the number of wave cycles per unit distance. It is often used in spectroscopy to describe electromagnetic radiation's scale.
Given in units of reciprocal meters (m^-1), the wavenumber provides a convenient way to express the energy associated with electromagnetic waves.
In the context of the Rydberg formula, the wavenumber of a spectral line is directly related to the difference in energy between two electron transitions in a hydrogen atom.

• Higher wavenumbers indicate higher energy and shorter wavelengths.
• Wavenumber is calculated as the reciprocal of the wavelength: \( \bar{u} = 1/\lambda \).

Understanding wavenumber is crucial for interpreting spectra and determining the properties and behaviors of light interacting with matter.